Chapter 29: Understanding Inequalities

Learn Grade 6 mathematics with very-beginner explanations, original worked examples, practice, reasoning, and review.

Grade 6Beginner Friendly10 Examples Per Topic30 Q&APractice + Review

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Chapter Overview

This chapter teaches Understanding Inequalities in original, simple language with step-by-step reasoning, worked examples, practice exercises, common mistakes, and review questions.

Key Technical Terms

Meaning of inequality (An inequality compares values that may be greater than, less than, or equal within a range.)
Greater than (Greater than is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
Less than (Less than is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
Greater than or equal to (Greater than or equal to is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
Less than or equal to (Less than or equal to is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
One-step inequalities (One-step inequalities is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)

How to Learn This Chapter

Read the explanation first, follow each worked example slowly, cover the answer and retry the problem yourself, then use the practice and review questions to check understanding.

29.1 Meaning of inequality

An inequality compares values that may be greater than, less than, or equal within a range. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x + 7 = 18.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 11

Worked Example 2

Problem: Solve x - 9 = 14.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 23

Worked Example 3

Problem: Solve 4x = 28.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 7

Worked Example 4

Problem: Solve x ÷ 5 = 6.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 30

Worked Example 5

Problem: Solve 2x + 3x = 25.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 5

Worked Example 6

Problem: Solve 3x + 4 = 19.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 5

Worked Example 7

Problem: Solve 5x - 2 = 23.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 5

Worked Example 8

Problem: Solve 0.5x + 2 = 6.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 8

Worked Example 9

Problem: Solve 4x + 1 = 2x + 13.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 6

Worked Example 10

Problem: Solve 6x - 3 = 3x + 12.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 5

Practice Exercise

Create one new Grade 6 problem about Meaning of inequality. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is changing only one side of a relationship or forgetting to check the solution.

29.2 Greater than

Greater than is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x + 3 > 8.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 5

Worked Example 2

Problem: Solve 2x ≤ 12.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 6

Worked Example 3

Problem: Solve x - 4 ≥ 7.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≥ 11

Worked Example 4

Problem: Solve 3x < 15.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x < 5

Worked Example 5

Problem: Solve x + 9 ≤ 20.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 11

Worked Example 6

Problem: Solve 4x > 16.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 4

Worked Example 7

Problem: Solve x - 2 < 6.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x < 8

Worked Example 8

Problem: Solve 5x ≥ 25.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≥ 5

Worked Example 9

Problem: Solve x + 1 > 1.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 0

Worked Example 10

Problem: Solve 2x + 2 ≤ 10.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 4

Practice Exercise

Create one new Grade 6 problem about Greater than. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

29.3 Less than

Less than is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x + 3 > 8.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 5

Worked Example 2

Problem: Solve 2x ≤ 12.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 6

Worked Example 3

Problem: Solve x - 4 ≥ 7.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≥ 11

Worked Example 4

Problem: Solve 3x < 15.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x < 5

Worked Example 5

Problem: Solve x + 9 ≤ 20.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 11

Worked Example 6

Problem: Solve 4x > 16.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 4

Worked Example 7

Problem: Solve x - 2 < 6.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x < 8

Worked Example 8

Problem: Solve 5x ≥ 25.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≥ 5

Worked Example 9

Problem: Solve x + 1 > 1.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 0

Worked Example 10

Problem: Solve 2x + 2 ≤ 10.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 4

Practice Exercise

Create one new Grade 6 problem about Less than. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

29.4 Greater than or equal to

Greater than or equal to is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x + 3 > 8.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 5

Worked Example 2

Problem: Solve 2x ≤ 12.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 6

Worked Example 3

Problem: Solve x - 4 ≥ 7.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≥ 11

Worked Example 4

Problem: Solve 3x < 15.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x < 5

Worked Example 5

Problem: Solve x + 9 ≤ 20.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 11

Worked Example 6

Problem: Solve 4x > 16.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 4

Worked Example 7

Problem: Solve x - 2 < 6.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x < 8

Worked Example 8

Problem: Solve 5x ≥ 25.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≥ 5

Worked Example 9

Problem: Solve x + 1 > 1.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 0

Worked Example 10

Problem: Solve 2x + 2 ≤ 10.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 4

Practice Exercise

Create one new Grade 6 problem about Greater than or equal to. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

29.5 Less than or equal to

Less than or equal to is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x + 3 > 8.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 5

Worked Example 2

Problem: Solve 2x ≤ 12.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 6

Worked Example 3

Problem: Solve x - 4 ≥ 7.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≥ 11

Worked Example 4

Problem: Solve 3x < 15.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x < 5

Worked Example 5

Problem: Solve x + 9 ≤ 20.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 11

Worked Example 6

Problem: Solve 4x > 16.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 4

Worked Example 7

Problem: Solve x - 2 < 6.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x < 8

Worked Example 8

Problem: Solve 5x ≥ 25.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≥ 5

Worked Example 9

Problem: Solve x + 1 > 1.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 0

Worked Example 10

Problem: Solve 2x + 2 ≤ 10.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 4

Practice Exercise

Create one new Grade 6 problem about Less than or equal to. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

29.6 One-step inequalities

One-step inequalities is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x + 3 > 8.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 5

Worked Example 2

Problem: Solve 2x ≤ 12.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 6

Worked Example 3

Problem: Solve x - 4 ≥ 7.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≥ 11

Worked Example 4

Problem: Solve 3x < 15.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x < 5

Worked Example 5

Problem: Solve x + 9 ≤ 20.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 11

Worked Example 6

Problem: Solve 4x > 16.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 4

Worked Example 7

Problem: Solve x - 2 < 6.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x < 8

Worked Example 8

Problem: Solve 5x ≥ 25.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≥ 5

Worked Example 9

Problem: Solve x + 1 > 1.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 0

Worked Example 10

Problem: Solve 2x + 2 ≤ 10.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 4

Practice Exercise

Create one new Grade 6 problem about One-step inequalities. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is changing only one side of a relationship or forgetting to check the solution.

29.7 Testing values in inequalities

Testing values in inequalities is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x + 3 > 8.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 5

Worked Example 2

Problem: Solve 2x ≤ 12.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 6

Worked Example 3

Problem: Solve x - 4 ≥ 7.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≥ 11

Worked Example 4

Problem: Solve 3x < 15.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x < 5

Worked Example 5

Problem: Solve x + 9 ≤ 20.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 11

Worked Example 6

Problem: Solve 4x > 16.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 4

Worked Example 7

Problem: Solve x - 2 < 6.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x < 8

Worked Example 8

Problem: Solve 5x ≥ 25.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≥ 5

Worked Example 9

Problem: Solve x + 1 > 1.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 0

Worked Example 10

Problem: Solve 2x + 2 ≤ 10.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 4

Practice Exercise

Create one new Grade 6 problem about Testing values in inequalities. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is changing only one side of a relationship or forgetting to check the solution.

29.8 Representing solution sets

Representing solution sets is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x + 3 > 8.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 5

Worked Example 2

Problem: Solve 2x ≤ 12.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 6

Worked Example 3

Problem: Solve x - 4 ≥ 7.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≥ 11

Worked Example 4

Problem: Solve 3x < 15.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x < 5

Worked Example 5

Problem: Solve x + 9 ≤ 20.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 11

Worked Example 6

Problem: Solve 4x > 16.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 4

Worked Example 7

Problem: Solve x - 2 < 6.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x < 8

Worked Example 8

Problem: Solve 5x ≥ 25.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≥ 5

Worked Example 9

Problem: Solve x + 1 > 1.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 0

Worked Example 10

Problem: Solve 2x + 2 ≤ 10.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 4

Practice Exercise

Create one new Grade 6 problem about Representing solution sets. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

29.9 Inequalities on number lines

Inequalities on number lines is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x + 3 > 8.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 5

Worked Example 2

Problem: Solve 2x ≤ 12.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 6

Worked Example 3

Problem: Solve x - 4 ≥ 7.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≥ 11

Worked Example 4

Problem: Solve 3x < 15.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x < 5

Worked Example 5

Problem: Solve x + 9 ≤ 20.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 11

Worked Example 6

Problem: Solve 4x > 16.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 4

Worked Example 7

Problem: Solve x - 2 < 6.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x < 8

Worked Example 8

Problem: Solve 5x ≥ 25.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≥ 5

Worked Example 9

Problem: Solve x + 1 > 1.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 0

Worked Example 10

Problem: Solve 2x + 2 ≤ 10.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 4

Practice Exercise

Create one new Grade 6 problem about Inequalities on number lines. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is changing only one side of a relationship or forgetting to check the solution.

29.10 Real-life limits and constraints

Real-life limits and constraints is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Use Real-life limits and constraints to analyze the values 15 and 9. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Real-life limits and constraints is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 2

Problem: Use Real-life limits and constraints to analyze the values 29 and 14. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 3

Problem: Use Real-life limits and constraints to analyze the values 2 and 18. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 4

Problem: Use Real-life limits and constraints to analyze the values 24 and 18. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 5

Problem: Use Real-life limits and constraints to analyze the values 13 and 17. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 6

Problem: Use Real-life limits and constraints to analyze the values 21 and 13. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 7

Problem: Use Real-life limits and constraints to analyze the values 27 and 19. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 8

Problem: Use Real-life limits and constraints to analyze the values 25 and 12. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 9

Problem: Use Real-life limits and constraints to analyze the values 9 and 6. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 10

Problem: Use Real-life limits and constraints to analyze the values 12 and 12. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Answer: Check the meaning of the values and the rule required by the problem.

Practice Exercise

Create one new Grade 6 problem about Real-life limits and constraints. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

29.11 Writing inequalities from words

Writing inequalities from words is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x + 3 > 8.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 5

Worked Example 2

Problem: Solve 2x ≤ 12.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 6

Worked Example 3

Problem: Solve x - 4 ≥ 7.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≥ 11

Worked Example 4

Problem: Solve 3x < 15.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x < 5

Worked Example 5

Problem: Solve x + 9 ≤ 20.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 11

Worked Example 6

Problem: Solve 4x > 16.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 4

Worked Example 7

Problem: Solve x - 2 < 6.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x < 8

Worked Example 8

Problem: Solve 5x ≥ 25.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≥ 5

Worked Example 9

Problem: Solve x + 1 > 1.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x > 0

Worked Example 10

Problem: Solve 2x + 2 ≤ 10.

Use inverse operations to isolate x.
Keep track of the inequality sign.
Test one value from the solution set.

Very beginner explanation: An inequality can describe many possible values rather than one exact value.

Answer: x ≤ 4

Practice Exercise

Create one new Grade 6 problem about Writing inequalities from words. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is changing only one side of a relationship or forgetting to check the solution.

29.12 Checking inequality solutions

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x + 7 = 18.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 11

Worked Example 2

Problem: Solve x - 9 = 14.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 23

Worked Example 3

Problem: Solve 4x = 28.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 7

Worked Example 4

Problem: Solve x ÷ 5 = 6.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 30

Worked Example 5

Problem: Solve 2x + 3x = 25.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 5

Worked Example 6

Problem: Solve 3x + 4 = 19.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 5

Worked Example 7

Problem: Solve 5x - 2 = 23.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 5

Worked Example 8

Problem: Solve 0.5x + 2 = 6.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 8

Worked Example 9

Problem: Solve 4x + 1 = 2x + 13.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 6

Worked Example 10

Problem: Solve 6x - 3 = 3x + 12.

Simplify each side if needed.
Use inverse operations to isolate the variable.
Keep the relationship balanced.
Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 5

Practice Exercise

Create one new Grade 6 problem about Checking inequality solutions. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is changing only one side of a relationship or forgetting to check the solution.

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30 Review Questions and Answers

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

Answer: An inequality compares values that may be greater than, less than, or equal within a range.

Q2. What is important to understand about Greater than?

Answer: Greater than is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q3. What is important to understand about Less than?

Answer: Less than is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Answer: Greater than or equal to is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Answer: Less than or equal to is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Answer: One-step inequalities is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q7. What is important to understand about Testing values in inequalities?

Answer: Testing values in inequalities is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q8. What is important to understand about Representing solution sets?

Answer: Representing solution sets is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q9. What is important to understand about Inequalities on number lines?

Answer: Inequalities on number lines is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q10. What is important to understand about Real-life limits and constraints?

Answer: Real-life limits and constraints is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q11. What is important to understand about Writing inequalities from words?

Answer: Writing inequalities from words is a Grade 6 mathematics idea in Understanding Inequalities. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q12. What is important to understand about Checking inequality solutions?

Answer: An inequality compares values that may be greater than, less than, or equal within a range.

Q13. Why should you show your steps?

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

Q14. Why is estimation useful?

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

Q15. Why do units matter?

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

Q16. When should you round?

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

Q17. How can you check an answer?

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

Q18. What should you identify first in a word problem?

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

Q19. What makes a final answer complete?

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

Q20. What should you do if an answer seems unreasonable?

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

Q21. Why are diagrams useful?

Answer: A labelled diagram can reveal relationships that are harder to see in words alone.

Q22. Why are tables useful?

Answer: Tables organize values so patterns, rates, comparisons, and missing information are easier to see.

Q23. Why are graphs useful?

Answer: Graphs make relationships, changes, trends, and comparisons visible.

Q24. Why should you explain your reasoning?

Answer: An explanation shows why a method works instead of only giving a final number.

Q25. Why can more than one strategy be correct?

Answer: Different correct strategies can represent the same mathematics in different but equivalent ways.

Q26. What should you do before using a formula or rule?

Answer: Check what each quantity means and whether the rule matches the situation.

Q27. How does practice help in mathematics?

Answer: Practice builds accuracy and fluency while helping you recognize which strategies fit different problems.

Q28. What is a good way to learn from a mistake?

Answer: Find the exact step where the reasoning changed from correct to incorrect, then solve a similar problem again.

Q29. Why should you compare an exact answer with an estimate?

Answer: The estimate gives a quick reasonableness check for the exact calculation.

Q30. What does representing mathematics mean?

Answer: It means showing an idea with numbers, symbols, words, diagrams, tables, graphs, or models.