Two-step inequality word problems

Answers

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

Darcie wants to donate a minimum of \(3\) blankets.

Will Darcie meet her goal if she crochets.. Response
Less than \(3\) blankets? No
Exactly \(3\) blankets? Yes
More than \(3\) blankets? Yes
Conclusion Total blankets \(\geq 3\)






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.

Is the total number of articles..  Response
Less than \(22\) articles? No
Exactly \(22\) articles? No
More than \(22\) articles? Yes
Conclusion Total articles \( > 22\)






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.

Would Kim's team win with.. Response
Less than \(17\) runs? No
Exactly \(17\) runs? No
More than \(17\) runs? Yes
Conclusion Total runs \( > 17\)






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 \($13.50\)

Can Janey buy the CD if she has.. Response
Less than \($13.50\)? No
Exactly \($13.50\)? Yes
More than \($13.50\)? Yes
Conclusion Total dollars \(\geq 13.50\)






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: