EASYTUTORGUIDE

Practical tutorials, tools, courses, digital skills, and business promotion.

Free Learning
Google Translate — English / فارسی / العربية
EasyTutorGuide LogoLearn, Build, & Grow

Chapter 26: Inequalities

Learn Grade 7 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.

Grade 7Beginner Friendly10 Examples Per Topic30 Q&APractice + Review
Reading tools
Advertisement area — AdSense / Auto Ads

Chapter Overview

This chapter teaches Inequalities with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.

Key Technical Terms

  • Rational Number (a number that can be written as a fraction of integers with a nonzero denominator)
  • Prime Number (a whole number greater than 1 with exactly two positive factors)
  • Inequality (a comparison using symbols such as <, >, ≤, or ≥)
  • Circle Graph (a graph using sectors of a circle to show parts of a whole)
  • Estimate (a close approximation used to check whether an answer is reasonable)
  • Solution (a value or result that satisfies the problem)
  • Representation (a way to show mathematics using symbols, diagrams, tables, graphs, or words)
  • Reasonableness (whether an answer makes sense in the context of the problem)

How to Learn This Chapter

Read the explanation first, follow the examples one step at a time, cover the answer and retry the problem yourself, then use the practice and review questions to check understanding.

26.1 Meaning of inequality

An inequality compares quantities using symbols such as <, >, ≤, or ≥. This topic focuses on Meaning of inequality .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x+3>8.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality compares quantities using symbols such as <, >, ≤, or ≥. This topic focuses on Meaning of inequality .

Answer: x>5

Worked Example 2

Problem: Solve 2x≤10.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality compares quantities using symbols such as <, >, ≤, or ≥. This topic focuses on Meaning of inequality .

Answer: x≤5

Worked Example 3

Problem: Solve -3x>12.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality compares quantities using symbols such as <, >, ≤, or ≥. This topic focuses on Meaning of inequality .

Answer: x<-4

Worked Example 4

Problem: Solve x-7≥2.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality compares quantities using symbols such as <, >, ≤, or ≥. This topic focuses on Meaning of inequality .

Answer: x≥9

Worked Example 5

Problem: Solve 4x<20.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality compares quantities using symbols such as <, >, ≤, or ≥. This topic focuses on Meaning of inequality .

Answer: x<5

Worked Example 6

Problem: Solve 2x + 5 = 17.

  1. Simplify each side if needed.
  2. Use inverse operations to collect variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution.

Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.

Answer: x = 6

Worked Example 7

Problem: Solve 3x - 4 = 11.

  1. Simplify each side if needed.
  2. Use inverse operations to collect variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution.

Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.

Answer: x = 5

Worked Example 8

Problem: Solve 5x + 7 = 2x + 22.

  1. Simplify each side if needed.
  2. Use inverse operations to collect variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution.

Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.

Answer: x = 5

Worked Example 9

Problem: Solve 4(x + 2) = 24.

  1. Simplify each side if needed.
  2. Use inverse operations to collect variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution.

Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.

Answer: x = 4

Worked Example 10

Problem: Solve 7x - 3 = 4x + 12.

  1. Simplify each side if needed.
  2. Use inverse operations to collect variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution.

Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.

Answer: x = 5

Practice Exercise

Create one new question about Meaning of inequality. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

26.2 Greater than

Greater than is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x+3>8.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Greater than is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x>5

Worked Example 2

Problem: Solve 2x≤10.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Greater than is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x≤5

Worked Example 3

Problem: Solve -3x>12.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Greater than is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x<-4

Worked Example 4

Problem: Solve x-7≥2.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Greater than is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x≥9

Worked Example 5

Problem: Solve 4x<20.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Greater than is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x<5

Worked Example 6

Problem: Solve x + 3 > 8.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x > 5

Worked Example 7

Problem: Solve 2x ≤ 10.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x ≤ 5

Worked Example 8

Problem: Solve -3x > 12.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x < -4

Worked Example 9

Problem: Solve x - 7 ≥ 2.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x ≥ 9

Worked Example 10

Problem: Solve 4x < 20.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x < 5

Practice Exercise

Create one new question about Greater than. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

26.3 Less than

Less than is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x+3>8.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Less than is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x>5

Worked Example 2

Problem: Solve 2x≤10.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Less than is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x≤5

Worked Example 3

Problem: Solve -3x>12.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Less than is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x<-4

Worked Example 4

Problem: Solve x-7≥2.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Less than is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x≥9

Worked Example 5

Problem: Solve 4x<20.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Less than is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x<5

Worked Example 6

Problem: Solve x + 3 > 8.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x > 5

Worked Example 7

Problem: Solve 2x ≤ 10.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x ≤ 5

Worked Example 8

Problem: Solve -3x > 12.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x < -4

Worked Example 9

Problem: Solve x - 7 ≥ 2.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x ≥ 9

Worked Example 10

Problem: Solve 4x < 20.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x < 5

Practice Exercise

Create one new question about Less than. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

26.4 Greater than or equal to

Greater than or equal to is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x+3>8.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Greater than or equal to is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x>5

Worked Example 2

Problem: Solve 2x≤10.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Greater than or equal to is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x≤5

Worked Example 3

Problem: Solve -3x>12.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Greater than or equal to is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x<-4

Worked Example 4

Problem: Solve x-7≥2.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Greater than or equal to is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x≥9

Worked Example 5

Problem: Solve 4x<20.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Greater than or equal to is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x<5

Worked Example 6

Problem: Solve x + 3 > 8.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x > 5

Worked Example 7

Problem: Solve 2x ≤ 10.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x ≤ 5

Worked Example 8

Problem: Solve -3x > 12.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x < -4

Worked Example 9

Problem: Solve x - 7 ≥ 2.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x ≥ 9

Worked Example 10

Problem: Solve 4x < 20.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x < 5

Practice Exercise

Create one new question about Greater than or equal to. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

26.5 Less than or equal to

Less than or equal to is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x+3>8.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Less than or equal to is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x>5

Worked Example 2

Problem: Solve 2x≤10.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Less than or equal to is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x≤5

Worked Example 3

Problem: Solve -3x>12.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Less than or equal to is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x<-4

Worked Example 4

Problem: Solve x-7≥2.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Less than or equal to is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x≥9

Worked Example 5

Problem: Solve 4x<20.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Less than or equal to is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x<5

Worked Example 6

Problem: Solve x + 3 > 8.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x > 5

Worked Example 7

Problem: Solve 2x ≤ 10.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x ≤ 5

Worked Example 8

Problem: Solve -3x > 12.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x < -4

Worked Example 9

Problem: Solve x - 7 ≥ 2.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x ≥ 9

Worked Example 10

Problem: Solve 4x < 20.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x < 5

Practice Exercise

Create one new question about Less than or equal to. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

26.6 One-step inequalities

One-step inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: One-step inequalities: Identify a correct example.

  1. State the definition and show why the example fits.
  2. Show all important reasoning.

Very beginner explanation: One-step inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 2

Problem: One-step inequalities: Identify a non-example.

  1. Explain which requirement is missing.
  2. Show all important reasoning.

Very beginner explanation: One-step inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 3

Problem: One-step inequalities: Compare two cases.

  1. State one similarity and one difference.
  2. Show all important reasoning.

Very beginner explanation: One-step inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 4

Problem: One-step inequalities: Apply the idea in a real situation.

  1. Translate the situation into mathematical language.
  2. Show all important reasoning.

Very beginner explanation: One-step inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 5

Problem: One-step inequalities: Create and check your own example.

  1. Use the definition, then verify each condition.
  2. Show all important reasoning.

Very beginner explanation: One-step inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 6

Problem: Explain One-step inequalities in one simple sentence.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: One-step inequalities becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: State the meaning, then give one small example.

Worked Example 7

Problem: Give one example that does not satisfy One-step inequalities and explain why.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: One-step inequalities becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Identify the rule or condition that fails.

Worked Example 8

Problem: Show One-step inequalities using a table, diagram, number line, graph, or equation.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: One-step inequalities becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Choose the representation that makes the relationship easiest to see.

Worked Example 9

Problem: After solving a One-step inequalities problem, how can you check the answer?

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: One-step inequalities becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Use estimation, an inverse operation, substitution, or a second representation.

Worked Example 10

Problem: Give one real-life situation where One-step inequalities could be useful.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: One-step inequalities becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Connect the idea to money, measurement, data, design, or technology.

Practice Exercise

Create one new question about One-step inequalities. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

26.7 Two-step inequalities

Two-step inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Two-step inequalities: Identify a correct example.

  1. State the definition and show why the example fits.
  2. Show all important reasoning.

Very beginner explanation: Two-step inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 2

Problem: Two-step inequalities: Identify a non-example.

  1. Explain which requirement is missing.
  2. Show all important reasoning.

Very beginner explanation: Two-step inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 3

Problem: Two-step inequalities: Compare two cases.

  1. State one similarity and one difference.
  2. Show all important reasoning.

Very beginner explanation: Two-step inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 4

Problem: Two-step inequalities: Apply the idea in a real situation.

  1. Translate the situation into mathematical language.
  2. Show all important reasoning.

Very beginner explanation: Two-step inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 5

Problem: Two-step inequalities: Create and check your own example.

  1. Use the definition, then verify each condition.
  2. Show all important reasoning.

Very beginner explanation: Two-step inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 6

Problem: Explain Two-step inequalities in one simple sentence.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Two-step inequalities becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: State the meaning, then give one small example.

Worked Example 7

Problem: Give one example that does not satisfy Two-step inequalities and explain why.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Two-step inequalities becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Identify the rule or condition that fails.

Worked Example 8

Problem: Show Two-step inequalities using a table, diagram, number line, graph, or equation.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Two-step inequalities becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Choose the representation that makes the relationship easiest to see.

Worked Example 9

Problem: After solving a Two-step inequalities problem, how can you check the answer?

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Two-step inequalities becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Use estimation, an inverse operation, substitution, or a second representation.

Worked Example 10

Problem: Give one real-life situation where Two-step inequalities could be useful.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Two-step inequalities becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Connect the idea to money, measurement, data, design, or technology.

Practice Exercise

Create one new question about Two-step inequalities. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

26.8 Graphing inequalities

Graphing inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Graphing inequalities: Identify a correct example.

  1. State the definition and show why the example fits.
  2. Show all important reasoning.

Very beginner explanation: Graphing inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 2

Problem: Graphing inequalities: Identify a non-example.

  1. Explain which requirement is missing.
  2. Show all important reasoning.

Very beginner explanation: Graphing inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 3

Problem: Graphing inequalities: Compare two cases.

  1. State one similarity and one difference.
  2. Show all important reasoning.

Very beginner explanation: Graphing inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 4

Problem: Graphing inequalities: Apply the idea in a real situation.

  1. Translate the situation into mathematical language.
  2. Show all important reasoning.

Very beginner explanation: Graphing inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 5

Problem: Graphing inequalities: Create and check your own example.

  1. Use the definition, then verify each condition.
  2. Show all important reasoning.

Very beginner explanation: Graphing inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 6

Problem: Explain Graphing inequalities in one simple sentence.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Graphing inequalities becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: State the meaning, then give one small example.

Worked Example 7

Problem: Give one example that does not satisfy Graphing inequalities and explain why.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Graphing inequalities becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Identify the rule or condition that fails.

Worked Example 8

Problem: Show Graphing inequalities using a table, diagram, number line, graph, or equation.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Graphing inequalities becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Choose the representation that makes the relationship easiest to see.

Worked Example 9

Problem: After solving a Graphing inequalities problem, how can you check the answer?

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Graphing inequalities becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Use estimation, an inverse operation, substitution, or a second representation.

Worked Example 10

Problem: Give one real-life situation where Graphing inequalities could be useful.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Graphing inequalities becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Connect the idea to money, measurement, data, design, or technology.

Practice Exercise

Create one new question about Graphing inequalities. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

26.9 Open circles

Open circles is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x+3>8.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Open circles is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x>5

Worked Example 2

Problem: Solve 2x≤10.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Open circles is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x≤5

Worked Example 3

Problem: Solve -3x>12.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Open circles is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x<-4

Worked Example 4

Problem: Solve x-7≥2.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Open circles is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x≥9

Worked Example 5

Problem: Solve 4x<20.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Open circles is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x<5

Worked Example 6

Problem: Solve x + 3 > 8.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x > 5

Worked Example 7

Problem: Solve 2x ≤ 10.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x ≤ 5

Worked Example 8

Problem: Solve -3x > 12.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x < -4

Worked Example 9

Problem: Solve x - 7 ≥ 2.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x ≥ 9

Worked Example 10

Problem: Solve 4x < 20.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x < 5

Practice Exercise

Create one new question about Open circles. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

26.10 Closed circles

Closed circles is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x+3>8.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Closed circles is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x>5

Worked Example 2

Problem: Solve 2x≤10.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Closed circles is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x≤5

Worked Example 3

Problem: Solve -3x>12.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Closed circles is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x<-4

Worked Example 4

Problem: Solve x-7≥2.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Closed circles is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x≥9

Worked Example 5

Problem: Solve 4x<20.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Closed circles is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x<5

Worked Example 6

Problem: Solve x + 3 > 8.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x > 5

Worked Example 7

Problem: Solve 2x ≤ 10.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x ≤ 5

Worked Example 8

Problem: Solve -3x > 12.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x < -4

Worked Example 9

Problem: Solve x - 7 ≥ 2.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x ≥ 9

Worked Example 10

Problem: Solve 4x < 20.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x < 5

Practice Exercise

Create one new question about Closed circles. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

26.11 Inequalities with negative numbers

Inequalities with negative numbers is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Inequalities with negative numbers: Identify a correct example.

  1. State the definition and show why the example fits.
  2. Show all important reasoning.

Very beginner explanation: Inequalities with negative numbers is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 2

Problem: Inequalities with negative numbers: Identify a non-example.

  1. Explain which requirement is missing.
  2. Show all important reasoning.

Very beginner explanation: Inequalities with negative numbers is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 3

Problem: Inequalities with negative numbers: Compare two cases.

  1. State one similarity and one difference.
  2. Show all important reasoning.

Very beginner explanation: Inequalities with negative numbers is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 4

Problem: Inequalities with negative numbers: Apply the idea in a real situation.

  1. Translate the situation into mathematical language.
  2. Show all important reasoning.

Very beginner explanation: Inequalities with negative numbers is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 5

Problem: Inequalities with negative numbers: Create and check your own example.

  1. Use the definition, then verify each condition.
  2. Show all important reasoning.

Very beginner explanation: Inequalities with negative numbers is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 6

Problem: Explain Inequalities with negative numbers in one simple sentence.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Inequalities with negative numbers becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: State the meaning, then give one small example.

Worked Example 7

Problem: Give one example that does not satisfy Inequalities with negative numbers and explain why.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Inequalities with negative numbers becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Identify the rule or condition that fails.

Worked Example 8

Problem: Show Inequalities with negative numbers using a table, diagram, number line, graph, or equation.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Inequalities with negative numbers becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Choose the representation that makes the relationship easiest to see.

Worked Example 9

Problem: After solving a Inequalities with negative numbers problem, how can you check the answer?

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Inequalities with negative numbers becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Use estimation, an inverse operation, substitution, or a second representation.

Worked Example 10

Problem: Give one real-life situation where Inequalities with negative numbers could be useful.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Inequalities with negative numbers becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Connect the idea to money, measurement, data, design, or technology.

Practice Exercise

Create one new question about Inequalities with negative numbers. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

26.12 Real-life constraints

Real-life constraints is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x+3>8.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Real-life constraints is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x>5

Worked Example 2

Problem: Solve 2x≤10.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Real-life constraints is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x≤5

Worked Example 3

Problem: Solve -3x>12.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Real-life constraints is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x<-4

Worked Example 4

Problem: Solve x-7≥2.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Real-life constraints is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x≥9

Worked Example 5

Problem: Solve 4x<20.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: Real-life constraints is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: x<5

Worked Example 6

Problem: Solve x + 3 > 8.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x > 5

Worked Example 7

Problem: Solve 2x ≤ 10.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x ≤ 5

Worked Example 8

Problem: Solve -3x > 12.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x < -4

Worked Example 9

Problem: Solve x - 7 ≥ 2.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x ≥ 9

Worked Example 10

Problem: Solve 4x < 20.

  1. Use inverse operations to isolate x.
  2. Reverse the inequality sign if multiplying or dividing by a negative number.
  3. Graph the result if required.

Very beginner explanation: An inequality usually represents a range of possible solutions rather than one exact number.

Answer: x < 5

Practice Exercise

Create one new question about Real-life constraints. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

26.13 Checking inequality solutions

An inequality compares quantities using symbols such as <, >, ≤, or ≥. This topic focuses on Checking inequality solutions .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x+3>8.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality compares quantities using symbols such as <, >, ≤, or ≥. This topic focuses on Checking inequality solutions .

Answer: x>5

Worked Example 2

Problem: Solve 2x≤10.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality compares quantities using symbols such as <, >, ≤, or ≥. This topic focuses on Checking inequality solutions .

Answer: x≤5

Worked Example 3

Problem: Solve -3x>12.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality compares quantities using symbols such as <, >, ≤, or ≥. This topic focuses on Checking inequality solutions .

Answer: x<-4

Worked Example 4

Problem: Solve x-7≥2.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality compares quantities using symbols such as <, >, ≤, or ≥. This topic focuses on Checking inequality solutions .

Answer: x≥9

Worked Example 5

Problem: Solve 4x<20.

  1. Use inverse operations.
  2. If multiplying or dividing by a negative, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality compares quantities using symbols such as <, >, ≤, or ≥. This topic focuses on Checking inequality solutions .

Answer: x<5

Worked Example 6

Problem: Solve 2x + 5 = 17.

  1. Simplify each side if needed.
  2. Use inverse operations to collect variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution.

Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.

Answer: x = 6

Worked Example 7

Problem: Solve 3x - 4 = 11.

  1. Simplify each side if needed.
  2. Use inverse operations to collect variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution.

Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.

Answer: x = 5

Worked Example 8

Problem: Solve 5x + 7 = 2x + 22.

  1. Simplify each side if needed.
  2. Use inverse operations to collect variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution.

Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.

Answer: x = 5

Worked Example 9

Problem: Solve 4(x + 2) = 24.

  1. Simplify each side if needed.
  2. Use inverse operations to collect variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution.

Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.

Answer: x = 4

Worked Example 10

Problem: Solve 7x - 3 = 4x + 12.

  1. Simplify each side if needed.
  2. Use inverse operations to collect variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution.

Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.

Answer: x = 5

Practice Exercise

Create one new question about Checking inequality solutions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

Chapter Notes, Practice, and Common Mistakes

Chapter Notes

  • Read the complete question before choosing an operation, formula, graph, or model.
  • Write technical words together with their meaning until you are comfortable using them.
  • Show all important steps so another student can follow your reasoning.
  • Keep units, labels, signs, axes, variables, and mathematical symbols clear.
  • Estimate before or after calculating when an estimate can help check reasonableness.
  • For real-life problems, explain what the final number means in the situation.

Extra Practice

  1. Create and solve a new question about Meaning of inequality . Show your reasoning and check your answer.
  2. Create and solve a new question about Greater than . Show your reasoning and check your answer.
  3. Create and solve a new question about Less than . Show your reasoning and check your answer.
  4. Create and solve a new question about Greater than or equal to . Show your reasoning and check your answer.
  5. Create and solve a new question about Less than or equal to . Show your reasoning and check your answer.
  6. Create and solve a new question about One-step inequalities . Show your reasoning and check your answer.
  7. Create and solve a new question about Two-step inequalities . Show your reasoning and check your answer.
  8. Create and solve a new question about Graphing inequalities . Show your reasoning and check your answer.
  9. Create and solve a new question about Open circles . Show your reasoning and check your answer.
  10. Create and solve a new question about Closed circles . Show your reasoning and check your answer.
  11. Create and solve a new question about Inequalities with negative numbers . Show your reasoning and check your answer.
  12. Create and solve a new question about Real-life constraints . Show your reasoning and check your answer.

Common Mistakes

  • Combining unlike terms.
  • Changing only one side of an equation.
  • Forgetting to distribute to every term.
  • Using a pattern rule that works only for the first few terms.
  • Writing code without tracing variable values.
  • Using a mathematical model without stating assumptions.
Advertisement area — AdSense / Auto Ads

30 Review Questions and Answers

Q1. What is important to remember about Meaning of inequality?

Answer: An inequality compares quantities using symbols such as , ≤, or ≥. This topic focuses on Meaning of inequality.

Q2. What is important to remember about Greater than?

Answer: Greater than is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q3. What is important to remember about Less than?

Answer: Less than is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q4. What is important to remember about Greater than or equal to?

Answer: Greater than or equal to is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q5. What is important to remember about Less than or equal to?

Answer: Less than or equal to is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q6. What is important to remember about One-step inequalities?

Answer: One-step inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q7. What is important to remember about Two-step inequalities?

Answer: Two-step inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q8. What is important to remember about Graphing inequalities?

Answer: Graphing inequalities is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q9. What is important to remember about Open circles?

Answer: Open circles is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q10. What is important to remember about Closed circles?

Answer: Closed circles is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q11. What is important to remember about Inequalities with negative numbers?

Answer: Inequalities with negative numbers is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q12. What is important to remember about Real-life constraints?

Answer: Real-life constraints is an important Grade 7 idea in Inequalities. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q13. What is important to remember about Checking inequality solutions?

Answer: An inequality compares quantities using symbols such as , ≤, or ≥. This topic focuses on Checking inequality solutions.

Q14. Why should you show your steps?

Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.

Q15. Why is estimation useful?

Answer: Estimation helps you judge whether a final answer is reasonable before accepting it.

Q16. Why do units matter?

Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.

Q17. When should you round?

Answer: Usually round near the end of a calculation unless the question specifically asks for earlier rounding.

Q18. How can you check an answer?

Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.

Q19. What should you identify first in a word problem?

Answer: Identify what is known, what is unknown, and the relationship connecting them.

Q20. What makes a final answer complete?

Answer: Give the value, correct units when needed, and a brief interpretation in context.

Q21. What should you do if an answer seems unreasonable?

Answer: Re-read the question, check copied values, signs, units, formulas, and calculations.

Q22. Why are diagrams useful?

Answer: A labelled diagram can reveal relationships between lengths, angles, areas, and other quantities.

Q23. Why are tables useful?

Answer: Tables organize values and help reveal patterns, relationships, rates, and missing information.

Q24. Why should you explain your reasoning?

Answer: Reasoning shows why a method works, not just what answer you obtained.

Q25. How do mistakes help learning?

Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent the same error later.

Q26. How should you study this chapter?

Answer: Review definitions, redo worked examples without looking, practise mixed questions, and explain solutions in your own words.

Q27. When is a calculator useful?

Answer: A calculator is useful for lengthy arithmetic after you understand the mathematical setup and can estimate the expected size of the answer.

Q28. Why compare more than one strategy?

Answer: Different strategies can make a problem easier and provide a way to verify the result.

Q29. What is a mathematical model?

Answer: It is a simplified mathematical representation of a real situation used to analyze, explain, or predict.

Q30. Why should assumptions be stated?

Answer: Assumptions show what conditions the solution depends on and help readers judge whether the model is reasonable.