Introduction

Solving Linear Inequalities is an important Grade 8 math skill because students are moving from simple answers toward explaining how the math works.

In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with solving linear inequalities.

What Is Solving Linear Inequalities?

Solving Linear Inequalities means choosing a model, naming what each number means, and explaining the strategy.

The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.

Understanding Solving Linear Inequalities

Before solving, students should slow down and decide what each number, shape, unit, or label represents.

  • Read the question carefully and identify what is being asked.
  • Choose a model, equation, table, or diagram that matches the situation.
  • Solve one step at a time and keep units or labels attached.
  • Use the answer explanation to check that the result makes sense.

Visual Models

Visual Model 1

Question: Which inequality has the solution graphed below?

Visual Model 1

  • A. \(x<2\)
  • B. \(x>2\)
  • C. \(x\leq2\)
  • D. \(x\geq2\)

Why it works: The open circle at \(2\) means \(2\) is not included. The ray points left to smaller values, so \(x<2\).

Answer: \(x<2\)

Visual Model 2

Question: Graph the solution to \(x+4<7\) on a number line. The inequality is:

Visual Model 2

  • A. \(x<3\)
  • B. \(x>3\)
  • C. \(x\leq3\)
  • D. \(x\geq3\)

Why it works: Subtract \(4\): \(x<3\). The open circle and left-pointing ray show all numbers less than \(3\).

Answer: \(x<3\)

Worked Examples

Example 1

Question: Which graph represents \(x\geq-1\)?

Example 1

  • A. Open circle at \(-1\), ray left
  • B. Open circle at \(-1\), ray right
  • C. Closed circle at \(-1\), ray right
  • D. Closed circle at \(0\), ray right
  1. The inequality \(x\geq-1\) includes \(-1\) (closed circle) and all values greater (ray points right).

Answer: Closed circle at \(-1\), ray right

Example 2

Question: Solve: \(x+7<10\). Which number line shows the solution?

Example 2

  • A. \(x<3\)
  • B. \(x>3\)
  • C. \(x\leq3\)
  • D. \(x\geq3\)
  1. Subtract \(7\): \(x<3\).
  2. The open circle and left ray match this solution.

Answer: \(x<3\)

Example 3

Question: Which shows the solution to \(x-3>-7\) on a number line?

Example 3

  • A. \(x>-4\)
  • B. \(x<-4\)
  • C. \(x\geq-4\)
  • D. \(x\leq-4\)
  1. Add \(3\): \(x>-4\).
  2. Open circle at \(-4\) with right ray confirms this.

Answer: \(x>-4\)

Real-World Word Problems

Problem 1

Question: A student solves \(-5x\leq15\) and gets \(x\leq-3\). Which explains the error?

  • A. Did not flip inequality
  • B. Divided incorrectly
  • C. Did not subtract first
  • D. No error

Why it works: Dividing by \(-5\) requires flipping: \(x\geq-3\), not \(x\leq-3\).

Answer: Did not flip inequality

Problem 2

Question: A temperature increases by 3° per hour. If it starts at \(12° C\) and must stay below \(30° C\), which inequality represents valid times (in hours)?

  • A. \(t<6\)
  • B. \(t>6\)
  • C. \(t\leq6\)
  • D. \(t\geq6\)

Why it works: Set up: \(12+3t<30\). Subtract \(12\): \(3t<18\). Divide by \(3\): \(t<6\) hours.

Answer: \(t<6\)

Common Mistakes

  • Rushing before identifying what the numbers represent.
  • Choosing an operation that does not match the situation.
  • Dropping labels, units, or context from the answer.
  • Skipping the estimate or reasonableness check.

Strategy Tips

  • Underline the question being asked.
  • Use a model before jumping to computation.
  • Write an equation that matches the story or picture.
  • Explain the final answer in a sentence.

Practice Questions

Question 1

Solve: \(-2x+5\geq11\)

  • A. \(x\geq-3\)
  • B. \(x\geq3\)
  • C. \(x\leq-3\)
  • D. \(x\leq3\)

Question 2

If \(x-8\leq-2\), what is the solution?

  • A. \(x\leq6\)
  • B. \(x\geq6\)
  • C. \(x\leq-10\)
  • D. \(x\geq-10\)

Question 3

Solve: \(-4x>12\)

  • A. \(x>-3\)
  • B. \(x>3\)
  • C. \(x<-3\)
  • D. \(x<3\)

Question 4

Which inequality is equivalent to \(2x+1\geq9\)?

  • A. \(x\leq4\)
  • B. \(x<4\)
  • C. \(x\geq4\)
  • D. \(x>4\)

Question 5

Solve: \(5-x>8\)

  • A. \(x>-3\)
  • B. \(x>13\)
  • C. \(x<-3\)
  • D. \(x<13\)

Question 6

Solve: \(2x-6\leq10\)

  • A. \(x\leq8\)
  • B. \(x\geq8\)
  • C. \(x\leq2\)
  • D. \(x\geq2\)
Full Answer Explanations Click to show all answers and explanations

Question 1

Answer: \(x\leq-3\)

Subtract \(5\): \(-2x\geq6\). Divide by \(-2\) (and FLIP the inequality): \(x\leq-3\).

Question 2

Answer: \(x\leq6\)

Add \(8\) to both sides: \(x\leq-2+8=6\).

Question 3

Answer: \(x<-3\)

Divide by \(-4\) (FLIP the inequality): \(x<\frac{12}{-4}=-3\).

Question 4

Answer: \(x\geq4\)

Subtract \(1\): \(2x\geq8\). Divide by \(2\): \(x\geq4\).

Question 5

Answer: \(x<-3\)

Subtract \(5\): \(-x>3\). Multiply by \(-1\) (FLIP): \(x<-3\).

Question 6

Answer: \(x\leq8\)

Add \(6\): \(2x\leq16\). Divide by \(2\): \(x\leq8\).

Timed practice quizzes

Related Quizzes

After studying Solving Linear Inequalities, open these two timed quizzes in a new tab for focused extra practice.

Connection to Standards

This lesson supports Grade 8 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.

Summary

Solving Linear Inequalities becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.

GOLDEN RULE

Understand the model before choosing the operation.