7.4 Equations with Rational Expressions and Graphs
Solving an Equation with Rational Expressions
Step 1 Determine the domain of the variable.
Step 2 Multiply each side of the equation by the LCDto clear the fractions.
Step 3 Solvethe resulting equation.
Step 4 Checkthat each proposed solution is in the domain, and discard
any values that are not. Check the remaining proposed solution(s) in
the original equation.Solving a Rational EquationSolve
Step 1 The domain, which excludes 0, was found in Example 1(a).
Step 2 Multiply by the LCD, 2x.
Step 3 Distributive property
Multiply.
Subtract 4.
Proposed solution Divide byStep 4 CHECK Original equation
Let✓ True
The solution set is 5 - 16. NOW TRY
-
7
2
=-
7
2
x=-1.
2
- 1
-
3
2
^7
21 - 12
2
x
-
3
2
=
7
2 x
x=- 1 - 3.
- 3 x= 3
4 - 3 x= 7
2 xa
2
x
b - 2 xa
3
2
b = 2 xa
7
2 x
b
2 xa
2
x
-
3
2
b = 2 xa
7
2 x
b
2
x
-
3
2
=
7
2 x
.
EXAMPLE 2
CAUTION When each side of an equation is multiplied by a variableexpres-
sion, the resulting “solutions” may not satisfy the original equation. You must either
determine and observe the domain or check all proposed solutions in the original
equation. It is wise to do both.
Solving a Rational Equation with No SolutionSolve
Step 1 From Example 1(b),we know that the domain excludes 3 and
Step 2 Factor. Then multiply each side by the LCD,
1 x+ 321 x- 32 a
2
x- 3
-
3
x+ 3
b = 1 x+ 321 x- 32 c
12
1 x+ 321 x- 32
d
x^2 - 9 1 x+ 321 x- 32.
- 3.
2
x- 3
-
3
x+ 3
=
12
x^2 - 9
.
EXAMPLE 3
NOW TRY
EXERCISE 2
Solve.
1
3 x-
3
4 x=
1
3
NOW TRY ANSWER
- E-^54 F