SOLVING RATIONAL EQUATIONS

Recall that when solving for fractions that are added or subtracted from each
other that a common denominator must be found. This is true even if the
denominator contains a variable. An extra condition that must always be
considered is that any value of the variable that makes the denominator zero
must be excluded.

Example 1. Solve

Step 3. Simplify .

Step 4. Solve for a.

This is the solution since a ≠ 0 and 32 yields a valid check.

Example 2. Solve

Step 1. Find the LCD.
The LCD for this problem is x .
Step 2. Multiply all terms by the LCD of x.

Step 3. Simplify.

Step 4. Solve for x.

This is the solution since x ≠ 0 and 5 yields a valid check.

Example 3. Solve

Step 1. Find the LCD.
The LCD for this problem is ( b – 4 ) ( b + 3 ).
Step 2. Multiply all terms by the LCD.

Step 3. Simplify.

Step 4. Solve for b.

This is the solution since b ≠ -3 or 4 and 25 yields a valid check.

Example 4. Solve

Step 1. Find the LCD.
The LCD for this problem is 5 ( x - 2 ).
Step 2. Multiply all terms by the LCD.

Step 3. Simplify

Step 4. Solve for x.

This is not the solution since 2 was excluded at the beginning of the problem.
This problem then has no real solutions .

Where two numbers are compared using division , a ratio is said to exist.
When two ratios are made equal to each other a proportion exists . These
ideas lead to the cross-multiplication property of proportions , which states:

If ( b ≠ 0 and d ≠ 0 ) then ad = bc

Example 5. Using cross-multiplication, solve

Step 1. Apply the property to the problem.

Step 2. Simplify.

Step 3. Solve for x.

This is the solution since x ≠ 2 or -4 and 11 yields a valid check.