Chapter 8 - Section 8.4 - Nonlinear Inequalities in One Variable - Exercise Set - Page 510: 37

$(−∞,−5)$ U $(-1,1)$ U$(5,∞)$

$x^4-26x^2+25\geq 0$ $x^2=y$ $(x^2)^2-26(x^2)+25\geq 0$ $y^2-26y+25\geq 0$ $(y-1)(y-25)\geq 0$ $(y-1)(y-25)\geq 0$ $(x^2-1)(x^2-25)\geq 0$ $(x-1)(x+1)(x-5)(x+5) \geq 0$ $x-1 \geq 0$ $x-1+1 \geq 0+1$ $x \geq 1$ $x+1 \geq 0$ $x+1-1 \geq 0-1$ $x \geq -1$ $x-5 \geq 0$ $x-5+5 \geq 0+5$ $x \geq 5$ $x+5 \geq 0$ $x+5-5 \geq 0-5$ $x \geq -5$ We have five sections: $(−∞,−5)$, $(−5,-1)$, $(-1,1)$, $(1,5)$ and $(5,∞)$. We need to test one value for x in each section to determine if the section would be a solution set. Since we have the $\geq$ sign, we exclude the end points and use parentheses instead of brackets. Let $x=−10$,$x=-3$, $x=0$, $x=3$, and $x=10$ $x=-10$ $x^4-26x^2+25\geq 0$ $(-10)^4-26(-10)^2+25\geq 0$ $10000-26*100+25 \geq 0$ $10025-2600 \geq 0$ $7425 \geq 0$ (true) $x=-3$ $x^4-26x^2+25\geq 0$ $(-3)^4-26(-3)^2+25\geq 0$ $81-26*9+25 \geq 0$ $106 -234 \geq 0$ $-128 \geq 0$ (false) $x=0$ $x^4-26x^2+25\geq 0$ $0^4-26*0^2+25\geq 0$ $0-26*0+25\geq 0$ $25 -0 \geq 0$ $25 \geq 0$ (true) $x=3$ $x^4-26x^2+25\geq 0$ $(3)^4-26(3)^2+25\geq 0$ $81-26*9+25 \geq 0$ $106 -234 \geq 0$ $-128 \geq 0$ (false) $x=10$ $x^4-26x^2+25\geq 0$ $(10)^4-26(10)^2+25\geq 0$ $10000-26*100+25 \geq 0$ $10025-2600 \geq 0$ $7425 \geq 0$ (true)