Answer
$(-∞, -5]$ U $[-4, 4]$ U $[5, ∞)$
Work Step by Step
$(x^2-16)(x^2-25) \geq 0$
$x^2-16=0$
$x^2-16+16=0+16$
$x^2=16$
$\sqrt {x^2} = \sqrt {16}$
$x=±4$
$x^2-25=0$
$x^2-25+25=0+25$
$x^2=25$
$\sqrt {x^2} = \sqrt {25}$
$x=±5$
Five regions to test: $(-∞, -5]$, $[-5, -4]$, $[-4, 4]$, $[4, 5]$, $[5, ∞)$
Let $x=-6$, $x=-4.5$, $x=0$, $x=4.5$, $x=6$
$x=-6$
$(x^2-16)(x^2-25) \geq 0$
$((-6)^2-16)((-6)^2-25) \geq 0$
$(36-16)(36-25) \geq 0$
$20*11 \geq 0$
$220 \geq 0$ (true)
$x=-4.5$
$(x^2-16)(x^2-25) \geq 0$
$((-4.5)^2-16)((-4.5)^2-25) \geq 0$
$(20.25-16)(20.25-25) \geq 0$
$4.25*-4.75 \geq 0$
$-20.1875 \geq 0$ (false)
$x=0$
$(x^2-16)(x^2-25) \geq 0$
$(0^2-16)(0^2-25) \geq 0$
$(0-16)(0-25) \geq 0$
$-16*-25 \geq 0$
$400 \geq 0$ (true)
$x=4.5$
$(x^2-16)(x^2-25) \geq 0$
$((4.5)^2-16)((4.5)^2-25) \geq 0$
$(20.25-16)(20.25-25) \geq 0$
$4.25*-4.75 \geq 0$
$-20.1875 \geq 0$ (false)
$x=6$
$(x^2-16)(x^2-25) \geq 0$
$((6)^2-16)((6)^2-25) \geq 0$
$(36-16)(36-25) \geq 0$
$20*11 \geq 0$
$220 \geq 0$ (true)
