example 1 solve a rational equation the lcd for the terms is 24(3 – x). original equation solve....
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Example 2 Solve a Rational Equation The LCD is (p + 1)(p – 1). Original equation Solve Check your solution. Multiply by the LCD.TRANSCRIPT
Solve a Rational Equation
The LCD for the terms is 24(3 – x).
Original equation
Solve . Check your solution.
Multiply each side by 24(3 – x).
Solve a Rational Equation
Distributive Property
Simplify.
Simplify.
Add 6x and –63 to each side.
Answer: The solution is –45.
Solve a Rational Equation
The LCD is (p + 1)(p – 1).
Original equation
Solve Check your solution.
Multiply by the LCD.
Solve a Rational Equation
(p – 1)(p2 – p – 5)
= (p2 – 7)(p + 1) + p(p + 1)(p – 1)
p3 – p2 – 5p – p2 + p + 5 = p3 + p2 – 7p – 7 + p3 – pp3 – 2p2 – 4p + 5
= 2p3 + p2 – 8p – 7
0
= p3 + 3p2 – 4p – 12
Divide commonfactors.
DistributiveProperty
Simplify.
Subtract p3 – 2p2 – 4p + 5from each side.
Solve a Rational Equation
Zero ProductProperty
0 = (p + 3)(p + 2)(p – 2)Factor.
0 = p + 3 or 0 = p + 2 or 0 = p – 2
Answer: The solutions are –3, –2 and 2.
Mixture Problem
BRINE Aaron adds an 80% brine (salt and water) solution to 16 ounces of solution that is 10% brine. How much of the solution should be added to create a solution that is 50% brine?
Understand Aaron needs to know how much of asolution needs to be added to anoriginal solution to create a newsolution.
Mixture Problem
Plan Each solution has a certainpercentage that is salt. Thepercentage of brine in the finalsolution must equal the amount ofbrine divided by the total solution.
Percentage of brine in solution
Mixture Problem
Substitute.
Simplify numerator.
LCD is 100(16 + x).
Solve Write a proportion.
Mixture Problem
Distribute.
Subtract 50x and160.
Divide each side by 30.
Answer: Aaron needs to add ounces of 80% brine solution.
Simplify.
Divide common factors.
Distance Problem
SWIMMING Lilia swims for 5 hours in a stream that has a current of 1 mile per hour. She leaves her dock and swims upstream for 2 miles and then back to her dock. What is her swimming speed in still water?Understand We are given the speed of the current,
the distance she swims upstream, andthe total time.
Plan She swam 2 miles upstream against thecurrent and 2 miles back to the dock withthe current. The formula that relates
distance, time, and rate is d = rt or
Distance Problem
Solve
Original equation
Time going withthe current plus
time going againstthe current equals
totaltime.
5
Let r equal her speed in still water. Thenher speed with the current is r + 1, andher speed against the current is r – 1.
Distance Problem
Divide Common Factors
Distribute.
Simplify.
Subtract 4r from each side.
(r + 1)2 + (r – 1)2 = 5(r2 – 1) Simplify.
Multiply each side by r2 – 1.
Distance Problem
Use the Quadratic Formula to solve for r.
Quadratic Formula
x = r, a = 5, b = – 4, and c = –5
Simplify.
Simplify.