Graph Inequalities and Write Them from Graphs | Grade 7 Math | LevelKite

You already know how to test whether one value is a solution of an inequality: substitute it and see whether the statement is true. An inequality such as $x < 3$ , where the variable $x$ can be any number, has endlessly many solutions, such as $2$ , $0$ , $-10$ and $2.9$ . You cannot list them all, so you graph the inequality instead. Graphing means you show every solution at once on a number line.

A graph works because a number line keeps numbers in order. Numbers increase from left to right, so a number to the right of another is greater, and a number to the left is less. That holds for negative numbers too: $-3 < -1$ because $-3$ is farther left.

Key concept

The circle and the ray show every solution

The graph of an inequality has two parts.

A circle on the boundary. The boundary is the number that separates solutions from non-solutions. For $\le$ or $\ge$ , the circle is filled: the boundary makes the inequality true, so it is included. For $<$ or $>$ , the circle is hollow: the boundary makes the inequality false, so it is excluded.
A ray. A ray starts at the circle and runs forever in one direction, which its arrowhead shows. It covers, or shades, the side where the inequality is true. To find that side, test one value on each side of the boundary.

With no context, every number on the ray is a solution, including negatives and decimals. A real situation may limit solutions to whole numbers or nonnegative values (zero or greater).

Worked example

Graph an inequality with a whole-number boundary

Graph $x < 3$ , where $x$ can be any number.

The variable $x$ is compared with $3$ , so $3$ is the boundary. The circle goes on the tick labeled $3$ .
Decide the circle by testing the boundary itself. The statement is false, because $3$ is not less than itself. So $3$ is not a solution: $<$ excludes the boundary, and the circle is hollow.
$3 < 3 \text{ is false}$
Test one value on the left of $3$ . Try $0$ : the statement is true, so the left side holds solutions.
$0 < 3 \text{ is true}$
Test one value on the right of $3$ . Try $5$ : the statement is false, so the right side holds no solutions.
$5 < 3 \text{ is false}$
The ray covers the true side. It starts at the hollow circle and runs left forever, because every number to the left of $3$ , such as $-10$ , is less than $3$ . Its arrowhead shows that it never stops.

Answer

The graph of $x < 3$ is a hollow circle at $3$ with a ray running to the left, as shown below.

The hollow circle at

3

and the ray to the left show

x < 3

.

Worked example

Graph an inequality with a negative decimal boundary

Graph $y \ge -2.5$ , where $y$ can be any number.

The boundary is $-2.5$ , so the circle goes on the tick labeled $-2.5$ . Testing the boundary gives a true statement, because $\ge$ means greater than or equal to. So $-2.5$ is a solution: $\ge$ includes the boundary, and the circle is filled.
$-2.5 \ge -2.5 \text{ is true}$
Test a value on the left. Try $-3$ : it lies to the left of $-2.5$ , so it is less than $-2.5$ , and the statement is false.
$-3 \ge -2.5 \text{ is false}$
Test a value on the right. Try $0$ : it lies to the right of $-2.5$ , so it is greater, and the statement is true.
$0 \ge -2.5 \text{ is true}$
The ray covers the true side. It starts at the filled circle and runs right forever, past $0$ , because every number to the right of $-2.5$ is greater than $-2.5$ .

Answer

The graph of $y \ge -2.5$ is a filled circle at $-2.5$ with a ray running to the right, as shown below.

The filled circle at

-2.5

and the ray to the right show

y \ge -2.5

.

Writing an inequality from a graph reverses the process. You start with a finished ray and read three things from it:

The boundary: the number printed at the tick where the circle sits.
The circle: filled means the boundary is a solution, so the symbol is $\le$ or $\ge$ . Hollow means it is not, so the symbol is $<$ or $>$ .
The direction: a ray to the left covers numbers less than the boundary, and a ray to the right covers numbers greater than it.

Write the variable the problem names first, then the symbol, then the boundary. With the variable first, a ray to the left gives $<$ or $\le$ , and a ray to the right gives $>$ or $\ge$ . Then check: a value on the shaded side must make your inequality true, and a value off it must make it false.

The example below reads this graph for the variable

p

: a filled circle at

-1.5

and a ray to the left.

Worked example

Write the inequality shown by a ray

The graph above shows every solution of an inequality in the variable $p$ , where $p$ can be any number. Write the inequality.

Read the boundary. The circle sits on the tick labeled $-1.5$ , so $p$ is compared with $-1.5$ .
Read the circle. It is filled, so $-1.5$ is a solution. The symbol must include the boundary, so it is $\le$ or $\ge$ .
Read the direction. The ray runs left from $-1.5$ through $-2$ , $-2.5$ and $-3$ , and on forever. Every number to the left of $-1.5$ is less than $-1.5$ , so with $p$ written first, the symbol is $\le$ .
$p \le -1.5$
Check a value on the shaded side. Try $-2$ : it lies to the left of $-1.5$ , so the statement is true, as it must be.
$-2 \le -1.5 \text{ is true}$
Check a value off the shaded side. Try $0$ : it lies to the right of $-1.5$ , so the statement is false, which matches $0$ not being shaded.
$0 \le -1.5 \text{ is false}$

Answer

The graph shows $p \le -1.5$ : every number less than or equal to $-1.5$ is a solution.

Takeaway: a graph and an inequality are two ways to show the same solutions. The circle shows whether the boundary is included, and the ray covers the side where the inequality is true. You graph by placing the circle and testing one value on each side; you write by reading the boundary, circle and direction, then checking one value on the ray and one off it. Next, you will write inequalities to represent real-world situations, where the context may limit solutions to whole or nonnegative values.