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3.2.2: Creating and Solving Rational Inequalities
Aaron works as a lawn care assistant at the Evergreen
Golf Course. He can mow all 18 holes on the course in 6
hours. The new hire, Devin, needs 12 hours to do the
same job. One day, Devin begins mowing the course.
Aaron joins him 3 hours later.
1. Using the variable t for time, create an equation that
can be solved to determine the total time it will take
for Aaron and Devin to finish mowing all 18 holes of
the course working together.
2. Together, how long will it take Aaron and Devin to
mow the entire course?
3. How long will Aaron work to mow the course? Devin?
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3.2.2: Creating and Solving Rational Inequalities
1. Using the variable t for time, create an equation that
can be solved to determine the total time it will take
for Aaron and Devin to finish mowing all 18 holes of
the course working together.
• If the variable t represents the time Devin is
mowing the course, and it takes him 12 hours to
t
complete the job by himself, then
represents
12
the portion of the course he mows in t hours.
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3.2.2: Creating and Solving Rational Inequalities
• It takes Aaron 6 hours to mow the entire course by
himself. If he starts 3 hours after Devin, then Aaron will
work 3 hours less than Devin. Therefore, the ratio
t -3
represents the portion of the course Aaron mows
6
in t hours when working with Devin.
• Together, Devin and Aaron have to mow the entire
course.
1
• The ratio representing the entire course is , or 1.
1
• Adding the ratios completed by each of the two
t t -3
mowers yields the rational equation +
= 1.
12
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3.2.2: Creating and Solving Rational Inequalities
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2. Together, how long will it take Aaron and Devin to mow
the entire course?
t t -3
• To solve the rational equation
+
= 1 requires a
12
6
common denominator.
• The denominator of the first term is 12, and the
denominator of the second term is 6; therefore, the
least common denominator is 12.
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3.2.2: Creating and Solving Rational Inequalities
t
12
+
12 ·
t -3
6
t
12
Equation from
previous step
=1
t - 3)
(
+ 12 ·
= 12 ·1
6
Multiply each term by a
factor that will result in
a denominator of 1.
12t 12(t - 3)
+
= 12
12
6
Simplify.
t + 2(t – 3) = 12
Continue to simplify.
t + 2t – 6 = 12
3t – 6 = 12
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3.2.2: Creating and Solving Rational Inequalities
• We can therefore find a solution for t by solving the
equation 3t – 6 = 12.
3t – 6 = 12
Equation
3t = 18
Add 6 to both sides.
t=6
Divide both sides by 3.
• It will take a total of 6 hours for Aaron and Devin to
mow the entire course working together.
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3.2.2: Creating and Solving Rational Inequalities
3. How long will Aaron work to mow the course? Devin?
• Remember that t is the time it takes Devin to
complete the task on his own.
• Devin will have to work 6 hours and Aaron will
have to work for t – 3, or 3 hours.
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3.2.2: Creating and Solving Rational Inequalities