This section will discuss several types of applications. No matter how simple or complex the application is, the following steps will help to translate and solve the problem.
Read through the entire problem
Organize the information
Write the inequality
Solve the inequality
Check the answer
Look for key words and phrases. The following table gives some common translations for inequalities.
Words or Phrase
Symbol
Is less than
<
Is more than
>
At least
≥
At most
≤
Minimum
≥
Maximum
≤
No more than
≤
No less than
≥
Below is a video example of an application problem involving linear inequalities.
Example 1
Maria’s job provides a yearly base salary of $25,000 plus a commission of 7% of her sales. If her goal is to earn a yearly income of at least $39,000, what will her total sales need to be to achieve that goal?
Show Answer
Salary is $25,000
Let x be the total sales, so that Commission is 0.07 of x, or 0.07x
Income is at least $39,000 or ≥ $39,000
[latex]25000 + 0.07x≥ 39000[/latex] Salary + Commission ≥ Income
[latex]0.07x≥ 14000[/latex] Subtract 25,000 from 39,000
[latex]\frac{0.07x}{0.07}≥\frac{14000}{0.07}[/latex] Divide both sides by 0.07
[latex]x≥200000[/latex]
If Maria’s total sales are $200,000, she will earn $39,000.
If her total sales are more than $200,000, she will earn more than $39,000.
Answer
Maria’s total sales must be at least $200,000 in order to earn a yearly income of at least $39,000.
Here is another similar example:
Example 2
Leslie wants to rent a car for the day. The rental company charges a daily rate of $18.95 plus 19 cents for each mile driven. If Leslie only has $50 to spend on the car rental, what is the maximum number of miles she can drive? Round the answer to the nearest mile.
Show Answer
The cost per day is $18.95
Let x be the miles driven, so the mileage charge is 0.19[latex]x[/latex]
If Leslie drives 163 miles, she will spend $50 on the car rental. If Leslie drives less than 163 miles, she will spend less than $50 on the car rental.
Answer
In order to spend at most $50 on the car rental, Leslie must drive no more than 163 miles.
Special Consideration
What if the solution to our inequality had been [latex]x≤163.8210526…[/latex]? Normally, in this case we would round our answer to 164. However, let’s look at what happens if we substitute 164 for the variable in our inequality.
[latex]18.95 + 0.19(164)≤50[/latex]
[latex]18.95 + 31.16≤50[/latex]
[latex]50.11≤50[/latex]
This is a false statement.
Substituting [latex]x=164[/latex] into the inequality results in a false statement. Therefore, 164 is not a solution to the inequality. Because Leslie has a maximum of $50 to spend, we cannot round our solution up or the cost of the car rental will exceed $50.
Example 3
Leslie found a second rental company that charges 75 cents per mile with no additional fee. Recall the first rental company (from Example 2) charged $18.95 plus 19 cents for each mile driven. How many miles can she drive so that the second rental company is a better deal? Round down to the nearest whole mile.
Show Answer
The cost for the first rental company is
[latex]18.95+0.19x[/latex]
while the cost for the second rental company is
[latex].75x[/latex].
To determine the number of miles Leslie can drive so that the second company is cheaper, we solve the following inequality:
Initially, [latex]\frac{18.95}{0.56}\approx 33.8[/latex]. If we round down, we determine that if Leslie drives up to 33 miles, the second rental company will cost less.
Answer
Leslie can drive up to 33 miles.
In order to do the next example, you need to know how to find the average of a set of numbers. Recall that to find an average, you need to first add all the numbers together. Then you divide that sum by the number of values in your set.
Example 4
A student has scores of 85, 65, and 82 on his first three tests. He needs an average of at least 80 to earn a grade of B in the class. What is the minimum score that the student needs on the fourth test to ensure a grade of B?
Show Answer
Recall that to find an average, add the items and then divide by the number of items. Do not forget to include the fourth test score, which we can denote by [latex]x[/latex].
If the student earns an 88 on the fourth test, he will have an average of 80. If the student earns a score higher than 88, his grade will be higher than 80.
Answer
The student must get a minimum score of 88 on the fourth test to earn an average of at least 80.
Licenses and Attributions
CC licensed content, Original
Explanations and Examples . Authored by: Carla Kulinsky. Provided by: Salt Lake Community College 2016. License: Public Domain: No Known Copyright