A ** rational equation ** is an equation that involves
one or more rational expressions. For example,

the equation is a rational equation.

To solve a rational equation, the denominators can be
eliminated by multiplying each side of the

equation by the LCD of all of the rational expressions in the equation .

The steps are :

1. De termine the LCD of all of the rational expressions in the equation.

2. Multiply each side of the equation by the LCD.

3. Use the distributive law so that each in dividual expression is multiplied by
the LCD.

4. Simplify by dividing out common factors.

5. Solve the resulting equation.

6.

Check the answer(s) in the original equation. This is very important because it
is

possible to find a solution for the equation in step #5 that is not a solution
of the original

equation. This happens if the value found in step #5 makes one or more of the

denominators of the original equation equal to zero .

1. Use the steps given above to solve each of the fol lowing equations . For each
equation,

first write down the LCD.

2. Let . Find all
values of a such that f (a) =1.

3. ** Summary :**

The LCD ( least common denominator) is used when adding or subtracting rational
expressions

and when solving rational equations.

**When the LCD is used in adding or subtracting rational expressions, **the
numerator and

denominator of each fraction must be multiplied only by the factors that are
necessary to turn the

denominator into the LCD. That is, compare each denominator to the LCD and then
multiply

both the numerator and denominator of that fraction by the factors in the LCD
that are missing

from that denominator.

**When the LCD is used in solving rational equations,** multiply the entire
left side and the entire

right side of each equation by the LCD. This will result in each term being
multiplied by the

LCD. If the LCD has been chosen correctly, each fraction in the equation can
then be reduced

and the new equation should contain no fractions. Remember that it is very
important to check

your answer or answers in the original equation since extraneous solutions might
be obtained.

Keeping these guidelines in mind, add, subtract, or solve as indicated.

(a) Subtract:

(b) Solve:

**Answers**

1.

(a) x = 3

(b) t = 2, t = 5

(c) No solutions

2. a = -3/2, a = 2