Essential Questions Solving Rational Equations and Inequalities How do we solve rational equations and inequalities? Holt McDougal Algebra 2 Holt Algebra 2

A rational inequality is an inequality that contains one or more rational expressions. You can solve rational inequalities algebraically by multiplying each term by the least common denominator (LCD) of all the expressions in the inequality. However, you must consider two cases: the solution and the undefined variable.

Step 1: Find Your Critical Values Make sure your inequality has 0 on the right hand side (you may have to move some terms over). For polynomials, find the zeros (factor or use the techniques from Section 2.5). They are the critical values. For rational functions, set both the numerator and the denominator equal to zero (recall that the numerator equal to zero gives you x-intercepts and the denominator equal to zero gives you the location of vertical asymptotes).

Step 2: The Number Line Plot all critical values on the number line (make sure you put them in the right place!) Your critical values will divide the number line into intervals. Label them using interval notation. Important for rational functions: critical values obtained from asymptotes cannot be solutions (use parentheses on your interval notation for those regardless of the other values).

Step 3: Test Your Intervals Choose a value in each interval and plug that value into the function. If the output value is positive, write a “+” on the interval. If the output value is negative, write a “-” on the interval.

Step 4: Choose Your Solution Intervals Now look back at your original problem. Numbers greater than zero are positive. If the inequality sign is > or >, choose the intervals you labeled as “+”. Numbers less than zero are negative. If the inequality sign is < or <, choose the intervals you labeled as “-”. Write the interval notation for the interval(s) you have chosen.

Solving Rational Inequalities Algebraically Solve the inequality. Note that x ≠ 6. Multiply each term by x - 6. Solve for x. l l Check 0 Check 7 Check 10

Solving Rational Inequalities Algebraically Solve the inequality. Note that x ≠ 3. Multiply each term by x - 3. Solve for x. l l Check 0 Check 3.5 Check 5

Solving Rational Inequalities Algebraically Solve the inequality. Note that x ≠ 8. Multiply each term by x - 8. Solve for x. l l Check 0 Check 9 Check 11

Solving Rational Inequalities Algebraically Solve the inequality. Note that x ≠ 2. Multiply each term by x - 2. Solve for x. l l Check 0 Check 1 Check 3

Solving Rational Inequalities Algebraically Solve the inequality. Note that x ≠ -3. Multiply each term by x + 3. Solve for x. l l Check -4 Check -2 Check 0

Lesson 6.5 Practice C