Solving Equations That Contain Rational
Expressions
Let’s consider the case where some of the rational expressions have a
variable in the denominator.
Example
Solve:
| Solution |
 |
 |
| Multiply each side of the equation
by 4x, the LCD of the rational
expressions.
|
 |
 |
| Distribute 4x to each term on the
left side. |
 |
 |
| Reduce by canceling common factors. |
 |
 |
| |
8 + x(x - 2) |
= 23 |
| Distribute x. |
8 + x2 - 2x |
= 23 |
| The fractions have been eliminated and
we are left with the quadratic equation 8 + x2 - 2x = 23. |
|
|
| To solve this equation, first write it
in standard form, ax2 + bx + c = 0. |
x2 - 2x - 15 |
= 0 |
| To factor x2 - 2x - 15, find two
integers whose product is -15 and
whose sum is -2. They are -5 and 3. |
|
|
| Use the Zero Product Property.
Solve each equation. |
x - 5 x |
= 0
= 5 |
or
or |
x + 3 = 0 x = -3 |
So, the equation has two solutions: 5 and -3.
To check the solutions, we substitute each value of x in the original
equation and simplify.
