174 CHAPTER 1 Equations and Inequalities The values in Interval B, 11, 52, make the inequality true. The projectile is greater than 80 ft above ground level between 1 and 5 sec after it is launched. S Now Try Exercise 89. Interval A (–∞, 1) Interval B (1, 5) Interval C (5, ∞) 0 3 7 5 1 Test Value 02 – 6(0) + 5 , 0 5 , 0 False ? Test Value 32 – 6(3) + 5 , 0 –4 , 0 True ? Test Value 72 – 6(7) + 5 , 0 12 , 0 False ? Figure 16 Rational Inequalities Inequalities involving one or more rational expressions are rational inequalities. 5 x + 4 Ú 1 and 2x - 1 3x + 4 65 Rational inequalities Solving a Rational Inequality Step 1 Rewrite the inequality, if necessary, so that 0 is on one side and there is a single fraction on the other side. Step 2 Determine the values that will cause either the numerator or the denominator of the rational expression to equal 0. These values determine the intervals on the number line to consider. Step 3 Use a test value from each interval to determine which intervals form the solution set. A value causing a denominator to equal 0 will never be included in the solution set. If the inequality is strict, any value causing the numerator to equal 0 will be excluded. If the inequality is nonstrict, any such value will be included. CAUTION Solving a rational inequality such as 5 x + 4 Ú 1 by multiplying each side by x + 4 requires considering two cases, because the sign of x + 4 depends on the value of x. If x + 4 is negative, then the inequality symbol must be reversed. The procedure described in the box addresses this issue. EXAMPLE 8 Solving a Rational Inequality Solve 5 x + 4 Ú 1. SOLUTION Step 1 5 x + 4 - 1 Ú 0 Subtract 1 so that 0 is on one side. 5 x + 4 - x + 4 x + 4 Ú 0 Use x + 4 as the common denominator. 5 - 1x + 42 x + 4 Ú 0 Write as a single fraction. 1 - x x + 4 Ú 0 Combine like terms in the numerator, being careful with signs. Note the careful use of parentheses.
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