College Algebra (6th Edition)

Published by Pearson
ISBN 10: 0-32178-228-3
ISBN 13: 978-0-32178-228-1

Chapter 3 - Polynomial and Rational Functions - Exercise Set 3.6 - Page 420: 32

Answer

Solution set: $[0,4]\cup[6,\infty)$
1510355817

Work Step by Step

Follow the "Procedure for Solving Polynomial lnequalities",\ p.412: 1. Express the inequality in the form $f(x)<0, f(x)>0, f(x)\leq 0$, or $f(x)\geq 0,$ where $f$ is a polynomial function. $x(4-x)(x-6) \leq 0$ $f(x)=x(4-x)(x-6)$ 2. Solve the equation $f(x)=0$. The real solutions are the boundary points. $x(4-x)(x-6)=0$ $x=0$ or $x=4$ or $x=6$ 3. Locate these boundary points on a number line, thereby dividing the number line into intervals. 4. Test each interval's sign of $f(x)$ with a test value, $a$, from that interval, $\begin{array}{llll} Interval & a & f(a),signs & f(a) \leq 0 ? \\ & & a(4-a)(a-6) & \\ (-\infty,0) & -1 & (-)(+)(-) & F\\ (0,4) & 1 & (+)(+)(-) & T\\ (4,6) & 5 & (+)(-)(-) & F\\ (6,\infty) & 10 & (+)(-)(+) & T \end{array}$ 5. Write the solution set, selecting the interval or intervals that satisfy the given inequality. If the inequality involves $\leq$ or $\geq$, include the boundary points. Solution set: $[0,4]\cup[6,\infty)$
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