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Solving Rational Equations
ALGEBRA 2 LESSON 9-6
Solve
1 = 6x .
x – 3 x2– 9
1
= 6x
x – 3 x 2– 9
x2 – 9 = 6x(x – 3)
Write the cross products.
x2 – 9 = 6x2 – 18x
Distributive Property
–5x2 + 18x – 9 = 0
Write in standard form.
5x2 – 18x + 9 = 0
Multiply each side by –1.
(5x – 3)(x – 3) = 0
Factor.
5x – 3 = 0 or x – 3 = 0
x=3
5
or
x=3
Zero-Product Property
9-6
Solving Rational Equations
ALGEBRA 2 LESSON 9-6
(continued)
Check: When x = 3, both denominators in the original
equation are zero.
The original equation is undefined at x = 3.
So x = 3 is not a solution.
When 3 is substituted for x in the original equation,
5
5
both sides equal – 12 .
9-6
Solving Rational Equations
ALGEBRA 2 LESSON 9-6
Solve 3 – 4 = 1.
5x
3x
3
3
– 4 = 1.
5x 3x 3
1
15x 3 – 4 = 15x 3
5x
3x
45x – 60x = 15x
3
5x
3x
9 – 20 = 5x
–
Multiply each side by the LCD, 15x.
Distributive Property
Simplify.
11
=x
5
Since – 11 makes the original equation true, the solution is x = – 11.
5
5
9-6
Solving Rational Equations
ALGEBRA 2 LESSON 9-6
Josefina can row 4 miles upstream in a river in the same time it
takes her to row 6 miles downstream. Her rate of rowing in still water is 2
miles per hour. Find the speed of the river current.
Relate: speed with the current = speed in still water + speed of the current,
speed against the current = speed in still water – speed of the current,
time to row 4 miles upstream = time to row 6 miles downstream
Define:
Write:
Distance (mi)
Rate (mi/h)
With current
6
2+r
Against current
4
2–r
6
4
=
(2 + r )
(2 – r )
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Time (h)
6
(2 + r )
4
(2 – r )
Solving Rational Equations
ALGEBRA 2 LESSON 9-6
(continued)
6
4
=
(2 + r )
(2 – r )
4
6
(2 + r )(2 – r ) (2 + r ) = (2 + r )(2 – r ) (2 – r )
(2 – r )(6) = (2 + r )(4)
Multiply by the LCD
(2 + r )(2 – r ).
Simplify.
12 – 6r = 8 + 4r
Distributive Property
4 = 10r
Solve for r.
0.4 = r
Simplify.
The speed of the river current is 0.4 mi/h.
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Solving Rational Equations
ALGEBRA 2 LESSON 9-6
Jim and Alberto have to paint 6000 square feet of hallway in an
office building. Alberto works twice as fast as Jim. Working together, they
can complete the job in 15 hours. How long would it take each of them
working alone?
Relate:
Jim’s work speed + Alberto’s work speed = combined work speed
Define:
Time (hours)
Jim
2x
Alberto
x
Combined
Write:
Rate (square feet per hour)
6000
2x
6000
x
6000
= 400
15
15
6000 + 6000 = 400
2x
x
9-6
Solving Rational Equations
ALGEBRA 2 LESSON 9-6
(continued)
6000
+ 6000
2x
x
2x
= 400
6000
6000
+
= 2x(400)
2x
x
Multiply by the LCD, 2x.
2x(6000)
+ 2x(6000) = 2x(400)
2x
x
Distributive Property
6000 + 12000 = 800x
Simplify.
18000 = 800x
Simplify.
22.5 = x
Solve for x.
Alberto could paint the hallway in 22.5 hours.
Jim could paint the hallway in 2(22.5) hours or 45 hours.
9-6