# Solving Rational Equations

In section 2–3 we solved equations that had fractions with
whole numbers for the denominators. In

that section we solved the equations by first using the multiplication property
of equality to multiply

**every term in an equation ** by the Least Common Denominator. When we
multiply each term by

the LCD the resulting equation will not have any fractions in it. We can then
solve this equation by the

methods of chapter 2.

**Example**

multiply all 3 terms by 10 | |

subtract 3x from both sides | |

5x +15 = 0 | subtract 15 from both sides |

( divide both sides by 5) | |

x = –3 |

**Solving Rational Equations **

All of the **Rational Equations** in this section will contain terms with
denominators that are contain

polynomials instead of whole numbers.

When we solve the rational equations in this section we
will use the same method as before. We start

by multiplying **every term in an equation** by the Least Common Denominator.
The difference in

this section is that the LCD will not be a whole number as it was in section
2–3.

The LCD is 4(x + 2) | The LCD is 6x(x + 4) |

**How to solve a Rational Equation**

1. Factor each denominator if possible

2. Find the LCD for all the denominators.

3. Multiply every term in the equation by the LCD. Cancel and reduce . This will
leave you with an

equation without fractions .

4. Solve the equation.

5. Check your solution. If the solution makes any of the denominators of the
original rational equation

zero then they cannot be used and there is no solution or

Example 1 |
Example 2 |

Solve for x | Solve for x |

the LCD is 6x so mutiply each term of the equation by 6x |
the LCD is 12x so mutiply each term of the equation by 12x |

(2 does not cause 0 in denominator so) | (−2 does not cause 0 in denominator so) |

x = 2 | x = −2 |

Example 3 |
Example 4 |

Solve for x | Solve for x |

the LCD is 12x so mutiply each term of the equation by 6x |
the LCD is 12x so mutiply each term of the equation by 12x |

5/2does not cause 0 in denominator so | 1 does not cause 0 in denominator so |

x =5/2 | 1 = x |

Example 5 |
Example 6 |

Solve for x | Solve for x |

the LCD is (x − 2) so mutiply each term of the equation by (x − 2) |
the LCD is 2x(x − 3) so mutiply each term of the equation by 2x(x − 3) |

2 does cause 0 in denominator so | 3 does cause 0 in denominator so |

or No Solution | or No Solution |

**Example 7**

Solve for x

the LCD is (x +1)(x −1) so

mutiply each term of

the equation by (x +1)(x −1)

2 does not cause 0 in denominator so

x = 2

**Example 8**

Solve for x

factor each denominator

the LCD is 6(x − 3) so

mutiply each term of

the equation by 6(x − 3)

3 does cause 0 in denominator so

or No Solution

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