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: 18


Solution set: $(-\infty,\infty)$

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. $4x^{2}+1 \geq 4x$ $4x^{2}-4x+1 \geq 0$ $f(x)=4x^{2}-4x+1 \quad $...factor the trinomial... find factors of $4(1)=4$ that add to $-4:$ ($-2$ and $-2$ ) $4x^{2}-4x+1=4x^{2}-2x-2x+1= \quad$ ... in pairs ... $=2x(2x-1)-(2x-1)=(2x-1)(2x-1)$ (we also could/should have recognized this square of a difference) $f(x)=(2x-1)^{2} \geq 0 $ ... There is no need to follow the steps any further, because the above inequality "asks": "When is a square of a real number greater or equal to zero?" The answer: "Always." That is, for all x $\in \mathbb{R}$ Solution set: $(-\infty,\infty)$
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