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
is at least
Sample Sentence
Bill is at least 21 years old.
Equivalent Form
Bill's age is greater than or equal to
is at most
At most 5 students dropped 5 or fewer students dropped the cou
the course
cannot exceed earnings cannot exceed
earnings must be less than or equal
$1200.
must exceed The speed must exceed 15
The speed is greater than 15 mph
mph
is less than
Spot's weight is less than 50
lb.
is more
Boston is more than 200
than,
is greater miles away.
than
is between
The film was between 90
and 100 minutes long.
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):
The square of m is greater than zero
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.
1.
2.
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 ?