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## Section3.2Solving Multistep Linear Inequalities

###### ObjectivesPCC Course Content and Outcome Guide

We have learned how to solve one-step inequalities in Section. In this section, we will learn how to solve multistep inequalities.

### Subsection3.2.1Solving Multistep Inequalities

When solving a linear inequality, we follow the same steps in List 3.1.4. The only difference in our steps to solving is that when we multiply or divide by a negative number on both sides of an inequality, the direction of the inequality symbol must switch. We will look at some examples.

###### Example3.2.3

Solve for $t$ in the inequality $-3t+5\geq11\text{.}$ Write the solution set in both set-builder notation and interval notation.

Explanation
\begin{align*} -3t+5\amp\geq11\\ -3t+5\subtractright{5}\amp\geq11\subtractright{5}\\ -3t\amp\geq6\\ \divideunder{-3t}{-3}\amp\mathbin{\highlight{\le}}\divideunder{6}{-3}\\ t\amp\leq-2 \end{align*}

Note that when we divided both sides of the inequality by $-3\text{,}$ we had to switch the direction of the inequality symbol.

The solution set in set-builder notation is $\{t\mid t\leq-2\}\text{.}$

The solution set in interval notation is $(-\infty,-2]\text{.}$

###### Remark3.2.4

Since the inequality solved in Example 3.2.3 has infinitely many solutions, it's difficult to check. We found that all values of $t$ for which $t\leq-2$ are solutions, so one approach is to check if $-2$ is a solution and additionally if one other number less than $-2$ is a solution.

Here, we'll check that $-2$ satisfies this inequality:

\begin{align*} -3t+5\amp\geq11\\ -3(\substitute{-2})+5\amp\stackrel{?}{\geq}11\\ 6+5\amp\stackrel{?}{\geq}11\\ 11\amp\stackrel{\checkmark}{\geq}11 \end{align*}

Next, we can check another number smaller than $-2\text{,}$ such as $-5\text{:}$

\begin{align*} -3t+5\amp\geq11\\ -3(\substitute{-5})+5\amp\stackrel{?}{\geq}11\\ 15+5\amp\stackrel{?}{\geq}11\\ 20\amp\stackrel{\checkmark}{\geq}11 \end{align*}

Thus both $-2$ and $-5$ are solutions. It's important to note that this doesn't directly verify that all solutions to this inequality check. It's valuable though in that it would likely help us catch an error if we had made one. Consult your instructor to see if you're expected to check your answer in this manner.

###### Example3.2.5

Solve for $z$ in the inequality $(6z+5)-(2z-3)\lt-12\text{.}$ Write the solution set in both set-builder notation and interval notation.

Explanation
\begin{align*} (6z+5)-(2z-3)\amp\lt-12\\ 6z+5-2z+3\amp\lt-12\\ 4z+8\amp\lt-12\\ 4z+8\subtractright{8}\amp\lt-12\subtractright{8}\\ 4z\amp\lt-20\\ \divideunder{4z}{4}\amp\lt\divideunder{-20}{4}\\ z\amp\lt -5 \end{align*}

Note that we divided both sides of the inequality by $4$ and since this is a positive number we did not need to switch the direction of the inequality symbol.

The solution set in set-builder notation is $\{z\mid z\lt-5\}\text{.}$

The solution set in interval notation is $(-\infty,-5)\text{.}$

###### Example3.2.6

Solve for $x$ in $-2-2(2x+1)\gt4-(3-x)\text{.}$ Write the solution set in both set-builder notation and interval notation.

Explanation
\begin{align*} -2-2(2x+1)\amp\gt4-(3-x)\\ -2-4x-2\amp\gt4-3+x\\ -4x-4\amp\gt x+1\\ -4x-4\subtractright{x}\amp\gt x+1\subtractright{x}\\ -5x-4\amp\gt1\\ -5x-4\addright{4}\amp\gt1\addright{4}\\ -5x\amp\gt5\\ \divideunder{-5x}{-5}\amp\mathbin{\highlight{\lt}}\divideunder{5}{-5}\\ x\amp\lt-1 \end{align*}

Note that when we divided both sides of the inequality by $-5\text{,}$ we had to switch the direction of the inequality symbol.

The solution set in set-builder notation is $\{x\mid x\lt-1\}\text{.}$

The solution set in interval notation is $(-\infty,-1)\text{.}$

###### Example3.2.7

When a stopwatch started, the pressure inside a gas container was $4.2$ atm (standard atmospheric pressure). As the container was heated, the pressure increased by $0.7$ atm per minute. The maximum pressure the container can handle was $21.7$ atm. Heating must be stopped once the pressure reaches $21.7$ atm. In what time interval was the container safe?

Explanation

The pressure increases by $0.7$ atm per minute, so it increases by $0.7m$ after $m$ minutes. Counting in the original pressure of $4.2$ atm, pressure in the container can be modeled by $0.7m+4.2\text{,}$ where $m$ is the number of minutes since the stop watch started.

The container is safe when the pressure is $21.7$ atm or lower. We can write and solve this inequality:

\begin{align*} 0.7m+4.2\amp\leq21.7\\ 0.7m+4.2\subtractright{4.2}\amp\leq21.7\subtractright{4.2}\\ 0.7m\amp\leq17.5\\ \divideunder{0.7m}{0.7}\amp\leq\divideunder{17.5}{0.7}\\ m\amp\leq25 \end{align*}

In summary, the container was safe as long as $m\leq25\text{.}$ Assuming that $m$ also must be greater than or equal to zero, this means $0\leq m\leq 25\text{.}$ We can write this as the time interval as $[0,25]\text{.}$ Thus the container was safe between 0 minutes and 25 minutes.

### Subsection3.2.2Exercises

###### 1

Solve this inequality.

$\displaystyle{ {x+3} > {9} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 2

Solve this inequality.

$\displaystyle{ {x+3} > {7} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 3

Solve this inequality.

$\displaystyle{ {4} > {x-10} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 4

Solve this inequality.

$\displaystyle{ {5} > {x-8} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 5

Solve this inequality.

$\displaystyle{ {5x} \leq {10} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 6

Solve this inequality.

$\displaystyle{ {2x} \leq {8} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 7

Solve this inequality.

$\displaystyle{ {6} \geq {-2x} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 8

Solve this inequality.

$\displaystyle{ {6} \geq {-3x} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 9

Solve this inequality.

$\displaystyle{ {{\frac{4}{9}}x} > {12} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 10

Solve this inequality.

$\displaystyle{ {{\frac{5}{6}}x} > {5} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 11

A swimming pool is being filled with water from a garden hose at a rate of $5$ gallons per minute. If the pool already contains $60$ gallons of water and can hold $160$ gallons, after how long will the pool overflow?

Assume $m$ minutes later, the pool would overflow. Write an equation to model this scenario. There is no need to solve it.

###### 12

An engineer is designing a cylindrical springform pan. The pan needs to be able to hold a volume of $305$ cubic inches and have a diameter of $12$ inches. What’s the minimum height it can have? (Hint: The formula for the volume of a cylinder is $V=\pi r^2h$).

Assume the pan’s minimum height is $h$ inches. Write an equation to model this scenario. There is no need to solve it.

###### 13

Solve this inequality.

$\displaystyle{ {9x+6} > {42} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 14

Solve this inequality.

$\displaystyle{ {10x+3} > {83} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 15

Solve this inequality.

$\displaystyle{ {4} \geq {3x-5} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 16

Solve this inequality.

$\displaystyle{ {25} \geq {4x-3} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 17

Solve this inequality.

$\displaystyle{ {41} \leq {1-4x} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 18

Solve this inequality.

$\displaystyle{ {33} \leq {8-5x} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 19

Solve this inequality.

$\displaystyle{ {-6x-4} \lt {-58} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 20

Solve this inequality.

$\displaystyle{ {-7x-1} \lt {-29} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 21

Solve this inequality.

$\displaystyle{ {4} \geq {-8x+4} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 22

Solve this inequality.

$\displaystyle{ {3} \geq {-9x+3} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 23

Solve this inequality.

$\displaystyle{ {-5} > {5-x} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 24

Solve this inequality.

$\displaystyle{ {-8} > {1-x} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 25

Solve this inequality.

$\displaystyle{ {3\!\left(x+4\right)} \geq {24} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 26

Solve this inequality.

$\displaystyle{ {4\!\left(x+8\right)} \geq {72} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 27

Solve this inequality.

$\displaystyle{ {7t+7} \lt {1t+37} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 28

Solve this inequality.

$\displaystyle{ {8t+4} \lt {3t+19} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 29

Solve this inequality.

$\displaystyle{ {-9z+6} \leq {-z-58} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 30

Solve this inequality.

$\displaystyle{ {-8z+7} \leq {-z-28} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 31

Solve this inequality.

$\displaystyle{ {a-9-3a} > {-9-4a+12} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 32

Solve this inequality.

$\displaystyle{ {a-9-9a} > {-10-10a+7} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 33

Solve this inequality.

$\displaystyle{ {-8p+6-8p} \geq {2p+6} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 34

Solve this inequality.

$\displaystyle{ {-2p+3-6p} \geq {3p+3} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 35

Solve this inequality.

$\displaystyle{ {64} \lt {-4\!\left(p-9\right)} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 36

Solve this inequality.

$\displaystyle{ {-20} \lt {-5\!\left(p-5\right)} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 37

Solve this inequality.

$\displaystyle{ {-\left(x-2\right)} \geq {8} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 38

Solve this inequality.

$\displaystyle{ {-\left(x-8\right)} \geq {13} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 39

Solve this inequality.

$\displaystyle{ {47} \leq {7-5\!\left(z-1\right)} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 40

Solve this inequality.

$\displaystyle{ {20} \leq {8-2\!\left(z-8\right)} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 41

Solve this inequality.

$\displaystyle{ {5-\left(y+9\right)} \lt {6} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 42

Solve this inequality.

$\displaystyle{ {1-\left(y+7\right)} \lt {-4} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 43

Solve this inequality.

$\displaystyle{ {1+7\!\left(x-9\right)} \lt {-15-\left(7-3x\right)} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 44

Solve this inequality.

$\displaystyle{ {2+9\!\left(x-5\right)} \lt {-20-\left(7-5x\right)} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

###### 45

You are riding in a taxi and can only pay with cash. You have to pay a flat fee of ${\25}\text{,}$ and then pay ${\2.80}$ per mile. You have a total of ${\193}$ in your pocket.

Let $x$ be the number of miles the taxi will drive you. You want to know how many miles you can afford. Write an inequality to represent this situation in terms of how many miles you can afford:

Solve this inequality. At most how many miles can you afford?

You can afford at most miles.

Use interval notation to express the number of miles you can afford.

###### 46

You are riding in a taxi and can only pay with cash. You have to pay a flat fee of ${\30}\text{,}$ and then pay ${\3.50}$ per mile. You have a total of ${\135}$ in your pocket.

Let $x$ be the number of miles the taxi will drive you. You want to know how many miles you can afford. Write an inequality to represent this situation in terms of how many miles you can afford:

Solve this inequality. At most how many miles can you afford?

You can afford at most miles.

Use interval notation to express the number of miles you can afford.

###### 47

A car rental company offers the following two plans for renting a car:

Plan A: $\31$ per day and $16$ cents per mile

Plan B: $\49$ per day with free unlimited mileage

How many miles must one drive in order to justify choosing Plan B?

One must drive more than miles to justify choosing Plan B. In other words, it’s more economical to use plan B if your number of miles driven will be in the interval (answer with interval notation).

###### 48

A car rental company offers the following two plans for renting a car:

Plan A: $\29$ per day and $17$ cents per mile

Plan B: $\52$ per day with free unlimited mileage

How many miles must one drive in order to justify choosing Plan B?

One must drive more than miles to justify choosing Plan B. In other words, it’s more economical to use plan B if your number of miles driven will be in the interval (answer with interval notation).

###### 49

You are offered two different sales jobs. The first company offers a straight commission of $9\%$ of the sales. The second company offers a salary of $\300$ per week plus $4\%$ of the sales. How much would you have to sell in a week in order for the straight commission offer to be at least as good?

You’d have to sell more than worth of goods for the straight commission to be better for you. In other words, the dollar amount of goods sold would have to be in the interval (answer using interval notation).

###### 50

You are offered two different sales jobs. The first company offers a straight commission of $8\%$ of the sales. The second company offers a salary of $\430$ per week plus $4\%$ of the sales. How much would you have to sell in a week in order for the straight commission offer to be at least as good?

You’d have to sell more than worth of goods for the straight commission to be better for you. In other words, the dollar amount of goods sold would have to be in the interval (answer using interval notation).