Two-step Inequalities and Their Applications

1. $\frac{1}{15}(60-s) \geq 3$

Darcie wants to donate a minimum of $3$ blankets.

If Darcie skips $s$ days, then she will crochet for $60 -s$ days. Darcie crochets at a rate of $\frac {1}{15}$ of a blanket per day, so she can crochet a total of $\frac{1}{15}(60-s)$ blankets.

This amount must be greater than or equal to $3$ blankets, so the inequality is:

$\frac{1}{15}(60-s) \geq 3$

To solve the inequality, let's start by multiplying both sides by $15$. Remember that when we multiply an inequality by a negative number, the inequality sign reverses.

$\begin{aligned} \frac{1}{15}(60-s) & \geq 3 \\ 15 \cdot \frac{1}{15}(60-s) & \geq 15 \cdot 3 \\ 60-s & \geq 45 \\ 60-s-60 & \geq 45-60 \\ -s & \geq-15 \\ -s \cdot-1 & \leq-15 \cdot-1 \\ s & \leq 15 \\ \end{aligned}$

Darcie can skip no more than $15$ days of crocheting to meet her goal.

To graph the inequality $s \leq 15$, we first draw a circle at $15$. This circle divides the number line into two sections: one that contains solutions to the inequality and one that does not.

The solution includes the point $s=15$, so the circle at $15$ filled in.

Because the solution to the inequality says that $s \leq 15$, this means that solutions are numbers to the left of $15$.

The inequality is:

$\frac{1}{15}(60-s) \geq 3$

The graph of the solution of the inequality, $s \leq 15$, looks like this:

2. The inequality is: $m+\frac{1}{4} m+\frac{3}{2} m > 22$. The solution set is C: $m > 8$.

Mustafa, Heloise, and Gia have written more than a combined total of $22$ articles for the school newspaper.

Mustafa has written $m$ articles. Heloise has written $\frac{1}{4} m$ articles. Gia has written $\frac{3}{2} m$ articles.

They have written a combined total of $m+\frac{1}{4} m+\frac{3}{2} m$ articles.

This must be more than $22$ articles, so the inequality is:

$m+\frac{1}{4} m+\frac{3}{2} m > 22$

Let's start by combining like terms.

$\begin{aligned} m+\frac{1}{4} m+\frac{3}{2} m & > 22 \\ \frac{4}{4} m+\frac{1}{4} m+\frac{6}{4} m & > 22 \\ \frac{11}{4} m & > 22 \\ \frac{4}{11} \cdot \frac{11}{4} m & > \frac{4}{11} \cdot 22 \\ m & > 8 \end{aligned}$

Mustafa has written more than $8$articles.

The inequality is:

$m+\frac{1}{4} m+\frac{3}{2} m > 22$

The solution set is $m > 8$.

3. The inequality is: $6 +4 p > 17$. Kim's team scored a minimum of $3$ runs per inning.

Kim's team needed more than $17$ runs to win.

Kim's team already had $6$ runs. Since they scored $p$ runs per inning for $4$ innings, they scored an additional $4p$ runs, so Kim's team had a total $6 + 4p$ runs.

This must be greater than $17$, so the inequality is:

$6+4 p > 17$

We can start by subtracting $6$ from both sides of the inequality:

$\begin{aligned} 6+4 p & > 17 \\ 6-6+4 p & > 17-6 \\ 4 p & > 11 \\ \frac{4 p}{4} & > \frac{11}{4} \\ p & > 2 \frac{3}{4} \end{aligned}$

Kim's team must score more than $2 \frac{3}{4}$ runs per inning to win the game. So Kim's team must have scored a minimum of $3$ runs per inning.

The inequality is:

$6+4 p > 17$

Kim's team scored a minimum of $3$ runs per inning.

4. The inequality is: $3+1.2 c \geq 13.50$

Janey wants to earn enough money to buy a CD for $$13.50$

If Janie does $c$ chores, she will earn $$1.2c$ dollars for doing chores, plus $3$ dollars she already has. The amount she has is:

$3+1.2 c$

This amount must be greater than or equal to $$13.50$, so the inequality is:

$3+1.2 c \geq 13.50$

To solve the inequality, let's start by subtracting $3$ from both sides of the inequality:

$\begin{aligned} 3+1.2 c & \geq 13.5 \\ 3+1.2 c-3 & \geq 13.5-3 \\ 1.2 c & \geq 10.5 \\ \frac{1.2 c}{1.2} & \geq \frac{10.5}{1.2} \\ c & \geq 8.75 \end{aligned}$

Janie must do $8.75$ or more chores to have enough money to purchase the CD.

To graph the inequality $c \geq 8.75$, we first draw a circle at $8.75$. This circle divides the number line into two sections: one that contains solutions to the inequality and one that does not.

The solution includes the point $c=8.75$, so we fill in the circle at $8.75$.

Because the solution to the inequality says that $c \geq 8.75$, this means that solutions are numbers to the right of $8.75$.

The inequality is:

$3+1.2 c \geq 13.50$

The graph of the solution of the inequality $c \geq 8.75$ looks like this: