Solving Rational Equations
Digital Lesson
Rational Expression
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A rational expression is a fraction with polynomials for the numerator and denominator.
are rational expressions.
For example,
If x is replaced by a number making the denominator of a rational expression zero, the value of the rational expression is undefined.
Example: Evaluate for x = –3, 0, and 1.
x
undefined
1
1
− 3
0
undefined
0
undefined
9
Rational Equation
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A rational equation is an equation between rational expressions.
For example, and are rational equations.
4. Check the solutions.
3. Solve the resulting polynomial equation.
2. Clear denominators by multiplying both sides of the � equation by the LCM.
1. Find the LCM of the denominators.
To solve a rational equation:
Examples: Solve
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Examples: 1. Solve: .
Find the LCM.
Multiply by LCM = (x – 3).
Solve for x.
LCM = x – 3.
1 = x + 1
x = 0
Check.
Substitute 0.
Simplify.
(0)
(0)
(0)
True.
2. Solve: .
x – 1 = 2x
Find the LCM.
LCM = x(x – 1).
Multiply by LCM.
Simplify.
x = –1
Solve.
Example: Solve
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After clearing denominators, a solution of the polynomial equation may make a denominator of the rational equation zero.
Since x2 – 1 = (x – 1)(x + 1),
Since – 1 makes both denominators zero, the rational equation has no solutions.
Example: Solve: .
2x = – 2 → x = – 1
3x + 1 = x – 1
Check.
It is critical to check all solutions.
In this case, the value is not a solution of the rational equation.
LCM = (x – 1)(x + 1).
Example: Solve
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Example: Solve: .
Factor.
Polynomial Equation.
Simplify.
Factor.
The LCM is (x – 3)(x – 5).
x2 – 8x + 15 = (x – 3)(x – 5)
x(x – 5) = – 6
x2 – 5x + 6 = 0
(x – 2)(x – 3) = 0
x = 2 or x = 3
Check. x = 2 is a solution.
Check. x = 3 is not a solution since both sides would be undefined.
Original Equation.
Example: Using Work Formula
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To solve problems involving work, use the formula,
Example: If it takes 5 hours to paint a room, what part of the � work is completed after 3 hours?
Three-fifths of the work is completed after three hours.
If one room can be painted in 5 hours then the rate of work is � (rooms/hour). The time worked is 3 hours.
part of work completed = rate of work time worked.
Therefore, part of work completed = rate of work time worked
part of work completed .
Example: Word Problem
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Example: If a painter can paint a room in 4 hours and her � assistant can paint the room in 6 hours, how many hours � will it take them to paint the room working together?
Let t be the time it takes them to paint the room together.
LCM = 12.
Multiply by 12.
Simplify.
Working together they will paint the room in 2.4 hours.
painter
assistant
rate of work
time worked
part of work completed
t
t
Examples: Using Motion Formulas
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distance = rate × time and time = .
To solve problems involving motion, use the formulas,
Examples: 1. If a car travels at 60 miles per hour for 3 hours, � what distance has it traveled?
2. How long does it take an airplane to travel 1200 � miles flying at a speed of 250 miles per hour?
It takes 4.8 hours for the plane make its trip.
time = = = 4.8.
Since distance = 1200 (mi) and rate = 250 (mi/h),
Since rate = 60 (mi/h) and time = 3 h, then
The car travels 180 miles.
distance = rate time = 60 3 = 180.
Example: Word Problem
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Example: A traveling salesman drives from home to a client’s �store 150 miles away. On the return trip he drives 10 miles per hour slower and adds one-half hour in driving time.
Let r be the rate of travel (speed) in miles per hour.
300r – 300(r – 10) = r(r – 10)
Trip to client
Trip home
distance
rate
time
150
r
150
r – 10
LCM = 2r (r – 10).
Example continued
At what speed was the salesperson driving on the way to the client’s store?
Multiply by LCM.
Example Continued
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0 = r2 – 10r – 3000
The salesman drove from home to the client’s store at �60 miles per hour.
The return trip took one-half hour longer.
At 60 mph the time taken to drive the 150 miles �from the salesman’s home to the clients store is = 2.5 h.
At 50 mph (ten miles per hour slower) the �time taken to make the return trip of 150 miles is = 3 h.
r = 60 or – 50
Example continued
Check:
(–50 is irrelevant.)
300r – 300r + 3000 = r2 – 10r
0 = (r – 60)(r + 50)