ORCCA Solving Multistep Linear Inequalities

Section 2.2 Solving Multistep Linear Inequalities

Objectives: PCC Course Content and Outcome Guide

MTH 60 CCOG 2.3
MTH 60 CCOG 2.5

We learned how to solve one-step inequalities in Section 1.6. In this section, we will solve linear inequalities that need more than one step.

Figure 2.2.1. Alternative Video Lessons

Subsection 2.2.1 Solving Multistep Inequalities

When solving a linear inequality, we follow the same steps in Process 2.1.4. The only difference 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.

Process 2.2.2. Steps to Solve Linear Inequalities.

Simplify: Simplify the expressions on each side of the inequality by distributing and combining like terms.
Separate: Use addition or subtraction to isolate the variable terms and constant terms (numbers) so that they are on different sides of the inequality symbol.
Clear Coefficient: Use multiplication or division to eliminate the variable term's coefficient. If you multiply or divide each side by a negative number, switch the direction of the inequality symbol.
Check: A solution to a linear inequality has a “boundary number”. In the original inequality, check a number less than the boundary number, the boundary number itself, and a number greater than the boundary number to confirm what should be a solution is indeed a solution, and what should not is not. (This can be time-consuming, so use your judgment about when you might get away with skipping this.)
Summarize: State the solution set or (in the case of an application problem) summarize the result in a complete sentence using appropriate units.

Example 2.2.3.

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. At this point we think that the solution set in set-builder notation is $\{t\mid t\leq-2\}\text{,}$ and the solution set in interval notation is $(-\infty,-2]\text{.}$

Since there are infinitely many solutions, it's impossible to literally check them all. We believe that all values of $t$ for which $t\leq-2$ are solutions. We check that one number less than $-2$ (any number, your choice) satisfies the inequality. And that $-2$ satisfies the inequality. And that one number greater than $-2$ (any number, your choice) does not satisfy the inequality. We choose to check the values $-10\text{,}$ $-2\text{,}$ and $0\text{.}$

eps pdf png svg tex

\begin{align*} -3t+5\amp\geq12\amp -3t+5\amp\ge12\amp -3t+5\amp\ge11\\ -3(\substitute{-10})+5\amp\stackrel{?}{\geq}12\amp -3(\substitute{-2})+5\amp\stackrel{?}{\geq}12\amp -3(\substitute{0})+5\amp\stackrel{?}{\geq}11\\ 30+5\amp\stackrel{?}{\geq}11\amp 6+5\amp\stackrel{?}{\geq}11\amp 0+5\amp\stackrel{?}{\geq}11\\ 35\amp\stackrel{\checkmark}{\geq}11\amp 11\amp\stackrel{\checkmark}{\geq}11\amp 5\amp\stackrel{\text{no}}{\geq}11 \end{align*}

So both $-10$ and $-2$ are solutions as expected, while $0$ is not. This is evidence that our solution set is correct, and it's valuable in that making these checks would likely help us catch an error if we had made one. While it certainly does take time and space to make three checks like this, it has its value.

Example 2.2.4.

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

Explanation

\begin{align*} (6z+5)-(2z-3)\amp\gt-12\\ 6z+5-2z+3\amp\gt-12\\ 4z+8\amp\gt-12\\ 4z+8\subtractright{8}\amp\gt-12\subtractright{8}\\ 4z\amp\gt-20\\ \divideunder{4z}{4}\amp\gt\divideunder{-20}{4}\\ z\amp\gt -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. At this point we think that the solution set in set-builder notation is $\{z\mid z\gt-5\}\text{,}$ and the solution set in interval notation is $(-5,\infty)\text{.}$

Since there are infinitely many solutions, it's impossible to literally check them all. We believe that all values of $z$ for which $z\gt-5$ are solutions. We check that one number less than $-5$ (any number, your choice) does not satisfy the inequality. And that $-5$ does not satisfy the inequality. And that one number greater than $-5$ (any number, your choice) does satisfy the inequality. We choose to check the values $-10\text{,}$ $-5\text{,}$ and $0\text{.}$

eps pdf png svg tex

\begin{align*} (6z+5)-(2z-3)\amp\gt-12\amp(6z+5)-(2z-3)\amp\gt-12\amp(6z+5)-(2z-3)\amp\gt-12\\ (6(\substitute{-10})+5)-(2(\substitute{-10})-3)\amp\stackrel{?}{\gt}-12\amp(6(\substitute{-5})+5)-(2(\substitute{-5})-3)\amp\stackrel{?}{\gt}-12\amp(6(\substitute{0})+5)-(2(\substitute{0})-3)\amp\stackrel{?}{\gt}-12\\ (-60+5)-(-20-3)\amp\stackrel{?}{\gt}-12\amp(-30+5)-(-10-3)\amp\stackrel{?}{\gt}-12\amp(0+5)-(0-3)\amp\stackrel{?}{\gt}-12\\ -55-(-23)\amp\stackrel{?}{\gt}-12\amp-25-(-13)\amp\stackrel{?}{\gt}-12\amp5-(-3)\amp\stackrel{?}{\gt}-12\\ -32\amp\stackrel{\text{no}}{\gt}-12\amp-12\amp\stackrel{\text{no}}{\gt}-12\amp8\amp\stackrel{\checkmark}{\gt}-12 \end{align*}

So both $-10$ and $-5$ are not solutions as expected, while $0$ is a solution. This is evidence that our solution set is correct. The solution set in set-builder notation is $\{z\mid z\gt-5\}\text{.}$ The solution set in interval notation is $(-5,\infty)\text{.}$

Checkpoint 2.2.5.

Example 2.2.6.

When a stopwatch started, the pressure inside a gas container was $4.2$ atm (one atm is 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. Over what time interval was the container in a safe state?

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.

Reading Questions 2.2.2 Reading Questions

1.

When solving an inequality, what are the conditions when you have to reverse the direction of the inequality symbol?

2.

How is the solution set to a linear inequality different from the solution set to a linear equation?

Exercises 2.2.3 Exercises

Review and Warmup

1.

Solve this inequality.

$\displaystyle{ {x+3} > {8} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

2.

Solve this inequality.

$\displaystyle{ {x+4} > {6} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

3.

Solve this inequality.

$\displaystyle{ {4} > {x-9} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

4.

Solve this inequality.

$\displaystyle{ {5} > {x-7} }$

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.

$\displaystyle{ {{\frac{4}{7}}x} > {8} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

10.

Solve this inequality.

$\displaystyle{ {{\frac{5}{4}}x} > {20} }$

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 $10$ gallons per minute. If the pool already contains $70$ gallons of water and can hold $230$ 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 $248$ 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.

Solving Multistep Linear Inequalities

13.

Solve this inequality.

$\displaystyle{ {9x+4} > {31} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

14.

Solve this inequality.

$\displaystyle{ {10x+10} > {70} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

15.

Solve this inequality.

$\displaystyle{ {26} \geq {3x-4} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

16.

Solve this inequality.

$\displaystyle{ {18} \geq {4x-2} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

17.

Solve this inequality.

$\displaystyle{ {45} \leq {9-4x} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

18.

Solve this inequality.

$\displaystyle{ {21} \leq {6-5x} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

19.

Solve this inequality.

$\displaystyle{ {-6x-2} \lt {-44} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

20.

Solve this inequality.

$\displaystyle{ {-7x-9} \lt {-23} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

21.

Solve this inequality.

$\displaystyle{ {3} \geq {-8x+3} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

22.

Solve this inequality.

$\displaystyle{ {2} \geq {-9x+2} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

23.

Solve this inequality.

$\displaystyle{ {-4} > {5-x} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

24.

Solve this inequality.

$\displaystyle{ {-7} > {1-x} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

25.

Solve this inequality.

$\displaystyle{ {3\!\left(x+2\right)} \geq {12} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

26.

Solve this inequality.

$\displaystyle{ {4\!\left(x+6\right)} \geq {56} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

27.

Solve this inequality.

$\displaystyle{ {7t+5} \lt {5t+11} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

28.

Solve this inequality.

$\displaystyle{ {8t+2} \lt {2t+62} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

29.

Solve this inequality.

$\displaystyle{ {-7z+8} \leq {-z-46} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

30.

Solve this inequality.

$\displaystyle{ {-8z+5} \leq {-z-16} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

31.

Solve this inequality.

$\displaystyle{ {a-8-5a} > {-6-6a+10} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

32.

Solve this inequality.

$\displaystyle{ {a-10-7a} > {-2-9a+4} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

33.

Solve this inequality.

$\displaystyle{ {-6p+4-6p} \geq {2p+4} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

34.

Solve this inequality.

$\displaystyle{ {-9p+10-4p} \geq {3p+10} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

35.

Solve this inequality.

$\displaystyle{ {44} \lt {-4\!\left(p-7\right)} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

36.

Solve this inequality.

$\displaystyle{ {-10} \lt {-5\!\left(p-3\right)} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

37.

Solve this inequality.

$\displaystyle{ {-\left(x-10\right)} \geq {16} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

38.

Solve this inequality.

$\displaystyle{ {-\left(x-6\right)} \geq {11} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

39.

Solve this inequality.

$\displaystyle{ {4} \leq {8-4\!\left(z-9\right)} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

40.

Solve this inequality.

$\displaystyle{ {81} \leq {9-9\!\left(z-6\right)} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

41.

Solve this inequality.

$\displaystyle{ {5-\left(y+8\right)} \lt {-9} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

42.

Solve this inequality.

$\displaystyle{ {1-\left(y+6\right)} \lt {1} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

43.

Solve this inequality.

$\displaystyle{ {1+10\!\left(x-8\right)} \lt {-38-\left(6-3x\right)} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

44.

Solve this inequality.

$\displaystyle{ {2+8\!\left(x-3\right)} \lt {-19-\left(6-5x\right)} }$

In set-builder notation, the solution set is .

In interval notation, the solution set is .

Applications

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 ${\$2.60}$ per mile. You have a total of ${\$155}$ 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?
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 ${\$3.30}$ per mile. You have a total of ${\$261}$ 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?
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: $\$30$ per day and $16$ cents per mile

Plan B: $\$47$ 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 .

48.

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

Plan A: $\$27$ per day and $18$ cents per mile

Plan B: $\$50$ 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 .

49.

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 $\$240$ 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 .

50.

You are offered two different sales jobs. The first company offers a straight commission of $7\%$ of the sales. The second company offers a salary of $\$370$ 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 .