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Chapter 30: Graphing Inequalities with Two Variables

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

Grade 6Beginner Friendly10 Examples Per Topic30 Q&APractice + Review
Estimated reading time0% read
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Chapter Overview

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

Key Technical Terms

  • Two-variable inequality (A variable is a symbol used to represent a number that may change or may be unknown.)
  • Coordinate pairs as possible solutions (Coordinate pairs as possible solutions is a Grade 6 mathematics idea in Graphing Inequalities with Two Variables. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
  • Testing an ordered pair (Testing an ordered pair is a Grade 6 mathematics idea in Graphing Inequalities with Two Variables. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
  • Boundary line idea (Boundary line idea is a Grade 6 mathematics idea in Graphing Inequalities with Two Variables. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
  • Points that satisfy an inequality (An inequality compares values that may be greater than, less than, or equal within a range.)
  • Simple x plus y constraints (Simple x plus y constraints is a Grade 6 mathematics idea in Graphing Inequalities with Two Variables. 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.

30.1 Two-variable inequality introduction

A variable is a symbol used to represent a number that may change or may be unknown. 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: Evaluate 3x + 4 when x = 5.

  1. Replace x with 5.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 19

Worked Example 2

Problem: Evaluate 2n + 7 when n = 6.

  1. Replace n with 6.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 19

Worked Example 3

Problem: Evaluate 5a - 3 when a = 4.

  1. Replace a with 4.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 17

Worked Example 4

Problem: Evaluate 4p + 1 when p = 8.

  1. Replace p with 8.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 33

Worked Example 5

Problem: Evaluate 6m - 5 when m = 3.

  1. Replace m with 3.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 13

Worked Example 6

Problem: Evaluate 2.5x + 1 when x = 4.

  1. Replace x with 4.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 11

Worked Example 7

Problem: Evaluate 7q + 2 when q = 2.

  1. Replace q with 2.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 16

Worked Example 8

Problem: Evaluate 3r - 1 when r = 9.

  1. Replace r with 9.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 26

Worked Example 9

Problem: Evaluate 4y + 6 when y = 5.

  1. Replace y with 5.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 26

Worked Example 10

Problem: Evaluate 8k - 4 when k = 3.

  1. Replace k with 3.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 20

Practice Exercise

Create one new Grade 6 problem about Two-variable inequality introduction. 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.

30.2 Coordinate pairs as possible solutions

Coordinate pairs as possible solutions is a Grade 6 mathematics idea in Graphing Inequalities with Two Variables. 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: Where is the point (2, 3) located?

  1. Read x first: move horizontally.
  2. Read y second: move vertically.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are written (x, y), horizontal coordinate first.

Answer: Quadrant I

Worked Example 2

Problem: Where is the point (-4, 5) located?

  1. Read x first: move horizontally.
  2. Read y second: move vertically.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are written (x, y), horizontal coordinate first.

Answer: Quadrant II

Worked Example 3

Problem: Where is the point (-3, -2) located?

  1. Read x first: move horizontally.
  2. Read y second: move vertically.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are written (x, y), horizontal coordinate first.

Answer: Quadrant III

Worked Example 4

Problem: Where is the point (6, -1) located?

  1. Read x first: move horizontally.
  2. Read y second: move vertically.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are written (x, y), horizontal coordinate first.

Answer: Quadrant IV

Worked Example 5

Problem: Where is the point (0, 4) located?

  1. Read x first: move horizontally.
  2. Read y second: move vertically.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are written (x, y), horizontal coordinate first.

Answer: on an axis

Worked Example 6

Problem: Where is the point (5, 0) located?

  1. Read x first: move horizontally.
  2. Read y second: move vertically.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are written (x, y), horizontal coordinate first.

Answer: on an axis

Worked Example 7

Problem: Where is the point (-7, 2) located?

  1. Read x first: move horizontally.
  2. Read y second: move vertically.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are written (x, y), horizontal coordinate first.

Answer: Quadrant II

Worked Example 8

Problem: Where is the point (1, -6) located?

  1. Read x first: move horizontally.
  2. Read y second: move vertically.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are written (x, y), horizontal coordinate first.

Answer: Quadrant IV

Worked Example 9

Problem: Where is the point (-2, -8) located?

  1. Read x first: move horizontally.
  2. Read y second: move vertically.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are written (x, y), horizontal coordinate first.

Answer: Quadrant III

Worked Example 10

Problem: Where is the point (8, 7) located?

  1. Read x first: move horizontally.
  2. Read y second: move vertically.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are written (x, y), horizontal coordinate first.

Answer: Quadrant I

Practice Exercise

Create one new Grade 6 problem about Coordinate pairs as possible solutions. 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.

30.3 Testing an ordered pair

Testing an ordered pair is a Grade 6 mathematics idea in Graphing Inequalities with Two Variables. 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: Does the point (2, 3) satisfy x + y ≤ 6?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 2

Problem: Does the point (4, 4) satisfy x + y ≤ 6?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 3

Problem: Does the point (1, 5) satisfy x + y < 8?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 4

Problem: Does the point (3, 6) satisfy y ≥ 5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 5

Problem: Does the point (7, 2) satisfy x < 5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 6

Problem: Does the point (0, 0) satisfy x+y≥0?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 7

Problem: Does the point (-1, 4) satisfy y>2?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 8

Problem: Does the point (5, 1) satisfy x≤5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 9

Problem: Does the point (2, 2) satisfy x+y>5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 10

Problem: Does the point (6, 3) satisfy x-y≥2?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Practice Exercise

Create one new Grade 6 problem about Testing an ordered pair. 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.

30.4 Boundary line idea

Boundary line idea is a Grade 6 mathematics idea in Graphing Inequalities with Two Variables. 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: Does the point (2, 3) satisfy x + y ≤ 6?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 2

Problem: Does the point (4, 4) satisfy x + y ≤ 6?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 3

Problem: Does the point (1, 5) satisfy x + y < 8?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 4

Problem: Does the point (3, 6) satisfy y ≥ 5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 5

Problem: Does the point (7, 2) satisfy x < 5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 6

Problem: Does the point (0, 0) satisfy x+y≥0?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 7

Problem: Does the point (-1, 4) satisfy y>2?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 8

Problem: Does the point (5, 1) satisfy x≤5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 9

Problem: Does the point (2, 2) satisfy x+y>5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 10

Problem: Does the point (6, 3) satisfy x-y≥2?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Practice Exercise

Create one new Grade 6 problem about Boundary line idea. 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.

30.5 Points that satisfy an 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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 Points that satisfy an 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.

30.6 Points that do not satisfy an 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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 Points that do not satisfy an 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.

30.7 Simple x plus y constraints

Simple x plus y constraints is a Grade 6 mathematics idea in Graphing Inequalities with Two Variables. 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: Does the point (2, 3) satisfy x + y ≤ 6?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 2

Problem: Does the point (4, 4) satisfy x + y ≤ 6?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 3

Problem: Does the point (1, 5) satisfy x + y < 8?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 4

Problem: Does the point (3, 6) satisfy y ≥ 5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 5

Problem: Does the point (7, 2) satisfy x < 5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 6

Problem: Does the point (0, 0) satisfy x+y≥0?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 7

Problem: Does the point (-1, 4) satisfy y>2?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 8

Problem: Does the point (5, 1) satisfy x≤5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 9

Problem: Does the point (2, 2) satisfy x+y>5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 10

Problem: Does the point (6, 3) satisfy x-y≥2?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Practice Exercise

Create one new Grade 6 problem about Simple x plus y 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.

30.8 Simple x less-than constraints

Simple x less-than constraints is a Grade 6 mathematics idea in Graphing Inequalities with Two Variables. 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: Does the point (2, 3) satisfy x + y ≤ 6?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 2

Problem: Does the point (4, 4) satisfy x + y ≤ 6?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 3

Problem: Does the point (1, 5) satisfy x + y < 8?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 4

Problem: Does the point (3, 6) satisfy y ≥ 5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 5

Problem: Does the point (7, 2) satisfy x < 5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 6

Problem: Does the point (0, 0) satisfy x+y≥0?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 7

Problem: Does the point (-1, 4) satisfy y>2?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 8

Problem: Does the point (5, 1) satisfy x≤5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 9

Problem: Does the point (2, 2) satisfy x+y>5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 10

Problem: Does the point (6, 3) satisfy x-y≥2?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Practice Exercise

Create one new Grade 6 problem about Simple x less-than 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.

30.9 Simple y greater-than constraints

Simple y greater-than constraints is a Grade 6 mathematics idea in Graphing Inequalities with Two Variables. 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: Does the point (2, 3) satisfy x + y ≤ 6?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 2

Problem: Does the point (4, 4) satisfy x + y ≤ 6?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 3

Problem: Does the point (1, 5) satisfy x + y < 8?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 4

Problem: Does the point (3, 6) satisfy y ≥ 5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 5

Problem: Does the point (7, 2) satisfy x < 5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 6

Problem: Does the point (0, 0) satisfy x+y≥0?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 7

Problem: Does the point (-1, 4) satisfy y>2?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 8

Problem: Does the point (5, 1) satisfy x≤5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 9

Problem: Does the point (2, 2) satisfy x+y>5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 10

Problem: Does the point (6, 3) satisfy x-y≥2?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Practice Exercise

Create one new Grade 6 problem about Simple y greater-than 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.

30.10 Graphing solution regions conceptually

Graphing solution regions conceptually is a Grade 6 mathematics idea in Graphing Inequalities with Two Variables. 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: Does the point (2, 3) satisfy x + y ≤ 6?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 2

Problem: Does the point (4, 4) satisfy x + y ≤ 6?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 3

Problem: Does the point (1, 5) satisfy x + y < 8?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 4

Problem: Does the point (3, 6) satisfy y ≥ 5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 5

Problem: Does the point (7, 2) satisfy x < 5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 6

Problem: Does the point (0, 0) satisfy x+y≥0?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 7

Problem: Does the point (-1, 4) satisfy y>2?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 8

Problem: Does the point (5, 1) satisfy x≤5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 9

Problem: Does the point (2, 2) satisfy x+y>5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 10

Problem: Does the point (6, 3) satisfy x-y≥2?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Practice Exercise

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

Common Mistake

A common mistake is ignoring the scale, labels, units, or the type of data shown.

30.11 Real-life two-variable constraints

A variable is a symbol used to represent a number that may change or may be unknown. 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: Evaluate 3x + 4 when x = 5.

  1. Replace x with 5.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 19

Worked Example 2

Problem: Evaluate 2n + 7 when n = 6.

  1. Replace n with 6.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 19

Worked Example 3

Problem: Evaluate 5a - 3 when a = 4.

  1. Replace a with 4.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 17

Worked Example 4

Problem: Evaluate 4p + 1 when p = 8.

  1. Replace p with 8.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 33

Worked Example 5

Problem: Evaluate 6m - 5 when m = 3.

  1. Replace m with 3.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 13

Worked Example 6

Problem: Evaluate 2.5x + 1 when x = 4.

  1. Replace x with 4.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 11

Worked Example 7

Problem: Evaluate 7q + 2 when q = 2.

  1. Replace q with 2.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 16

Worked Example 8

Problem: Evaluate 3r - 1 when r = 9.

  1. Replace r with 9.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 26

Worked Example 9

Problem: Evaluate 4y + 6 when y = 5.

  1. Replace y with 5.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 26

Worked Example 10

Problem: Evaluate 8k - 4 when k = 3.

  1. Replace k with 3.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 20

Practice Exercise

Create one new Grade 6 problem about Real-life two-variable 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.

30.12 Interpreting inequality graphs

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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. 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 Interpreting inequality graphs. 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 Two-variable inequality introduction?

Answer: A variable is a symbol used to represent a number that may change or may be unknown.

Q2. What is important to understand about Coordinate pairs as possible solutions?

Answer: Coordinate pairs as possible solutions is a Grade 6 mathematics idea in Graphing Inequalities with Two Variables. 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 Testing an ordered pair?

Answer: Testing an ordered pair is a Grade 6 mathematics idea in Graphing Inequalities with Two Variables. 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 Boundary line idea?

Answer: Boundary line idea is a Grade 6 mathematics idea in Graphing Inequalities with Two Variables. 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 Points that satisfy an inequality?

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

Q6. What is important to understand about Points that do not satisfy an inequality?

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

Q7. What is important to understand about Simple x plus y constraints?

Answer: Simple x plus y constraints is a Grade 6 mathematics idea in Graphing Inequalities with Two Variables. 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 Simple x less-than constraints?

Answer: Simple x less-than constraints is a Grade 6 mathematics idea in Graphing Inequalities with Two Variables. 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 Simple y greater-than constraints?

Answer: Simple y greater-than constraints is a Grade 6 mathematics idea in Graphing Inequalities with Two Variables. 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 Graphing solution regions conceptually?

Answer: Graphing solution regions conceptually is a Grade 6 mathematics idea in Graphing Inequalities with Two Variables. 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 Real-life two-variable constraints?

Answer: A variable is a symbol used to represent a number that may change or may be unknown.

Q12. What is important to understand about Interpreting inequality graphs?

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.