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Rational Equations Help

By — McGraw-Hill Professional
Updated on Sep 27, 2011

Introduction to Rational Equations

Some rational equations (an equation with one or more fractions as terms) become quadratic equations once each term has been multiplied by the least common denominator. Remember, you must be sure that any solutions do not lead to a zero in any denominator in the original equation.

There are two main approaches to clearing the denominator(s) in a rational equation. If the equation is in the form of “fraction = fraction,” cross multiply. If the equation is not in this form, find the least common denominator (LCD). Finding the least common denominator often means you need to factor each denominator completely. We learned in Chapter 7 to multiply both sides of the LCD, then to distribute the LCD. In this chapter we will simply multiply each term on each side of the equation by the LCD. If the new equation is a quadratic equation, collect all terms on one side of the equal sign and leave a zero on the other. In the examples and practice problems below, the solutions that lead to a zero in a denominator will be stated.

Examples

Rational Equations This is in the form “fraction = fraction” so we will cross multiply.

Rational Equations

Rational Equations

First factor each denominator, second find the LCD, and third multiply all three terms by the LCD.

Rational Equations

Rational Equations

We cannot let x be 4 because x = 4 leads to a zero in the denominator of Rational Equations . The only solution is x = 2.

Rational Equations Practice Problems

Practice

Because all of these problems factor, factoring is used in the solutions. If factoring takes too long on some of these, you may use the quadratic formula.

Rational Equations

Rational Equations

Solutions

Rational Equations

Rational Equations

Rational Equations

Rational Equations

Rational Equations

Using the fact that 1 – x = – (1 – x ) allows us to write the second denominator the same as the first.

Rational Equations

  1. Rational Equations

    Denominator factored: Rational Equations LCD = x ( x + 1)

Rational Equations

Rational Equations

Rational Equations

Rational Equations

Rational Equations

Practice problems for this concept can be found at: Algebra Quadratic Equations Practice Test.

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