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10-7 Solving Rational Equations
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Warm Up
California Standards
Lesson Presentation
10-7 Solving Rational Equations
Warm Up
1. Find the LCM of x, 2x2, and 6.
2. Find the LCM of p2 – 4p and p2 – 16.
Multiply. Simplify your answer.
3.
5.
4.
10-7 Solving Rational Equations
California
Standards
Preparation for
15.0 Students apply
algebraic techniques to solve rate problems,
work problems, and percent mixture
problems.
10-7 Solving Rational Equations
Vocabulary
rational equation
extraneous solutions
10-7 Solving Rational Equations
A rational equation is an equation that contains
one or more rational expressions. If a rational
equation is a proportion, it can be solved using
the Cross Product Property.
10-7 Solving Rational Equations
Additional Example 1: Solving Rational Equations
by Using Cross Products
Solve
. Check your answer.
5x = (x – 2)(3)
Use cross products.
5x = 3x – 6
Distribute 3 on the right side.
2x = –6
Subtract 3x from both sides.
x = –3
Divide both sides by 2.
10-7 Solving Rational Equations
Additional Example 1 Continued
Check
Substitute –3 for x in the
original equation.
–1 –1 
10-7 Solving Rational Equations
Check It Out! Example 1a
Solve
. Check your answer.
3n = (n + 4)(1)
Use cross products.
3n = n + 4
Distribute 1 on the right side.
2n = 4
Subtract n from both sides.
n=2
Divide both sides by 2.
10-7 Solving Rational Equations
Check It Out! Example 1a Continued
Check
Substitute 2 for n in the
original equation.

10-7 Solving Rational Equations
Check It Out! Example 1b
Solve
. Check your answer.
4h = (h + 1)(2)
Use cross products.
4h = 2h + 2
Distribute 2 on the right side.
2h = 2
Subtract 2h from both sides.
h=1
Divide both sides by 2.
10-7 Solving Rational Equations
Check It Out! Example 1b Continued
Check
Substitute 1 for h in the
original equation.

10-7 Solving Rational Equations
Check It Out! Example 1c
Solve
. Check your answer.
21x = (x – 7)(3)
Use cross products.
21x = 3x – 21
Distribute 3 on the right side.
18x = –21
Subtract 3x from both sides.
x=
Divide both sides by 18.
10-7 Solving Rational Equations
Check It Out! Example 1c Continued
Check
Substitute
for x in the
original equation.

10-7 Solving Rational Equations
Some rational equations contain sums or
differences of rational expressions. To solve
these, you must find a common denominator for
all the rational expressions in the equation.
10-7 Solving Rational Equations
Additional Example 2A: Solving Rational Equations
by Using the LCD
Solve each equation. Check your answer.
Step 1 Find the LCD.
2x(x + 1)
Include every factor of the denominator.
Step 2 Multiply both sides by the LCD.
Distribute
on the left
side.
10-7 Solving Rational Equations
Additional Example 2A Continued
Step 3 Simplify and solve.
Divide out common factors.
(2x)(2) + 6(x + 1) = 5(x + 1)
Simplify.
4x + 6x + 6 = 5x + 5
Distribute and multiply.
10x + 6 = 5x + 5
Combine like terms.
Subtract 5x and 6
from both sides.
Divide both sides by 5.
5x = –1
10-7 Solving Rational Equations
Additional Example 2A Continued
Check

10-7 Solving Rational Equations
Additional Example 2B: Solving Rational
Equations by Using the LCD
Solve each equation. Check your answer.
Step 1 Find the LCD.
(x2)
Include every factor of the denominator.
Step 2 Multiply both sides by the LCD.
Distribute on
the left side.
10-7 Solving Rational Equations
Additional Example 2B Continued
Step 3 Simplify and solve.
Divide out common
factors.
4x – 3 = x2
–x2 + 4x – 3 = 0
x2 – 4x + 3 = 0
(x – 3)(x – 1) = 0
Simplify.
Subtract x2 from both
sides.
Multiply by – 1.
Factor.
x = 3 or 1 Solve.
10-7 Solving Rational Equations
Additional Example 2B Continued
Check


10-7 Solving Rational Equations
Check It Out! Example 2a
Solve each equation. Check your answer.
Step 1 Find the LCD.
a(a +1)
Include every factor of the denominator.
Step 2 Multiply both sides by the LCD.
Distribute on
the left side.
10-7 Solving Rational Equations
Check It Out! Example 2a Continued
Step 3 Simplify and solve.
Divide out
common
factors.
3a = 4(a + 1)
Simplify.
3a = 4a + 4
Distribute the 4.
–4 = a
Subtract the 4 and 3a from
both sides.
10-7 Solving Rational Equations
Check It Out! Example 2a Continued
Check

10-7 Solving Rational Equations
Check It Out! Example 2b
Solve each equation. Check your answer.
Step 1 Find the LCD.
2j(j +2)
Include every factor of the denominator.
Step 2 Multiply both sides by the LCD.
Distribute on
the left side.
10-7 Solving Rational Equations
Check It Out! Example 2b Continued
Solve each equation. Check your answer.
Divide out common
factors.
12j – 10(2j + 4) = 4j + 8
12j – 20j – 40 = 4j + 8
–12j = 48
j = –4
Simplify.
Distribute 10.
Combine like terms.
Divide both sides by
–12.
10-7 Solving Rational Equations
Check It Out! Example 2b Continued
Check

10-7 Solving Rational Equations
When you multiply each side of an equation
by the LCD, you may get an extraneous
solution. An extraneous solution is a
solution to a resulting equation that is not a
solution to the original equation. Because of
extraneous solutions, it is especially important
to check your answers.
10-7 Solving Rational Equations
Additional Example 3: Extraneous Solutions
Solve
. Check your answer.
2(x2 – 1) = (x + 1)(x – 6) Use cross products.
Distribute 2 on the left side.
2x2 – 2 = x2 – 5x – 6
Multiply the right side.
Subtract x2 from both sides.
x2 + 5x + 4 = 0
Add 5x and 6 to both sides.
Factor the quadratic expression.
(x + 1)(x + 4) = 0
x = –1 or x = –4
Use the Zero Product Property.
Solve.
10-7 Solving Rational Equations
Additional Example 3 Continued
Solve
. Check your answer.
Check
Because
and
are undefined, –1 is
not a solution.


–1 is an extraneous solution. The only solution is –4.
10-7 Solving Rational Equations
Check It Out! Example 3a
Solve. Check your answer.
(x – 2)(x – 7) = 3(x – 7) Use cross products.
x2 – 9x + 14 = 3x – 21 Distribute 3 on the right side.
Multiply the left side.
x2 – 12x + 35 = 0
Subtract 3x from both sides.
Add 21 to both sides.
(x – 7)(x – 5) = 0
x = 7 or x = 5
Factor the quadratic expression.
Use the Zero Product Property.
Solve.
10-7 Solving Rational Equations
Check It Out! Example 3a Continued
Check


Because
and
are undefined, 7 is
not a solution.
7 is an extraneous solution. The only solution is 5.
10-7 Solving Rational Equations
Check It Out! Example 3b
Solve. Check your answer.
(x + 1)(x – 3) = 4(x – 2) Use cross products.
x2 – 2x – 3 = 4x – 8 Distribute 4 on the right side.
Multiply the left side.
x2 – 6x + 5 = 0
Subtract 4x from both sides.
Add 8 to both sides.
(x – 1)(x – 5) = 0
x = 1 or x = 5
Factor the quadratic expression.
Use the Zero Product Property.
Solve.
10-7 Solving Rational Equations
Check It Out! Example 3b Continued
Check
1 and 5 are
solutions.


There are no extraneous solutions. The solutions
are 1 and 5.
10-7 Solving Rational Equations
Check It Out! Example 3c
Solve. Identify any extraneous solutions.
6(x2 + 2x) = 9(x2)
6x2 + 12x) = 9x2
3x2 – 12x = 0
3x(x – 4) = 0
3x = 0, or x – 4 = 0
x = 0 or x = 4
Use cross products.
Distribute 6 on the left side.
Multiply the right side.
Subtract 9x2 from both sides.
Multiply through with – 1.
Factor the quadratic expression.
Use the Zero Product Property.
Solve.
10-7 Solving Rational Equations
Check It Out! Example 3c Continued
Check

Because
and
are undefined, 0 is
not a solution.

0 is an extraneous solution. The only solution is 4.
10-7 Solving Rational Equations
Lesson Quiz
Solve each equation. Check your answer.
1.
24
–4, 3
2.
3.
4. Solve
Check your answer.
–5
.