Two-step Inequalities and Their Applications

1. $x > 4$

Let’s start by subtracting $2$ from both sides of the inequality:

$8 x+2 > 34$

$8 x+2-2 > 34-2$

$8 x > 32$

Next, let's divide both sides by $8$:

$8 x > 32$

$\frac{8 x}{8} > \frac{32}{8}$

$x > 4$

The solution set of the inequality is $x > 4$.

2. C.

Let’s start by subtracting $18$ from both sides of the inequality:

$5 a+18 < -27$

$5 a+18-18 < -27-18$

$5 a < -45$

To isolate $a$, we need to divide both sides by $5$.

$5 a < -45$

$\frac{5 a}{5} < \frac{-45}{5}$

$a < -9$

To graph the inequality $a < -9$, we first draw a circle at $-9$. This circle divides the number line into two sections: one that contains solutions to the inequality and one that does not.

Since the solution uses a less than sign, the solution does not include the point where $a= - 9$. So the circle at $-9$ is not filled in.

Because the solution to the inequality says that $a < -9$, this means that solutions are numbers to the left of $-9$.

The graph that represents the solution of the inequality $a < -9$ is shown in Pink:

3. C.

Let’s start by adding $11$ to both sides of the inequality:

$-11-2 d \geq 1$

$-11+11-2 d \geq 1+11$

$-2 d \geq 12$

To isolate $d$ we need to divide both sides by $-2$:

$-2 d \geq 12$

$\frac{-2 d}{-2} \leq \frac{12}{-2}$

$d \leq-6$

To graph the inequality $d \leq-6$, we first draw a circle at $-6$. This circle divides the number line into two sections: one that contains solutions to the inequality and one that does not.

Since the solution uses a less than or equal to sign, the solution includes the point where $d = -6$. So the circle at $-6$ is filled in.

Because the solution to the inequality says that $d \leq-6$, this means that solutions are numbers to the left of $-6$.

The graph that represents the solution of the inequality $d \leq-6$, is shown Pink:

4. $x > -\frac{7}{3}$

Let’s start by subtracting $8$ from both sides of the inequality:

$-3 x+8 < 15$

$-3 x+8-8 < 15-8$

$-3 x < 7$

Next, let's divide both sides by $-3$.When you divide an inequality by a negative number, the inequality sign must be reversed.

$-3 x < 7$

$\frac{-3 x}{-3} > \frac{7}{-3}$

$x > -\frac{7}{3}$

The solution set of the inequality is $x > -\frac{7}{3}$.

5. $x \geq-10$

Let’s start by subtracting $13$ from both sides of the inequality:

$5 x+13 \geq-37$

$5 x+13-13 \geq-37-13$

$5 x \geq-50$

Next, let's divide both sides by $5$:

$5 x \geq-50$

$\frac{5 x}{5} \geq \frac{-50}{5}$

$x \geq-10$

The solution set of the inequality is:

$x \geq-10$.

6. D.

Let’s start by adding $15$ to both sides of the inequality:

$12 b-15 > 21$

$12 b-15+15 > 21+15$

$12 b > 36$

To isolate $b$, we need to divide both sides by $12$:

$12 b > 36$

$\frac{12 b}{12} > \frac{36}{12}$

$b > \frac{36}{12}$

$b > 3$

To graph the inequality $b > 3$, we first draw a circle at $3$. This circle divides the number line into two sections: one that contains solutions to the inequality and one that does not.

Since the solution uses a greater than sign, the solution does not include the point where $b=3$, So the circle at $3$ is not filled in.

Because the solution to the inequality says that $b > 3$, this means that solutions are numbers to the right of $3$.

The graph that represents the solution of the inequality $b > 3$, is shown in Red:

7. A.

Let’s start by adding $15$ to both sides of the inequality:

$-3 b-15 > -24$

$-3 b-15+15 > -24+15$

$-3 b > -9$

To isolate $b$, we need to divide both sides by $-3$. When you divide an inequality by a negative number, the inequality sign must be reversed.

$-3 b > -9$

$\frac{-3 b}{-3} < \frac{-9}{-3}$

$b < 3$

To graph the inequality $b < 3$, we first draw a circle at $3$. This circle divides the number line into two sections: one that contains solutions to the inequality and one that does not.

Since the solution uses a less than sign, the solution does not include the point where $b=3$. So the circle at $3$ is not filled in.

Because the solution to the inequality says that $b < 3$, this means that solutions are numbers to the left of $3$.

The graph that represents the solution of the inequality $b < 3$ is shown in Blue: