Lesson 8–6

Example 1  Solve a Rational Equation
Solve  –  = . Check your solution.

The LCD for the three denominators is 3x.
            –  = 	Original equation
           3x( – ) = 3x()	Multiply each side by 3x.
	Distributive Property
          29 – x = 24	Simplify.
           –x = –5	Subtract 29 from each side.
           x = 5	Divide each side by –1.

Check 	 –  = 	Original equation
 –   	x = 5
 –   	Simplify.
 =   	The solution is correct.

Example 2  Elimination of a Possible Solution
Solve  –  = . Check your solution.

The LCD is a2 – 25.
 –  = 	Original equation; the LCD is (a2 – 25).
(a2 – 25)( – ) = (a2 – 25)()	Multiply each side by the LCD.
(a2 – 25)() – (a2 – 25)() = (a2 – 25)()	Distributive Property
(2a2) – (a – 5)(a) = (a + 5)(7)	Simplify.
2a2 – a2 + 5a = 7a + 35	Simplify.
a2 – 2a – 35 = 0	Subtract (7a + 35) from each side.
(a – 7)(a + 5) = 0	Factor.
a – 7 = 0        or           a + 5 = 0	Zero Product Property
a = 7	a = –5	

Check:
 –  = 	Original equation	 –  = 	Original equation	
 –   	a = 7	–  	a = –5	
 –   	Simplify.	 –   	Simplify.	
 =   		Since a = –5 results in a zero in the denominator, eliminate –5 from the list of solutions.	
The solution is 7.

Example 3  Work Problem
CONSTRUCTION  The Belgrade Intermediate School needs a new roof. Since the weather is turning cold quickly, the school board decides to hire two construction companies to complete the project. Rick’s Roofing claims to be able to complete the job in 8.5 days. Phil’s Fix–It Inc. claims to be able to complete the job in 11 days. If the time estimates of both companies are accurate, how long will it take for the two companies to complete the job working together?

In 1 day, Rick’s Roofing can complete  of the roof.
In 2 days, Rick’s Roofing can complete  • 2 or  of the roof.
In d days, Rick’s Roofing can complete  • d or  of the roof.
Likewise, in d days, Phil’s Fix–It Inc. can complete  • d or  of the roof. 

Together, they will complete the whole roof.

	
	
	+ 		= 	1	
Solve the equation.
 +  = 1	Original equation
  93.5( + ) = 93.5(1)	Multiply each side by 93.5.
 11d + 8.5d = 93.5	Simplify.
 19.5d = 93.5	Simplify.
 d ≈ 4.8	Divide each side by 19.5.

It will take about 4.8 days to complete the roof.

Example 4  Rate Problem
TRAVEL  A direct flight from Chicago, IL, to Washington, D.C., is approximately 600 miles. On a particular day, the speed of the wind is 50 miles per hour at the cruising altitude of a passenger jet. A jet leaves for this flight flying with the wind. Then the jet makes the return flight against the wind. The total in–air time for the jet for both flights was 2.7 hours. What is the speed of the jet in still air?

Words  The formula that relates distance, time, and rate is d = rt or  = t. 
Variables  Let r be the speed of the jet in still air. Then the speed of the jet with the wind is r + 50, and the speed of the jet against the wind is r – 50.
Equation

	
	
	+		= 	2.7	
Solve the equation.
 +  = 2.7	Original equation.
10(r + 50)(r – 50)( + ) = 10(r + 50)(r – 50)(2.7)	Multiply each side by 10(r + 50)(r – 50).
10(r – 50)(600) + 10(r + 50)(600) = 10(r + 50)(r – 50)(2.7)	Distributive Property
6000r – 300,000 + 6000r + 300,000 = 27r2 – 67,500	Simplify.
12,000r = 27r2 – 67,500	Simplify.
0 = 27r2 – 12,000r – 67,500	Subtract 12,000r from each side.
0 = 3(9r2 – 4,000r – 22,500)	Factor out the GCF.

Use the Quadratic Formula to solve for r.
x = 	Quadratic Formula
r = 	x = r, a = 9, b = –4,000, and c = –22,500
r = 	Simplify.
r = 	Simplify.
r = 450 or –	Simplify.

Since the speed must be positive, the answer is 450 miles per hour.

Example 5  Solve a Rational Inequality
Solve  < 24.

Step 1	Values that make a denominator equal to 0 are excluded from the domain. For this inequality, the excluded value is 8.
Step 2	Solve the related equation.

 = 24	Related equation
(b – 8) = (b – 8)(24)	Multiply each side by (b – 8).
6 = 24b – 192	Simplify.
198 = 24b	Add 192 to each side.
8.25 = b	Divide each side by 24.

Step 3	Draw vertical lines at the excluded value and at the solution to separate the number line into regions.

Now test a sample value in each region to determine if the values in the region satisfy the inequality.

Test b = 0.	Test b = 8.1.	Test b = 10.	
  24	  24	  24	
      –  24	         60  24	          3  24	
      – < 24  	         60 24	           3 < 24  	
b < 8 is a solution.	8 < b < 8.25 is not a solution.	b > 8.25 is a solution.	

The solution is b < 8 or b > 8.25.