Learn on PengienVision, Mathematics, Grade 7Chapter 5: Solve Problems Using Equations and Inequalities

Lesson 4: Solve Inequalities Using Addition or Subtraction

In this Grade 7 lesson from enVision Mathematics Chapter 5, students learn to solve one-variable inequalities using the Addition and Subtraction Properties of Inequality, applying the same inverse operations used to solve equations while maintaining the inequality relationship. Students write, solve, and graph inequalities on a number line using real-world contexts such as baggage weight limits and temperature changes. The lesson also addresses inequalities with rational numbers, including decimals and fractions.

Section 1

Addition and Subtraction Property of Inequality

Property

For any numbers $a$ , $b$ , and $c$ , if $a < b$ , then

a + c < b + c \qquad a - c < b - c

For any numbers $a$ , $b$ , and $c$ , if $a > b$ , then

a + c > b + c \qquad a - c > b - c

We can add or subtract the same quantity from both sides of an inequality and still keep the inequality.

Examples

To solve $x + 7 < 15$ , subtract 7 from both sides: $x + 7 - 7 < 15 - 7$ , which simplifies to $x < 8$ .
To solve $y - 4 \geq -2$ , add 4 to both sides: $y - 4 + 4 \geq -2 + 4$ , which simplifies to $y \geq 2$ .
Given $12 > z + 5$ , subtract 5 from both sides: $12 - 5 > z + 5 - 5$ , so $7 > z$ , which means $z < 7$ .

Explanation

This property is just like it is for equations. You can add or subtract the same number on both sides of an inequality, and the relationship between the two sides stays the same. The inequality sign does not change.

Section 2

Graphing Inequality Solutions

Property

To show the solution set of an inequality on a number line:

For inequalities with $\leq$ or $\geq$ , use a filled-in dot to show the endpoint is included in the solution.
For inequalities with $<$ or $>$ , use an open circle to show the endpoint is not included.
Shade the region of the number line that represents all possible solutions, often with an arrow to show it continues infinitely.

Examples

To graph $x < 2$ , place an open circle at 2 and shade the number line to the left.
The graph for $x \geq -1$ has a filled-in dot at -1 and shading to the right.
The solution set for $x > 5$ is shown with an open circle at 5 and an arrow shading everything to the right.

Explanation

Think of the dot as a gate. A filled-in (closed) dot means the gate is part of your property (the solution). An open circle means it's just a post marking the boundary, but it isn't included.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Addition and Subtraction Property of Inequality

Property

For any numbers $a$ , $b$ , and $c$ , if $a < b$ , then

a + c < b + c \qquad a - c < b - c

For any numbers $a$ , $b$ , and $c$ , if $a > b$ , then

a + c > b + c \qquad a - c > b - c

We can add or subtract the same quantity from both sides of an inequality and still keep the inequality.

Examples

To solve $x + 7 < 15$ , subtract 7 from both sides: $x + 7 - 7 < 15 - 7$ , which simplifies to $x < 8$ .
To solve $y - 4 \geq -2$ , add 4 to both sides: $y - 4 + 4 \geq -2 + 4$ , which simplifies to $y \geq 2$ .
Given $12 > z + 5$ , subtract 5 from both sides: $12 - 5 > z + 5 - 5$ , so $7 > z$ , which means $z < 7$ .

Explanation

Section 2

Graphing Inequality Solutions

Property

To show the solution set of an inequality on a number line:

For inequalities with $\leq$ or $\geq$ , use a filled-in dot to show the endpoint is included in the solution.
For inequalities with $<$ or $>$ , use an open circle to show the endpoint is not included.
Shade the region of the number line that represents all possible solutions, often with an arrow to show it continues infinitely.

Examples

To graph $x < 2$ , place an open circle at 2 and shade the number line to the left.
The graph for $x \geq -1$ has a filled-in dot at -1 and shading to the right.
The solution set for $x > 5$ is shown with an open circle at 5 and an arrow shading everything to the right.

Explanation

Think of the dot as a gate. A filled-in (closed) dot means the gate is part of your property (the solution). An open circle means it's just a post marking the boundary, but it isn't included.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Addition and Subtraction Property of Inequality

Property

For any numbers $a$ , $b$ , and $c$ , if $a < b$ , then

a + c < b + c \qquad a - c < b - c

For any numbers $a$ , $b$ , and $c$ , if $a > b$ , then

a + c > b + c \qquad a - c > b - c

We can add or subtract the same quantity from both sides of an inequality and still keep the inequality.

Examples

To solve $x + 7 < 15$ , subtract 7 from both sides: $x + 7 - 7 < 15 - 7$ , which simplifies to $x < 8$ .
To solve $y - 4 \geq -2$ , add 4 to both sides: $y - 4 + 4 \geq -2 + 4$ , which simplifies to $y \geq 2$ .
Given $12 > z + 5$ , subtract 5 from both sides: $12 - 5 > z + 5 - 5$ , so $7 > z$ , which means $z < 7$ .

Explanation

Section 2

Graphing Inequality Solutions

Property

To show the solution set of an inequality on a number line:

For inequalities with $\leq$ or $\geq$ , use a filled-in dot to show the endpoint is included in the solution.
For inequalities with $<$ or $>$ , use an open circle to show the endpoint is not included.
Shade the region of the number line that represents all possible solutions, often with an arrow to show it continues infinitely.

Examples

To graph $x < 2$ , place an open circle at 2 and shade the number line to the left.
The graph for $x \geq -1$ has a filled-in dot at -1 and shading to the right.
The solution set for $x > 5$ is shown with an open circle at 5 and an arrow shading everything to the right.

Explanation

Think of the dot as a gate. A filled-in (closed) dot means the gate is part of your property (the solution). An open circle means it's just a post marking the boundary, but it isn't included.