Section 3.2 Solving Multistep Linear Inequalities
We have learned how to solve one-step inequalities in Section 2.6. In this section, we will learn how to solve multistep inequalities.
Subsection 3.2.1 Solving Multistep Inequalities
When solving a linear inequality, we follow the same steps in List 3.1.3. 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.
- Simplify
Simplify the expressions on each side of the inequality by distributing and combining like terms.
- Isolate
Use addition or subtraction to isolate the variable terms and constant terms (numbers) so that they are on different sides of the inequality symbol.
- Eliminate
Use multiplication or division to eliminate the variable term's coefficient. If each side of the inequality is multiplied or divided by a negative number, switch the direction of the inequality symbol.
- Check
When specified, verify the infinite solution set by checking multiple solutions.
- Summarize
State the solution set or (in the case of an application problem) summarize the result in a complete sentence using appropriate units.
Example 3.2.2.
Solve for \(t\) in the inequality \(-3t+5\geq11\text{.}\) Write the solution set in both set-builder notation and interval notation.
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 \((-\infty,-2]\text{.}\)
Remark 3.2.3.
Since the inequality solved in Example 3.2.2 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:
Next, we can check another number smaller than \(-2\text{,}\) such as \(-5\text{:}\)
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.
Example 3.2.4.
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.
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 \((-\infty,-5)\text{.}\)
Example 3.2.5.
Solve for \(x\) in \(-2-2(2x+1)\gt4-(3-x)\text{.}\) Write the solution set in both set-builder notation and interval notation.
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 \((-\infty,-1)\text{.}\)
Example 3.2.6.
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?
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:
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.
Exercises 3.2.2 Exercises
Review and Warmup
For each problem below, solve the inequality, and in addition to writing both set-builder and interval notations for each solution set, draw a graph of each solution set on a number line.
1.
\({x+4} > {8}\)
{ x | x > 4 };
2.
\({x+4} > {6}\)
;
3.
\({5} > {x-9}\)
;
4.
\({1} > {x-7}\)
;
5.
\({2x} \leq {4}\)
;
6.
\({2x} \leq {8}\)
;
7.
\({9} \geq {-3x}\)
;
8.
\({12} \geq {-3x}\)
;
9.
\({{\frac{6}{7}}x} > {18}\)
;
10.
\({{\frac{7}{4}}x} > {28}\)
;
11.
A wading pool is being filled with water from a garden hose at a rate of \(10\) gallons per minute. If the pool already contains \(80\) gallons of water and can hold \(250\) 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 \(338\) cubic inches and have a diameter of \(14\) inches. Whatâs the minimum height it can have? (Hint: The formula for the volume of a cylinder is \(V=\pi r^2h\)).
Assume the panâs minimum height is \(h\) inches. Write an equation to model this scenario. There is no need to solve it.
Solving Multistep Linear Inequalities
For each problem below, solve the inequality and, in addition to writing both set-builder and interval notations for each solution set, draw a graph of each solution set on a number line.
13.
\({2x+4} > {10}\)
;
14.
\({3x+10} > {31}\)
;
15.
\({4} \geq {4x-4}\)
;
16.
\({23} \geq {5x-2}\)
;
17.
\({63} \leq {9-6x}\)
;
18.
\({34} \leq {6-7x}\)
;
19.
\({-8x-2} \lt {-66}\)
;
20.
\({-9x-9} \lt {-27}\)
;
21.
\({3} \geq {-10x+3}\)
;
22.
\({2} \geq {-3x+2}\)
;
23.
\({-8} > {1-x}\)
;
24.
\({-6} > {2-x}\)
;
25.
\({5\!\left(x+2\right)} \geq {20}\)
;
26.
\({6\!\left(x+6\right)} \geq {84}\)
;
27.
\({8t+5} \lt {1t+68}\)
;
28.
\({9t+2} \lt {3t+44}\)
;
29.
\({-9z+8} \leq {-z-64}\)
;
30.
\({-10z+5} \leq {-z-22}\)
;
31.
\({a-2-6a} > {-9-7a+3}\)
;
32.
\({a-5-2a} > {-2-8a+11}\)
;
33.
\({-6p+4-3p} \geq {4p+4}\)
;
34.
\({-10p+10-10p} \geq {5p+10}\)
;
35.
\({60} \lt {-6\!\left(p-7\right)}\)
;
36.
\({-14} \lt {-7\!\left(p-3\right)}\)
;
37.
\({-\left(x-10\right)} \geq {14}\)
;
38.
\({-\left(x-6\right)} \geq {9}\)
;
39.
\({57} \leq {9-3\!\left(z-6\right)}\)
;
40.
\({46} \leq {1-9\!\left(z-3\right)}\)
;
41.
\({1-\left(y+8\right)} \lt {-14}\)
;
42.
\({2-\left(y+6\right)} \lt {1}\)
;
43.
\({2+10\!\left(x-8\right)} \lt {-83-\left(2-3x\right)}\)
;
44.
\({3+8\!\left(x-3\right)} \lt {-40-\left(2-5x\right)}\)
;
Applications
45.
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 \({\$2.60}\) per mile. You have a total of \({\$160}\) 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 \({\$35}\text{,}\) and then pay \({\$3.30}\) per mile. You have a total of \({\$266}\) 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 18 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 for a one-day rental?
48.
A car rental company offers the following two plans for renting a car.
Plan A: 27 per day and 19 cents per mile
Plan B: 51 per day with free unlimited mileage
How many miles must one drive in order to justify choosing Plan B for a one-day rental?
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 \(\$250\) 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?
Youâd 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 \(3\%\) of the sales. The second company offers a salary of \(\$380\) per week plus \(1\%\) 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?
Youâd 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 .