01 / 20
>

Solve & Graph
Linear Inequalities

Learn to solve inequalities of the form \( px + q > r \) and \( px + q < r \), then graph and interpret their solution sets on a number line.

7th Grade Math

What is an Inequality?

An inequality is a mathematical statement that compares two expressions using symbols like \( < \), \( > \), \( \leq \), or \( \geq \). Unlike an equation, which has one answer, an inequality usually has infinitely many solutions.

\( x = 5 \)
Equation: exactly one solution
\( x > 5 \)
Inequality: infinitely many solutions (6, 7, 5.1, 100, ...)

The Four Inequality Symbols

\( < \)
Less than
\( > \)
Greater than
\( \leq \)
Less than or equal to
\( \geq \)
Greater than or equal to

Why Do Inequalities Matter?

Real-world problems rarely have exact answers. You need to know limits, boundaries, and ranges. Inequalities let you express constraints like "at most," "at least," "no more than," and "fewer than."

Budget
"I can spend at most $50" means \( \text{cost} \leq 50 \)
Speed Limit
"Drive no more than 65 mph" means \( \text{speed} \leq 65 \)
Height Requirement
"Must be at least 48 in. tall" means \( \text{height} \geq 48 \)

Solving Inequalities: The Core Rule

You solve a linear inequality exactly like an equation with one critical exception:

1
Add or subtract to isolate the variable term: sign stays the same.
2
Multiply or divide by a positive number: sign stays the same.
3
Multiply or divide by a negative number: flip the sign.

Graphing on a Number Line

Once you solve for the variable, you represent the solution set on a number line. The circle type and arrow direction tell the full story.

Open circle, arrow right: \( x > 3 \) 1 2 3 4 5 Closed circle, arrow left: \( x \leq 3 \) 1 2 3 4 5

Equations vs. Inequalities

Equation Inequality
Symbol \( = \) \( <, >, \leq, \geq \)
Solutions Usually one value A range of values
Graph A single point A ray or segment
Multiply/divide by negative No special rule Flip the sign

How to Solve & Graph

1
Identify the inequality and the variable to isolate.
2
Subtract or add the constant term \( q \) from both sides.
3
Divide or multiply both sides by the coefficient \( p \). If \( p < 0 \), flip the sign.
4
Draw the number line: open or closed circle, shade the correct direction.
5
Check your answer by plugging in a value from the shaded region.

Example 1: Solve \( 2x + 3 < 11 \)

1
Subtract 3 from both sides: \( 2x + 3 - 3 < 11 - 3 \) → \( 2x < 8 \)
2
Divide both sides by 2 (positive, no flip): \( \frac{2x}{2} < \frac{8}{2} \) → \( x < 4 \)
3
Graph: open circle at 4, shade left.
1 2 3 4 5

Example 2: Solve \( -3x + 5 > 14 \)

1
Subtract 5 from both sides: \( -3x > 9 \)
2
Divide by \( -3 \) and flip the sign: \( x < -3 \)
3
Graph: open circle at \( -3 \), shade left.
-5 -4 -3 -2 -1

Example 3: Word Problem

1
Set up: Let \( t \) = number of tickets. Total cost: \( 8t + 4.50 \leq 30 \)
2
Subtract 4.50: \( 8t \leq 25.50 \)
3
Divide by 8: \( t \leq 3.1875 \)
4
Interpret: You can't buy a fraction of a ticket, so \( t \leq 3 \). You can buy at most 3 tickets.
0 1 2 3

Example 4: Fractional Coefficients

Solve \( \frac{1}{2}x - 7 \geq -3 \) and graph the solution.

1
Add 7: \( \frac{1}{2}x \geq 4 \)
2
Multiply by 2 (positive, no flip): \( x \geq 8 \)
3
Graph: closed circle at 8, shade right.
6 7 8 9 10

Common Mistakes to Avoid

✕ Forgetting to Flip
When dividing by a negative, you must reverse the inequality sign. This is the number-one error.
✕ Wrong Circle Type
Using a closed circle for \( < \) or \( > \), or an open circle for \( \leq \) or \( \geq \).
✕ Shading the Wrong Way
Shading left when the solution is \( x > \) (should shade right), or vice versa.

Quick Reference Guide

Symbol Meaning Circle Shade
\( < \) Less than Open Left
\( > \) Greater than Open Right
\( \leq \) Less than or equal to Closed Left
\( \geq \) Greater than or equal to Closed Right

Translating Words to Symbols

Phrase Symbol
"at most," "no more than" \( \leq \)
"at least," "no fewer than" \( \geq \)
"fewer than," "less than" \( < \)
"more than," "greater than" \( > \)

Question 1 of 5

Solve: \( 4x + 7 < 23 \). What is the solution?

Question 2 of 5

Solve: \( -2x + 6 \geq 12 \). What is the solution?

Question 3 of 5

The graph shows a closed circle at 5 with shading to the right. Which inequality matches?

3 4 5 6 7

Question 4 of 5

A gym charges $12 per visit plus a $25 membership fee. You want to spend less than $85 total. Which inequality represents the number of visits \( v \)?

Question 5 of 5

When solving \( -5x + 3 < 18 \), a student writes \( x < -3 \). What did they do wrong?