# 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 1. Find the LCD.
The LCD for this problem is 8a. Also a ≠ 0 .
Step 2. Multiply all terms by the LCD .**

**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.**

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