Introduction

Writing and Solving Inequalities is an important Grade 7 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 writing and solving inequalities.

What Is Writing and Solving Inequalities?

Writing and Solving 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 Writing and Solving 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 number line represents \(x<-2\)?

Visual Model 1

Why it works: \(x<-2\) means all values less than (but not equal to) \(-2\). Use an open circle and an arrow pointing left.

Answer: Open circle at \(-2\), arrow pointing left.

Visual Model 2

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

Visual Model 2

Why it works: \(x\geq-1\) includes \(-1\) (filled circle) and all values greater than \(-1\) (arrow right).

Answer: Filled circle at \(-1\), arrow pointing right.

Worked Examples

Example 1

Question: Which number line represents \(x\leq2\)?

Example 1

  1. \(x\leq2\) includes \(2\) (filled circle) and all values less than \(2\) (arrow left).

Answer: Filled circle at \(2\), arrow pointing left.

Example 2

Question: Solve and identify the key step: \(-5x\geq20\)

Example 2

  • A. \(x\leq-4\)
  • B. \(x\geq-4\)
  • C. \(x<4\)
  • D. \(x=-4\)
  1. When dividing by a negative number, always flip the inequality sign.
  2. Here: \(-5x\geq20 \Rightarrow x\leq-4\).

Answer: \(x\leq-4\)

Example 3

Question: Solve the inequality \(x-3<4\) and identify which number line is correct.

Example 3

  1. Add \(3\): \(x<7\).
  2. The boundary is at \(7\) (open circle, not included), and solutions extend left (all values less than \(7\)).

Answer: Open circle at \(7\), arrow pointing left.

Real-World Word Problems

Problem 1

Question: Maria earns $12 per hour. How many hours must she work to earn at least $144?

  • A. \(h<12\)
  • B. \(h\geq12\)
  • C. \(h\leq144\)
  • D. \(h>12\)

Why it works: "At least $144" means \(\geq\). Set up: \(12h\geq144\). Divide by \(12\): \(h\geq12\).

Answer: \(h\geq12\)

Problem 2

Question: A student has $50 and spends $12 on a book. Write an inequality to find how much can be spent on a pen.

  • A. \(12+p\leq50\)
  • B. \(p+50\geq12\)
  • C. \(p\geq62\)
  • D. \(p\leq50\)

Why it works: Total spending (book + pen) must not exceed $50, so \(12+p\leq50\).

Answer: \(12+p\leq50\)

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 the inequality: \(2x+5\geq13\)

  • A. \(x\geq3\)
  • B. \(x\geq4\)
  • C. \(x\leq4\)
  • D. \(x\geq9\)

Question 2

Which inequality represents "a number plus 8 is less than 15"?

  • A. \(x+8<15\)
  • B. \(x+8>15\)
  • C. \(x-8<15\)
  • D. \(8x<15\)

Question 3

Solve: \(x+7<12\)

  • A. \(x<5\)
  • B. \(x>5\)
  • C. \(x\leq5\)
  • D. \(x<19\)

Question 4

Solve: \(\frac{x}{3}\geq2\)

  • A. \(x\geq\frac{2}{3}\)
  • B. \(x\leq6\)
  • C. \(x\geq6\)
  • D. \(x\geq2\)

Question 5

Solve: \(4x<20\)

  • A. \(x>5\)
  • B. \(x<5\)
  • C. \(x\geq5\)
  • D. \(x<80\)

Question 6

Solve: \(3x-4\leq11\)

  • A. \(x\geq5\)
  • B. \(x\leq5\)
  • C. \(x>5\)
  • D. \(x\leq15\)
Full Answer Explanations Click to show all answers and explanations

Question 1

Answer: \(x\geq4\)

Subtract \(5\): \(2x\geq8\). Divide by \(2\): \(x\geq4\).

Question 2

Answer: \(x+8<15\)

"A number plus 8" translates to \(x+8\), and "is less than" translates to \(<\).

Question 3

Answer: \(x<5\)

Subtract \(7\) from both sides: \(x<12-7=5\).

Question 4

Answer: \(x\geq6\)

Multiply both sides by \(3\): \(x\geq6\). The inequality sign stays the same because we multiply by a positive.

Question 5

Answer: \(x<5\)

Divide both sides by \(4\): \(x<\frac{20}{4}=5\).

Question 6

Answer: \(x\leq5\)

Add \(4\): \(3x\leq15\). Divide by \(3\): \(x\leq5\).

Timed practice quizzes

Related Quizzes

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

Connection to Standards

This lesson supports Grade 7 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

Writing and Solving 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.