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Chapter 22: Inequalities

Learn Grade 8 Math from beginner foundations through advanced Grade 8 problem solving with detailed explanations, at least five examples per topic, practice exercises, and review questions.

Grade 8Beginner Friendly5+ Examples Per TopicPractice30 Q&A
Estimated reading time0% read
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What this chapter covers

This chapter contains 13 topics. Technical terms are followed by plain-language meanings where they first appear. Each topic includes at least five worked examples. Coding is included only where it naturally supports the Grade 8 coding expectations.

22.1 Meaning of inequality

An inequality compares values using symbols such as <, >, ≤, or ≥. In this section, the focus is Meaning of inequality.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve 2x + 5 = 17.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 6

Worked Example 2

Problem: Solve 3x - 4 = 11.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 3

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

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 4

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

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 4

Worked Example 5

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

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 6

Problem: Solve 0.5x + 2 = 7.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 10

Worked Example 7

Problem: Solve 2(x-3)+4=10.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 6

Worked Example 8

Problem: Solve 9 - 2x = 1.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 4

Worked Example 9

Problem: Solve 3x + 8 = 3x + 8.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: all real numbers

Worked Example 10

Problem: Solve 4x + 1 = 4x + 9.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: no solution

Practice exercise

Create one new Grade 8 problem involving Meaning of inequality. Show the important steps, include units when needed, and explain how you checked the answer.

22.2 Greater than

Greater than is an important Grade 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x + 3 > 8.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x > 5

Worked Example 2

Problem: Solve 2x ≤ 10.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≤ 5

Worked Example 3

Problem: Solve -3x > 12.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x < -4

Worked Example 4

Problem: Solve x - 7 ≥ 2.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≥ 9

Worked Example 5

Problem: Solve 4x < 20.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x < 5

Worked Example 6

Problem: Solve 5 - x ≥ 1.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≤ 4

Worked Example 7

Problem: Solve 0.5x > 3.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x > 6

Worked Example 8

Problem: Solve -2x ≤ 14.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≥ -7

Worked Example 9

Problem: Solve 3x+2<11.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x<3

Worked Example 10

Problem: Solve 7+x≥12.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x≥5

Practice exercise

Create one new Grade 8 problem involving Greater than. Show the important steps, include units when needed, and explain how you checked the answer.

22.3 Less than

Less than is an important Grade 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x + 3 > 8.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x > 5

Worked Example 2

Problem: Solve 2x ≤ 10.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≤ 5

Worked Example 3

Problem: Solve -3x > 12.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x < -4

Worked Example 4

Problem: Solve x - 7 ≥ 2.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≥ 9

Worked Example 5

Problem: Solve 4x < 20.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x < 5

Worked Example 6

Problem: Solve 5 - x ≥ 1.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≤ 4

Worked Example 7

Problem: Solve 0.5x > 3.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x > 6

Worked Example 8

Problem: Solve -2x ≤ 14.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≥ -7

Worked Example 9

Problem: Solve 3x+2<11.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x<3

Worked Example 10

Problem: Solve 7+x≥12.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x≥5

Practice exercise

Create one new Grade 8 problem involving Less than. Show the important steps, include units when needed, and explain how you checked the answer.

22.4 Greater than or equal to

Greater than or equal to is an important Grade 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x + 3 > 8.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x > 5

Worked Example 2

Problem: Solve 2x ≤ 10.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≤ 5

Worked Example 3

Problem: Solve -3x > 12.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x < -4

Worked Example 4

Problem: Solve x - 7 ≥ 2.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≥ 9

Worked Example 5

Problem: Solve 4x < 20.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x < 5

Worked Example 6

Problem: Solve 5 - x ≥ 1.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≤ 4

Worked Example 7

Problem: Solve 0.5x > 3.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x > 6

Worked Example 8

Problem: Solve -2x ≤ 14.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≥ -7

Worked Example 9

Problem: Solve 3x+2<11.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x<3

Worked Example 10

Problem: Solve 7+x≥12.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x≥5

Practice exercise

Create one new Grade 8 problem involving Greater than or equal to. Show the important steps, include units when needed, and explain how you checked the answer.

22.5 Less than or equal to

Less than or equal to is an important Grade 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x + 3 > 8.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x > 5

Worked Example 2

Problem: Solve 2x ≤ 10.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≤ 5

Worked Example 3

Problem: Solve -3x > 12.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x < -4

Worked Example 4

Problem: Solve x - 7 ≥ 2.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≥ 9

Worked Example 5

Problem: Solve 4x < 20.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x < 5

Worked Example 6

Problem: Solve 5 - x ≥ 1.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≤ 4

Worked Example 7

Problem: Solve 0.5x > 3.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x > 6

Worked Example 8

Problem: Solve -2x ≤ 14.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≥ -7

Worked Example 9

Problem: Solve 3x+2<11.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x<3

Worked Example 10

Problem: Solve 7+x≥12.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x≥5

Practice exercise

Create one new Grade 8 problem involving Less than or equal to. Show the important steps, include units when needed, and explain how you checked the answer.

22.6 One-step inequalities

One-step inequalities is an important Grade 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

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

  1. Look at the words in “One-step inequalities”.
  2. State what the idea is used for.
  3. Give one small mathematical example.

Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.

Answer: One-step inequalities is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: One-step inequalities: A student says, “I can use One-step inequalities without checking units or labels.” Is that a good method?

  1. Read the statement.
  2. Ask whether units, signs, labels, or conditions matter.
  3. Decide whether the method is safe.

Very beginner explanation: Many math mistakes come from using a correct calculation on the wrong quantity.

Answer: No. Important units, labels, signs, and conditions must be checked.

Worked Example 3

Problem: One-step inequalities: What is the first step when solving a problem about One-step inequalities?

  1. Read the whole question.
  2. Underline the known information.
  3. Identify what must be found.

Very beginner explanation: This prevents you from calculating before you understand the problem.

Answer: Identify the given information and the unknown before choosing a rule.

Worked Example 4

Problem: One-step inequalities: After solving a One-step inequalities problem, what should you do before accepting the answer?

  1. Estimate the expected size or direction.
  2. Check units and signs.
  3. Use an inverse method if possible.

Very beginner explanation: Checking is part of solving, not an optional extra.

Answer: Check whether the answer is reasonable and consistent with the problem.

Worked Example 5

Problem: One-step inequalities: Give one way One-step inequalities could appear outside a textbook.

  1. Think about money, measurements, data, maps, geometry, or technology.
  2. Connect the topic to one of those situations.

Very beginner explanation: Connecting math to real situations makes the rule easier to remember.

Answer: One-step inequalities can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: One-step inequalities: Which representation could help explain One-step inequalities: a table, graph, diagram, equation, or number line?

  1. Choose the representation that makes the relationship easiest to see.
  2. Label it clearly.

Very beginner explanation: Different representations show different features of the same mathematics.

Answer: Use the representation that best matches the problem; more than one may be valid.

Worked Example 7

Problem: One-step inequalities: A student gets an answer for One-step inequalities but cannot explain the steps. What should be improved?

  1. Rewrite the solution one step at a time.
  2. Name the rule used at each important step.

Very beginner explanation: A correct final number without reasoning may hide an error.

Answer: The reasoning should be shown so the solution can be checked.

Worked Example 8

Problem: One-step inequalities: Why can estimation help before a detailed One-step inequalities calculation?

  1. Round or use benchmark values.
  2. Predict the approximate answer.
  3. Compare the exact result with the estimate.

Very beginner explanation: If the exact answer is far from the estimate, recheck the work.

Answer: Estimation gives a target range for the final answer.

Worked Example 9

Problem: One-step inequalities: How can you test whether your rule for One-step inequalities works?

  1. Choose a very small easy example.
  2. Apply the rule.
  3. Check the result another way.

Very beginner explanation: Small examples expose mistakes quickly.

Answer: Test the rule on a simple case whose answer can be verified.

Worked Example 10

Problem: One-step inequalities: How would you teach One-step inequalities to someone seeing it for the first time?

  1. Define the idea.
  2. Show one easy example.
  3. Explain every step.
  4. Then let the learner try a similar problem.

Very beginner explanation: This sequence reduces memorization without understanding.

Answer: Teach meaning first, then a small worked example, then guided practice.

Practice exercise

Create one new Grade 8 problem involving One-step inequalities. Show the important steps, include units when needed, and explain how you checked the answer.

22.7 Two-step inequalities

Two-step inequalities is an important Grade 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

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

  1. Look at the words in “Two-step inequalities”.
  2. State what the idea is used for.
  3. Give one small mathematical example.

Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.

Answer: Two-step inequalities is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Two-step inequalities: A student says, “I can use Two-step inequalities without checking units or labels.” Is that a good method?

  1. Read the statement.
  2. Ask whether units, signs, labels, or conditions matter.
  3. Decide whether the method is safe.

Very beginner explanation: Many math mistakes come from using a correct calculation on the wrong quantity.

Answer: No. Important units, labels, signs, and conditions must be checked.

Worked Example 3

Problem: Two-step inequalities: What is the first step when solving a problem about Two-step inequalities?

  1. Read the whole question.
  2. Underline the known information.
  3. Identify what must be found.

Very beginner explanation: This prevents you from calculating before you understand the problem.

Answer: Identify the given information and the unknown before choosing a rule.

Worked Example 4

Problem: Two-step inequalities: After solving a Two-step inequalities problem, what should you do before accepting the answer?

  1. Estimate the expected size or direction.
  2. Check units and signs.
  3. Use an inverse method if possible.

Very beginner explanation: Checking is part of solving, not an optional extra.

Answer: Check whether the answer is reasonable and consistent with the problem.

Worked Example 5

Problem: Two-step inequalities: Give one way Two-step inequalities could appear outside a textbook.

  1. Think about money, measurements, data, maps, geometry, or technology.
  2. Connect the topic to one of those situations.

Very beginner explanation: Connecting math to real situations makes the rule easier to remember.

Answer: Two-step inequalities can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Two-step inequalities: Which representation could help explain Two-step inequalities: a table, graph, diagram, equation, or number line?

  1. Choose the representation that makes the relationship easiest to see.
  2. Label it clearly.

Very beginner explanation: Different representations show different features of the same mathematics.

Answer: Use the representation that best matches the problem; more than one may be valid.

Worked Example 7

Problem: Two-step inequalities: A student gets an answer for Two-step inequalities but cannot explain the steps. What should be improved?

  1. Rewrite the solution one step at a time.
  2. Name the rule used at each important step.

Very beginner explanation: A correct final number without reasoning may hide an error.

Answer: The reasoning should be shown so the solution can be checked.

Worked Example 8

Problem: Two-step inequalities: Why can estimation help before a detailed Two-step inequalities calculation?

  1. Round or use benchmark values.
  2. Predict the approximate answer.
  3. Compare the exact result with the estimate.

Very beginner explanation: If the exact answer is far from the estimate, recheck the work.

Answer: Estimation gives a target range for the final answer.

Worked Example 9

Problem: Two-step inequalities: How can you test whether your rule for Two-step inequalities works?

  1. Choose a very small easy example.
  2. Apply the rule.
  3. Check the result another way.

Very beginner explanation: Small examples expose mistakes quickly.

Answer: Test the rule on a simple case whose answer can be verified.

Worked Example 10

Problem: Two-step inequalities: How would you teach Two-step inequalities to someone seeing it for the first time?

  1. Define the idea.
  2. Show one easy example.
  3. Explain every step.
  4. Then let the learner try a similar problem.

Very beginner explanation: This sequence reduces memorization without understanding.

Answer: Teach meaning first, then a small worked example, then guided practice.

Practice exercise

Create one new Grade 8 problem involving Two-step inequalities. Show the important steps, include units when needed, and explain how you checked the answer.

22.8 Inequalities with integers

Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Inequalities with integers.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate -7 + (5).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -2

Worked Example 2

Problem: Calculate 4 + (-9).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -5

Worked Example 3

Problem: Calculate -6 + (-3).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -9

Worked Example 4

Problem: Calculate 12 + (-8).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 4

Worked Example 5

Problem: Calculate -15 + (20).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 6

Problem: Calculate -11 + (-4).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -15

Worked Example 7

Problem: Calculate 9 + (13).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 22

Worked Example 8

Problem: Calculate -2 + (7).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 9

Problem: Calculate 18 + (-25).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -7

Worked Example 10

Problem: Calculate -30 + (12).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -18

Practice exercise

Create one new Grade 8 problem involving Inequalities with integers. Show the important steps, include units when needed, and explain how you checked the answer.

22.9 Inequalities with decimals

A decimal uses place value to represent parts of one, including tenths, hundredths, thousandths, and smaller values. In this section, the focus is Inequalities with decimals.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Which is greater: 3.6 or 0.4?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 3.6

Worked Example 2

Problem: Which is greater: 8.25 or 1.5?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 8.25

Worked Example 3

Problem: Which is greater: 12.75 or 2.4?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 12.75

Worked Example 4

Problem: Which is greater: 0.96 or 0.3?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 0.96

Worked Example 5

Problem: Which is greater: 5.04 or 1.2?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 5.04

Worked Example 6

Problem: Which is greater: 14.8 or 2?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 14.8

Worked Example 7

Problem: Which is greater: 7.125 or 0.5?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 7.125

Worked Example 8

Problem: Which is greater: 20.05 or 3.75?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 20.05

Worked Example 9

Problem: Which is greater: 2.4 or 0.06?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 2.4

Worked Example 10

Problem: Which is greater: 100.5 or 4.02?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 100.5

Practice exercise

Create one new Grade 8 problem involving Inequalities with decimals. Show the important steps, include units when needed, and explain how you checked the answer.

22.10 Multiplying or dividing by a negative

Multiplying or dividing by a negative is an important Grade 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: An item costs $50.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 50.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $6.50

Worked Example 2

Problem: An item costs $100.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 100.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $13.00

Worked Example 3

Problem: An item costs $75.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 75.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $9.75

Worked Example 4

Problem: An item costs $120.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 120.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $15.60

Worked Example 5

Problem: An item costs $200.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 200.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $26.00

Worked Example 6

Problem: An item costs $150.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 150.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $19.50

Worked Example 7

Problem: An item costs $250.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 250.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $32.50

Worked Example 8

Problem: An item costs $80.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 80.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $10.40

Worked Example 9

Problem: An item costs $300.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 300.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $39.00

Worked Example 10

Problem: An item costs $60.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 60.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $7.80

Practice exercise

Create one new Grade 8 problem involving Multiplying or dividing by a negative. Show the important steps, include units when needed, and explain how you checked the answer.

22.11 Graphing inequalities

Graphing inequalities is an important Grade 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Graphing inequalities: Explain Graphing inequalities in one simple sentence.

  1. Look at the words in “Graphing inequalities”.
  2. State what the idea is used for.
  3. Give one small mathematical example.

Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.

Answer: Graphing inequalities is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Graphing inequalities: A student says, “I can use Graphing inequalities without checking units or labels.” Is that a good method?

  1. Read the statement.
  2. Ask whether units, signs, labels, or conditions matter.
  3. Decide whether the method is safe.

Very beginner explanation: Many math mistakes come from using a correct calculation on the wrong quantity.

Answer: No. Important units, labels, signs, and conditions must be checked.

Worked Example 3

Problem: Graphing inequalities: What is the first step when solving a problem about Graphing inequalities?

  1. Read the whole question.
  2. Underline the known information.
  3. Identify what must be found.

Very beginner explanation: This prevents you from calculating before you understand the problem.

Answer: Identify the given information and the unknown before choosing a rule.

Worked Example 4

Problem: Graphing inequalities: After solving a Graphing inequalities problem, what should you do before accepting the answer?

  1. Estimate the expected size or direction.
  2. Check units and signs.
  3. Use an inverse method if possible.

Very beginner explanation: Checking is part of solving, not an optional extra.

Answer: Check whether the answer is reasonable and consistent with the problem.

Worked Example 5

Problem: Graphing inequalities: Give one way Graphing inequalities could appear outside a textbook.

  1. Think about money, measurements, data, maps, geometry, or technology.
  2. Connect the topic to one of those situations.

Very beginner explanation: Connecting math to real situations makes the rule easier to remember.

Answer: Graphing inequalities can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Graphing inequalities: Which representation could help explain Graphing inequalities: a table, graph, diagram, equation, or number line?

  1. Choose the representation that makes the relationship easiest to see.
  2. Label it clearly.

Very beginner explanation: Different representations show different features of the same mathematics.

Answer: Use the representation that best matches the problem; more than one may be valid.

Worked Example 7

Problem: Graphing inequalities: A student gets an answer for Graphing inequalities but cannot explain the steps. What should be improved?

  1. Rewrite the solution one step at a time.
  2. Name the rule used at each important step.

Very beginner explanation: A correct final number without reasoning may hide an error.

Answer: The reasoning should be shown so the solution can be checked.

Worked Example 8

Problem: Graphing inequalities: Why can estimation help before a detailed Graphing inequalities calculation?

  1. Round or use benchmark values.
  2. Predict the approximate answer.
  3. Compare the exact result with the estimate.

Very beginner explanation: If the exact answer is far from the estimate, recheck the work.

Answer: Estimation gives a target range for the final answer.

Worked Example 9

Problem: Graphing inequalities: How can you test whether your rule for Graphing inequalities works?

  1. Choose a very small easy example.
  2. Apply the rule.
  3. Check the result another way.

Very beginner explanation: Small examples expose mistakes quickly.

Answer: Test the rule on a simple case whose answer can be verified.

Worked Example 10

Problem: Graphing inequalities: How would you teach Graphing inequalities to someone seeing it for the first time?

  1. Define the idea.
  2. Show one easy example.
  3. Explain every step.
  4. Then let the learner try a similar problem.

Very beginner explanation: This sequence reduces memorization without understanding.

Answer: Teach meaning first, then a small worked example, then guided practice.

Practice exercise

Create one new Grade 8 problem involving Graphing inequalities. Show the important steps, include units when needed, and explain how you checked the answer.

22.12 Open and closed circles

Open and closed circles is an important Grade 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x + 3 > 8.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x > 5

Worked Example 2

Problem: Solve 2x ≤ 10.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≤ 5

Worked Example 3

Problem: Solve -3x > 12.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x < -4

Worked Example 4

Problem: Solve x - 7 ≥ 2.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≥ 9

Worked Example 5

Problem: Solve 4x < 20.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x < 5

Worked Example 6

Problem: Solve 5 - x ≥ 1.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≤ 4

Worked Example 7

Problem: Solve 0.5x > 3.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x > 6

Worked Example 8

Problem: Solve -2x ≤ 14.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≥ -7

Worked Example 9

Problem: Solve 3x+2<11.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x<3

Worked Example 10

Problem: Solve 7+x≥12.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x≥5

Practice exercise

Create one new Grade 8 problem involving Open and closed circles. Show the important steps, include units when needed, and explain how you checked the answer.

22.13 Real-life constraints

Real-life constraints is an important Grade 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x + 3 > 8.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x > 5

Worked Example 2

Problem: Solve 2x ≤ 10.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≤ 5

Worked Example 3

Problem: Solve -3x > 12.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x < -4

Worked Example 4

Problem: Solve x - 7 ≥ 2.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≥ 9

Worked Example 5

Problem: Solve 4x < 20.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x < 5

Worked Example 6

Problem: Solve 5 - x ≥ 1.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≤ 4

Worked Example 7

Problem: Solve 0.5x > 3.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x > 6

Worked Example 8

Problem: Solve -2x ≤ 14.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x ≥ -7

Worked Example 9

Problem: Solve 3x+2<11.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x<3

Worked Example 10

Problem: Solve 7+x≥12.

  1. Use inverse operations to isolate x.
  2. If you multiply or divide by a negative number, reverse the inequality sign.
  3. Graph the solution if requested.

Very beginner explanation: An inequality can have many solutions rather than one exact number.

Answer: x≥5

Practice exercise

Create one new Grade 8 problem involving Real-life constraints. Show the important steps, include units when needed, and explain how you checked the answer.

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Chapter 22 Review Questions and Answers

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

Answer: An inequality compares values using symbols such as <, >, ≤, or ≥. In this section, the focus is Meaning of inequality.

Q2. What is important to remember about Greater than?

Answer: Greater than is an important Grade 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, 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 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, 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 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, 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 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, 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 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, 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 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q8. What is important to remember about Inequalities with integers?

Answer: Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Inequalities with integers.

Q9. What is important to remember about Inequalities with decimals?

Answer: A decimal uses place value to represent parts of one, including tenths, hundredths, thousandths, and smaller values. In this section, the focus is Inequalities with decimals.

Q10. What is important to remember about Multiplying or dividing by a negative?

Answer: Multiplying or dividing by a negative is an important Grade 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q11. What is important to remember about Graphing inequalities?

Answer: Graphing inequalities is an important Grade 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q12. What is important to remember about Open and closed circles?

Answer: Open and closed circles is an important Grade 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q13. What is important to remember about Real-life constraints?

Answer: Real-life constraints is an important Grade 8 concept in Inequalities. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

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 calculated answer is reasonable.

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 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 mathematical 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 and check copied values, signs, units, formulas, and calculations.

Q22. Why are diagrams useful?

Answer: A labelled diagram can reveal relationships among lengths, angles, areas, and coordinates.

Q23. Why are tables useful?

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

Q24. Why should you explain your reasoning?

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

Q25. How do mistakes help learning?

Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent repetition.

Q26. When is a calculator useful?

Answer: A calculator helps with lengthy arithmetic after the mathematical setup is understood and estimated.

Q27. Why compare more than one strategy?

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

Q28. What is mathematical communication?

Answer: It is presenting ideas clearly with words, symbols, diagrams, tables, graphs, and justified steps.

Q29. What is mathematical modelling?

Answer: It is representing a real situation with mathematics, testing the model, and revising it if needed.

Q30. Why should assumptions be stated?

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