Graph the solution set of a two-step inequality | Grade 7 Math | LevelKite

You already know how to solve a two-step inequality. Now you will graph its solutions, which means showing them on a number line.

An inequality usually has many solutions. Its solution set is the collection of every value of the variable that makes the inequality true. Here the variable is $x$ , a letter that stands for an unknown number. This lesson has no real-world situation, so $x$ can be any number: positive, negative, zero, a fraction or a decimal. In a real situation, $x$ might be limited to whole numbers or to values that are not negative, and then the solution set would be limited too.

Start with $2x + 3 > 7$ . Subtract $3$ from both sides: $2x + 3 - 3 > 7 - 3$ , so $2x > 4$ . The inequality stays true, because both sides change by the same amount. Then divide both sides by $2$ : $\frac{2x}{2} > \frac{4}{2}$ , so $x > 2$ . The symbol stays the same, because dividing both sides by the same positive number keeps their order.

The solved inequality $x > 2$ has the same solutions as $2x + 3 > 7$ , and it is easier to read. The table tests a few values of $x$ in the original inequality.

Test values in the original inequality

Value of x	Value of the left side	Is the left side greater than 7?
$0$	$2(0) + 3 = 0 + 3 = 3$	No
$1.9$	$2(1.9) + 3 = 3.8 + 3 = 6.8$	No
$2$	$2(2) + 3 = 4 + 3 = 7$	No: $7$ equals $7$ , so it is not greater
$2.1$	$2(2.1) + 3 = 4.2 + 3 = 7.2$	Yes
$5$	$2(5) + 3 = 10 + 3 = 13$	Yes

The table shows a pattern, and the solved inequality $x > 2$ confirms it. Numbers less than $2$ make $2x + 3 > 7$ false. Numbers greater than $2$ make it true, even $2.1$ , which is only a little more than $2$ . The number $2$ itself makes it false, because $7 > 7$ is not true. So $2$ is the boundary point: the number that separates the values that work from the values that do not.

The graph of $x > 2$ has two parts:

A hollow circle, an empty circle, sits on $2$ . Hollow means the boundary point is not in the solution set, as with $<$ or $>$ . A filled circle, a solid dot, means the boundary point is in the solution set, as with $\le$ or $\ge$ .
A ray, a thick line that starts at the circle and goes on forever in one direction, covers all the solutions. Here it runs to the right, because the numbers greater than $2$ lie to the right of $2$ . The arrowhead shows that it keeps going past the end of the line.

The hollow circle at

2

and the ray to the right show

x > 2

, the solution set of

2x + 3 > 7

.

Key concept

Solve first, then graph and check

Solve. Undo the addition or subtraction, then the multiplication or division. Reverse the symbol only if you multiply or divide both sides by a negative number.
Boundary and circle. The number in the solved inequality is the boundary point. A filled circle ( $\le$ or $\ge$ ) means the boundary is a solution. A hollow circle ( $<$ or $>$ ) means it is not.
Direction. With $x$ on the left, the ray runs right for $>$ or $\ge$ and left for $<$ or $\le$ .
Check. Pick one test value on each side of the boundary and substitute it into the original inequality. The one on the ray should make it true; the other should make it false.

When no situation limits $x$ to whole numbers or to values that are not negative, the finished ray shows the whole solution set.

Worked example

Dividing by a positive number

Graph the solution set of $3x - 5 \le 7$ .

Undo the subtraction first by adding 5 to both sides. Adding the same number to both sides keeps the inequality true.
$3x - 5 + 5 \le 7 + 5 \quad\Rightarrow\quad 3x \le 12$
Undo the multiplication by dividing both sides by 3. Dividing both sides by the same positive number keeps their order, so the symbol stays ≤.
$\frac{3x}{3} \le \frac{12}{3} \quad\Rightarrow\quad x \le 4$
The boundary point is 4. The symbol ≤ means less than or equal to, so 4 itself is a solution and the circle at 4 is filled. Substituting 4 into the original inequality confirms this.
$3(4) - 5 = 12 - 5 = 7,\ \text{and } 7 \le 7 \text{ is true.}$
With x on the left, x ≤ 4 describes 4 and every number less than 4. Those numbers lie to the left of 4, so the ray runs left.
Test a value on the ray side in the original inequality. Try 0. It should make the inequality true, and it does.
$3(0) - 5 = 0 - 5 = -5,\ \text{and } -5 \le 7 \text{ is true.}$
Test a value on the other side. Try 5. It should make the inequality false, and it does, so the ray points the right way.
$3(5) - 5 = 15 - 5 = 10,\ \text{and } 10 \le 7 \text{ is false.}$

Answer

The solution set is $x \le 4$ , shown by a filled circle at $4$ and a ray running left.

The filled circle at

4

and the ray to the left show

x \le 4

, with the test value

0

on the ray and

5

off it.

Worked example

Dividing by a negative number

Graph the solution set of $-4x + 3 > 11$ .

Subtract 3 from both sides to undo the addition. Both sides change by the same amount, so the inequality stays true.
$-4x + 3 - 3 > 11 - 3 \quad\Rightarrow\quad -4x > 8$
Divide both sides by -4 to undo the multiplication. Dividing by a negative number reverses the order of numbers, so the symbol changes from > to <.
$\frac{-4x}{-4} < \frac{8}{-4} \quad\Rightarrow\quad x < -2$
The boundary point is -2. The symbol < leaves the boundary out, so the circle at -2 is hollow. With x on the left, x < -2 describes the numbers less than -2. They lie to the left, so the ray runs left.
Test -3, on the ray side, in the original inequality. It should work, and it does.
$-4(-3) + 3 = 12 + 3 = 15,\ \text{and } 15 > 11 \text{ is true.}$
Test 0, on the other side. It should fail, and it does, so the ray points the right way.
$-4(0) + 3 = 0 + 3 = 3,\ \text{and } 3 > 11 \text{ is false.}$

Answer

The solution set is $x < -2$ , shown by a hollow circle at $-2$ and a ray running left.

The hollow circle at

-2

and the ray to the left show

x < -2

, with the test value

-3

on the ray and

0

off it.

Worked example

A boundary that is not a whole number

Graph the solution set of $4 - 3x \le -3$ .

Subtract 4 from both sides to undo the addition of 4. Both sides change by the same amount, so the inequality stays true.
$4 - 3x - 4 \le -3 - 4 \quad\Rightarrow\quad -3x \le -7$
Divide both sides by -3. You are dividing by a negative number, so the symbol reverses from ≤ to ≥.
$\frac{-3x}{-3} \ge \frac{-7}{-3} \quad\Rightarrow\quad x \ge \frac{7}{3}$
The boundary point is 7/3. Since 7 ÷ 3 is 2 with remainder 1, 7/3 is 2 and 1/3. A number line marked in whole numbers has no tick there, so place the boundary in words: one third of the way from 2 to 3.
$\frac{7}{3} = 2\frac{1}{3}$
The symbol ≥ includes the boundary, so the circle is filled. With x on the left, x ≥ 7/3 describes 7/3 and every number greater than it, so the ray runs right.
Test 3, on the ray side, in the original inequality. It should work, and it does, because -5 is less than -3.
$4 - 3(3) = 4 - 9 = -5,\ \text{and } -5 \le -3 \text{ is true.}$
Test 2, on the other side. It should fail, and it does, because -2 is greater than -3. So the ray points the right way.
$4 - 3(2) = 4 - 6 = -2,\ \text{and } -2 \le -3 \text{ is false.}$

Answer

The solution set is $x \ge \frac{7}{3}$ . Its graph is a filled circle at $2\frac{1}{3}$ , one third of the way from $2$ to $3$ , with a ray running right.

Takeaway: Solve first, reversing the symbol only if you multiply or divide both sides by a negative number. Then the solved inequality builds the graph: its number is the boundary point, its symbol picks a filled circle ( $\le$ or $\ge$ ) or a hollow one ( $<$ or $>$ ), and the ray covers the side where the solutions lie. A test value on each side, checked in the original inequality, confirms the graph. When $x$ can be any number, the finished ray shows the whole solution set.

Next, you will write and solve two-step inequalities from word problems and decide what the solution set means in a real situation, where only whole numbers or values that are not negative may make sense.