Procedure How to Solve a Rational Inequality Step 1: Write the inequality so that there is a 0 on one side and only one rational expression on the other side. If the inequality symbol is or , we are looking for a negative quantity in the interval on the number line. If the inequality symbol is or , we are looking for a positive quantity in the interval. Step 2: Find the numbers that make the numerator equal 0 and any numbers that make the denominator equal 0. Step 3: Put the numbers found in Step 2 on a number line. These values break up the number line into intervals. Step 4: Choose a test number in each interval to determine whether the rational inequality is positive or negative in each interval. Indicate this on the number line. Step 5: If the inequality is in the form p q 0 or p q 0, then the solution set contains the numbers in the interval where p q is negative. If the inequality is in the form p q 0 or p q 0, then the solution set contains the numbers in the interval where p q is positive. Step 6: Determine whether the endpoints of the intervals are included in or excluded from the solution set. Do not include any values that make the denominator equal 0. Summarize this procedure in your notes, in your own words. EXAMPLE 5 Solve 5 x 3 0. Graph the solution set, and write the solution in interval notation. Solution Step 1: The inequality is in the correct form—zero on one side and only one rational expression on the other side. Since the inequality symbol is 0, the solution set will contain the interval(s) where 5 x 3 is positive. Step 2: Find the numbers that make the numerator equal 0 and any numbers that make the denominator equal 0. Numerator: 5 Denominator: x 3 The numerator is a constant, 5, so it cannot equal 0. Step 3: Put 3 on a number line to break it up into intervals. 3 5 x 3 Set x 3 0 and solve for x. x 3 0 Step 4: Choose a test number in each interval to determine whether x 3 5 x 3 is positive or negative in each interval. Interval x 3 x 3 Test number x 4 x 0 Evaluate 5 x 3 5 4 3 5 1 5 5 0 3 5 3 Sign Negative Positive 686 CHAPTER 10 Quadratic Equations and Functions www.mhhe.com/messersmith
messersmith_power_intermediate_algebra_1e_ch4_7_10
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