Additional Problems: Using inequalities to Solve Word Problems

When you need to use an inequality to solve a word problem, you will probably encounter one of the phrases that follows. In the table each phrase is stated along with its mathematical translation into symbols.

Important Words

Sample Sentence

 Equivalent Form

Translation

is at least

Bill is at least 21 years old.

 Bill's age is greater than or equal to 21

b > 21

is at most

 At most 5 students dropped the course 5 or fewer students dropped the course n < 5

cannot exceed

 earnings cannot exceed $1200. earnings must be less than or equal to $1200 r < 1200

must exceed

 The speed must exceed 15 mph The speed is greater than 15 mph s > 15

is less than

 Spot's weight is less than 50 lb.

-

 w < 50

is more than,
is greater than

 Boston is more than 200 miles away.

-

 d > 200

is between

 The film was between 90 and 100 minutes long.

-

 90 < t < 100

 

Example 1: The length of Lisa's rectangular dining room is 12 ft. If the area of the room is at least 96 square feet, 
                    what is the smallest width the room could have?

Solution
We are trying to find width.  Let width be w. We know that Area = length X Width. So Area = 12 w. Since Area is at least 96, Area > 96.  Substituting 12w into the inequality for Area we have 12w > 96

Solve the inequality by dividing both side by 12 and we have w > 8

The length of the dining room must be at least 8 feet.

 

Example 2: Lucy earned $400 and $550 in interest the last 2 years. How much interest must she earn this year so that her average earnings over the three year period is more than $600?

Solution
We are trying to find how much she needs to earn this year .  Let x be the amount she needs to earn. An average is found by summing the amount each year and dividing by the number of years. So the average earnings would be

  .

The average must be more than $600. So we need to solve the inequality:   600 < .

Solve the inequality by multiplying both sides by 3, to clear the fraction, and we have 1800 < 400 + 550 + x.
Solving for x we have 850 < x.

Since her earnings needs to average more than $600 she needs to make more than $850 this year.

 

Exercises

For problems 1- 5 translate each statement into an inequality (do not solve):

  1. The square of m is greater than zero
  2. The absolute value of the opposite of k is greater than 2.
  3. The reciprocal of t is less than 0.
  4. The product of t and 5 is at most -6.
  5. The opposite of x is between -3 and 5.

Solve the following word problems by 1) Clearly identifying your variables 2) Setting up an inequality 3) Solving the inequality 4) Stating your answer.  This is not guess and check. You must represent the problem with an inequality.

6. Sarah has a 85 and a 89 on her first two math 23 tests.  What must she get on her third test to have at least a 90% test average.  Assume three tests and represent the problem with an inequality.  

7. Robert makes $3.50 per hour working at a convenience store. If he gets a bonus of $25 this week, how many hours must he work to make at least $165?

8. The low temperatures for the last two days were 28° and 15° . What must the low temperature for the next day be in order for the average temperature for the 3-day period to be less than 19° ?

Answers