Rational Expressions and Equations Chapter 6 Rational Expressions and Equations

Solving Rational Equations 6.6 Solving Rational Equations

Rational Equations A rational, or fractional, equation is an equation containing one or more rational expressions, often with the variable in the denominator. Here are some examples:

To Solve a Rational Equation 1. List any restrictions that exist. Numbers that make a denominator equal 0 cannot possibly be solutions. 2. Clear the equation of fractions by multiplying both sides by the LCM of the denominators. 3. Solve the resulting equation using the addition principle, the multiplication principle, and the principle of zero products, as needed. 4. Check the possible solution(s) in the original equation.

Solve: Solution Because no variable appears in the denominator, no restrictions exist. The LCM of 5, 2, and 4 is 20, so we multiply both sides by 20: Using the multiplication principle to multiply both sides by the LCM. Parentheses are important! Using the distributive law. Be sure to multiply EACH term by the LCM. Simplifying and solving for x. If fractions remain, we have either made a mistake or have not used the LCM of the denominators.

Checking Answers Since a variable expression could represent 0, multiplying both sides of an equation by a variable expression does not always produce an equivalent equation. Thus checking each solution in the original equation is essential.

Solve: Solution Note that x cannot equal 0. The LCM is 15x.

Check: The solution is 5.

Solve: Solution Note that x cannot equal 0. The LCM is x. x = 3 or x = 4

Check: For x = 3 For x = 4 Both of these check, so there are two solutions, 3 and 4.

Solve: Solution Note that x cannot equal 3 or 3. We multiply both sides of the equation by the LCM.

Solve: Solution Note that x cannot equal 1 or 1. Multiply both sides of the equation by the LCD. Because of the restriction above, 1 must be rejected as a solution. This equation has no solution.