466 CHAPTER 7 Rational Expressions and Applications
By contrast, consider the following equation.Divide out common factors.
Distributive property
Get 0 on one side.
Factor.
Zero-factor propertyx= Solve for x.2
3
or x= 4
3 x- 2 = 0 or x- 4 = 0
13 x- 221 x- 42 = 03 x^2 - 14 x+ 8 = 04 x- 8 + 4 x= 3 x^2 - 6 x41 x- 22 + 4 x= 3 x 1 x- 224 x 1 x- 221
x
+ 4 x 1 x- 221
x- 2= 4 x 1 x- 223
4
1
x+
1
x- 2=
3
4
Points to Remember When Working with Rational Expressions
and Equations
1.When simplifying rational expressions, the fundamental property is applied
only after numerators and denominators have been factored.
2.When adding and subtracting rational expressions, the common denominator
must be kept throughout the problem and in the final result.
3.When simplifying rational expressions, always check to see if the answer is
in lowest terms. If it is not, use the fundamental property.
4.When solving equations with rational expressions, the LCD is used to clear
the equation of fractions. Multiply each side by the LCD. (Notice how this
use differs from that of the LCD in Point 2.)
5.When solving equations with rational expressions, reject any proposed
solution that causes an original denominator to equal 0.For each exercise, indicate “expression”if an expression is to be simplified or “equation”
if an equation is to be solved. Then simplify the expression or solve the equation.
- 1
x
- 1
+
1
x- 3=-
5
44
p+ 2+
1
3 p+ 63 t^2 - t
6 t^2 + 15 t,6 t^2 +t- 1
2 t^2 - 5 t- 25x- 4
5=x+ 3
62
k^2 - 4 k+3
k^2 - 162 y^2 +y- 6
2 y^2 - 9 y+ 9#y
(^2) - 2 y- 3
y^2 - 1
8
t- 5
= 2
1
x^2 +x- 2
,
4 x^2
2 x- 2
x^3 y^2
x^2 y^4
#y
5
x^4
4
p
- 6
p
Multiply each side by the LCD,
4 x 1 x- 22 ,to clear fractions.
Both and 4 are solutions, since neither makes a denominator equal 0. Check to
confirm that the solution set is E.
2
3 , 4F
2
3
and
since a denominator
is 0 for these values.
xZ 0 xZ 2
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