Graph Inequalities and Write Them from Number Lines | Grade 6 Math | LevelKite

You already know how to read inequality symbols. In this lesson, $x$ is a variable: a letter that stands for a number. A solution of an inequality is a value of $x$ that makes the inequality true. For $x > 3$ , the numbers $4$ , $3.5$ and $100$ are all solutions, because each one is greater than $3$ . The solutions never run out: there are infinitely many of them, so you cannot list them all. Instead, you can show them with a graph: a picture on a number line that marks every solution at once.

The graph of an inequality like $x > 3$ has two parts.

A circle at the boundary. The boundary is the number that $x$ is compared with, like $3$ in $x > 3$ . A filled circle means the boundary is a solution: putting it in place of $x$ makes the inequality true. A hollow circle means the boundary is not a solution: putting it in makes the inequality false.
A ray. A ray is a part of a line that starts at a point and goes on forever in one direction. On a graph, the ray starts at the circle and looks like a thick line. An arrowhead at its end means it keeps going past the edge of the picture. Every point the ray covers is a solution, even a point between two tick marks. The boundary itself is a solution only when its circle is filled.

On the number lines in this lesson, numbers get greater as you move right.

Key concept

Test the boundary, then test each side

To graph an inequality that compares $x$ with one number, answer two questions by testing values.

Is the boundary a solution? Put it in place of $x$ . If the statement you get is true, use a filled circle; this happens with $\le$ and $\ge$ . If it is false, use a hollow circle; this happens with $<$ and $>$ .
Which side holds the other solutions? Test one value on each side of the boundary. The ray runs toward the side whose value makes the inequality true.

The finished graph shows every solution, even points between the tick marks. To write an inequality from a graph, read the circle and the ray, then check values on and off the ray.

Worked example

Graph an inequality with a filled circle

Graph $x \ge 2$ on a number line from $0$ to $10$ with tick marks $1$ apart.

The boundary is $2$ , the number that $x$ is compared with. Test it by putting $2$ in place of $x$ . The statement is true, because $2$ is equal to $2$ . So $2$ is a solution, and its circle is filled.
$2 \ge 2 \quad \text{true}$
Test a value on the right side of $2$ . Try $5$ . The statement is true, because $5$ is greater than $2$ .
$5 \ge 2 \quad \text{true}$
Test a value on the left side of $2$ . Try $1$ . The statement is false, because $1$ is less than $2$ . So the solutions lie on the right side of $2$ , not the left.
$1 \ge 2 \quad \text{false}$
So the graph has a filled circle at $2$ and a ray running right from it, toward the side that tested true.
Read the finished ray. It covers points between ticks, such as $3.5$ , and $3.5 \ge 2$ is true. Its arrowhead means it keeps going past $10$ , so $25$ is a solution too.
$3.5 \ge 2 \ \text{true}, \qquad 25 \ge 2 \ \text{true}$

Answer

The graph of $x \ge 2$ is a filled circle at $2$ with a ray to the right. It shows every number greater than or equal to $2$ .

The filled circle at

2

and the ray to the right show

x \ge 2

on a line with tick marks

1

apart.

Worked example

Graph an inequality with a fraction boundary

Graph $x > \frac{1}{4}$ on a number line from $0$ to $2$ with tick marks $0.25$ apart.

The tick labels are decimals, so write the boundary $\frac{1}{4}$ as a decimal. Multiply its top and bottom by $25$ to get hundredths. This keeps the value the same, because $\frac{25}{25} = 1$ . Since $\frac{1}{4}$ and $0.25$ name the same number, $x > \frac{1}{4}$ says the same thing as $x > 0.25$ , and the boundary sits at the tick printed $0.25$ .
$\frac{1}{4} = \frac{1 \times 25}{4 \times 25} = \frac{25}{100} = 0.25$
Test the boundary. No number is greater than itself, so the statement is false. The circle is hollow.
$0.25 > 0.25 \quad \text{false}$
Test one value on each side. Try $1$ on the right: true. Try $0$ on the left: false. So the solutions lie to the right of $0.25$ .
$1 > 0.25 \ \text{true}, \qquad 0 > 0.25 \ \text{false}$
The graph has a hollow circle at $0.25$ and a ray running right from it, toward the side that tested true.

Answer

The graph of $x > \frac{1}{4}$ is a hollow circle at the tick printed $0.25$ with a ray to the right.

The hollow circle at

0.25

, which equals

\frac{1}{4}

, and the ray to the right show

x > \frac{1}{4}

.

A boundary does not have to land on a printed tick. Take $x \le \frac{1}{3}$ . As a decimal, $\frac{1}{3} = 1 \div 3 = 0.333\ldots$ , with 3s that never end. On the number line from the last example, $\frac{1}{3}$ falls between the ticks for $0.25$ and $0.5$ , because $0.25 < 0.333\ldots < 0.5$ . The same tests still work:

Boundary: $\frac{1}{3} \le \frac{1}{3}$ is true, so the circle is filled.
Each side: on the left, $0 \le \frac{1}{3}$ is true; on the right, $1 \le \frac{1}{3}$ is false. So the ray runs left.

In words, the graph is a filled circle at $\frac{1}{3}$ , between the ticks for $0.25$ and $0.5$ , with a ray running left from it.

Now run the steps the other way: read a graph and write its inequality, with $x$ as the variable. For a graph with one circle and one ray, like every graph in this lesson:

Read the circle. Filled means the boundary is a solution, so use $\le$ or $\ge$ . Hollow means it is not, so use $<$ or $>$ .
Read the ray. A ray to the right covers numbers greater than the boundary. A ray to the left covers numbers less than the boundary.
Check values on and off the ray. A value on the ray should make your inequality true. A value off the ray should make it false. The boundary should test true for a filled circle and false for a hollow one.

A hollow circle at

6

and a ray to the left, on a line with tick marks

1

apart.

Worked example

Write the inequality a graph shows

The graph just above has a hollow circle at $6$ and a ray to the left, on a number line with tick marks $1$ apart. Write the inequality it shows, using $x$ for the variable.

Read the circle. It is hollow, so the boundary $6$ is not a solution. The symbol must be $<$ or $>$ , because those symbols leave the boundary out.
Read the ray. It runs left from $6$ , toward smaller numbers. So the solutions are the numbers less than $6$ .
$x < 6$
Check values on the ray. The ray covers $3$ . It also covers $5.5$ , which lies between the ticks for $5$ and $6$ . Both values make $x < 6$ true, as they should.
$3 < 6 \ \text{true}, \qquad 5.5 < 6 \ \text{true}$
Check a value off the ray. The ray does not cover $8$ , and $8$ makes $x < 6$ false, as it should.
$8 < 6 \quad \text{false}$
Check the boundary. The boundary $6$ makes $x < 6$ false, which matches the hollow circle.
$6 < 6 \quad \text{false}$

Answer

The graph shows $x < 6$ : every number less than $6$ is a solution, and $6$ is not.

Takeaway: for an inequality that compares $x$ with one number, the graph shows the same solutions as the symbols. The circle answers whether the boundary is a solution: filled for $\le$ and $\ge$ , hollow for $<$ and $>$ . The ray holds every other solution, even points between ticks. Graph by testing the boundary and one value on each side. Write from a graph by reading the circle and the ray, then checking values on and off the ray. Next, you will write inequalities to represent constraints: conditions that a number must meet.