Graph and Write Inequalities on Number Lines | Grade 8 Math | LevelKite

You already know how to test whether a number is a solution of an inequality: substitute it and check whether the statement is true. An inequality such as $x < 3$ has more solutions than you could ever list. The numbers $2$ , $0$ and $-10$ work, and so do $2.9$ and $2.99$ , but $3$ itself does not, because $3 < 3$ is false. A graph on a number line shows every solution at once.

In this lesson, $x$ can be any real number, so a graph shows every solution, whether it is an integer, a fraction or a decimal. When an inequality describes a real situation, the context may restrict the solution values: a number of people, for example, must be a whole number, so only some points on a graph would fit that situation.

Key concept

An inequality and its graph carry the same three facts

When an inequality compares $x$ with one number and $x$ can be any real number:

Boundary: the number $x$ is compared with, placed on its labelled tick.
Circle: filled for $\le$ or $\ge$ : these non-strict symbols allow equality, so the boundary is a solution. Hollow for $<$ or $>$ : these strict symbols rule out equality, so the boundary is not a solution.
Ray: a thick part of the line that runs from the circle, without end, to one side. It covers the side where a test value (a number you substitute) makes the inequality true. One test on each side is enough, because the inequality can switch between true and false only at the boundary.

Graphing finds these facts in an inequality; writing reads them back from a graph.

Worked example

Graphing with a filled circle

Graph $x \ge 1$ on a number line.

Find the boundary, the number that x is compared with, since that is where the inequality can switch between true and false. Here it is 1, so place the boundary on the tick labelled 1.
Test the boundary itself to choose the circle. Substituting x = 1 gives a true statement, because the symbol means greater than or equal to, and 1 is equal to 1. So 1 is a solution, and the circle is filled.
$1 \ge 1 \quad \text{(true)}$
Test a value on the left of the boundary, such as 0. The statement is false, so no number on the left side is a solution, since every number there gives the same result.
$0 \ge 1 \quad \text{(false)}$
Test a value on the right, such as 3. The statement is true, so every number on the right side is a solution.
$3 \ge 1 \quad \text{(true)}$
Finish the graph with a ray running from the filled circle to the right. Its arrowhead shows that the solutions continue past the end of the drawing, since every number greater than 1 works.

Answer

The graph of $x \ge 1$ is a filled circle at $1$ with a ray running to the right, as shown below.

The filled circle at

1

includes

1

, and the ray to the right covers every number greater than

1

, such as

1.5

and

3

: this is the graph of

x \ge 1

.

Worked example

Graphing when the variable is on the right

Graph $4 > x$ on a number line.

The boundary is 4, the number that x is compared with. Place it on the tick labelled 4.
Test the boundary. The symbol means greater than, which is strict and rules out equality, and 4 is not greater than itself. The boundary is not a solution, so leave the circle hollow.
$4 > 4 \quad \text{(false)}$
Test one value on each side. The value 0, on the left, makes the inequality true; the value 6, on the right, makes it false.
$4 > 0 \ \text{(true)} \qquad 4 > 6 \ \text{(false)}$
Run the ray from the hollow circle to the left, the side whose test value made the inequality true.
Confirm the side by reading the inequality from right to left: if 4 is greater than x, then x is less than 4, and numbers less than 4 lie to its left. Swapping the two sides and turning the symbol around states the same comparison, so both forms have the same solutions and the same graph.
$4 > x \ \text{ means the same as } \ x < 4$

Answer

The graph of $4 > x$ is a hollow circle at $4$ with a ray running to the left, as shown below; it is also the graph of $x < 4$ .

The hollow circle at

4

leaves

4

out, and the ray to the left covers every number less than

4

, such as

3.9

and

0

: this is the graph of

4 > x

.

Now go the other way: start from a finished graph and write its inequality. Read three facts from the picture.

Boundary: the number printed at the circle's tick. Take it from the label, not from where the circle seems to sit.
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 $>$ .
Direction: numbers increase to the right, so a ray running left covers the numbers less than the boundary, and a ray running right covers the numbers greater than it.

Then check your inequality with two values: one on the ray's side, which must make it true, and one on the other side of the boundary, which must make it false.

The ticks on this line are

0.5

apart, and the filled circle sits on the tick labelled

-1.5

.

Worked example

Writing an inequality from a graph

Write the inequality shown by the graph above, where $x$ can be any real number, then check it.

Read the boundary from its label. The circle sits on the tick labelled -1.5, so the boundary is exactly -1.5. The inequality compares x with -1.5, and its symbol is still to be chosen.
$x \ \square \ -1.5$
Read the circle. It is filled, so -1.5 is a solution and the symbol must allow equality.
$x \le -1.5 \quad \text{or} \quad x \ge -1.5$
Read the direction. Numbers increase to the right, so a ray running left covers the numbers less than -1.5. With the filled circle, that means less than or equal to.
$x \le -1.5$
Check with a value on the ray's side. The value -3 lies left of -1.5 and makes the inequality true, as it should.
$-3 \le -1.5 \quad \text{(true)}$
Check with a value on the other side. The value 0 lies right of -1.5 and makes the inequality false, which matches the part of the line the ray does not cover.
$0 \le -1.5 \quad \text{(false)}$

Answer

The graph shows $x \le -1.5$ . You could also write it as $-1.5 \ge x$ by swapping the sides and turning the symbol around, as in the second example.

Takeaway: for inequalities like the ones in this lesson, the inequality and its graph carry the same three facts: the boundary, whether the boundary is a solution, and which side holds the other solutions. To graph, place the boundary, test it to choose a filled or hollow circle, and test one value on each side to choose the ray's direction. To write, read those three facts from the graph, then check with one value on each side of the boundary.

Next, you will solve one-step, two-step and multi-step inequalities and explain when the inequality symbol reverses.