Stitz-Zeager_College_Algebra_e-book

# In doing so the amount of the element itself

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Unformatted text preview: e involved. We leave them to the reader. Since exponential functions are continuous on their domains, the Intermediate Value Theorem 3.1 applies. As with the algebraic functions in Section 5.3, this allows us to solve inequalities using sign diagrams as demonstrated below. Example 6.3.2. Solve the following inequalities. Check your answer graphically using a calculator. 1. 2x 2 −3x − 16 ≥ 0 2. ex ≤3 ex − 4 3. xe2x < 4x Solution. 2 1. Since we already have 0 on one side of the inequality, we set r(x) = 2x −3x − 16. The domain of r is all real numbers, so in order to construct our sign diagram, we seed to ﬁnd the zeros of 2 2 2 r. Setting r(x) = 0 gives 2x −3x − 16 = 0 or 2x −3x = 16. Since 16 = 24 we have 2x −3x = 24 , so by the one-to-one property of exponential functions, x2 − 3x = 4. Solving x2 − 3x − 4 = 0 gives x = 4 and x = −1. From the sign diagram, we see r(x) ≥ 0 on (−∞, −1] ∪ [4, ∞), which 2 corresponds to where the graph of y = r(x) = 2x −3x − 16, is on or above the x-...
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