Answer
$[-5, -2] U [2,5]$
Work Step by Step
$(x^2-4)(y^2-25) <=0$
$x^2-4<=0$
$x^2-4+4<=0+4$
$x^2<=4$
$\sqrt {x^2} <=\sqrt 4$
$x = ±2$
$y^2-25<=0$
$y^2-25+25<=0+25$
$y^2<=25$
$\sqrt {y^2} <= \sqrt {25}$
$y = ±5$
Five regions to test: $(-∞, -5]$, $[-5, -2]$, $[-2, 2]$, $[2, 5]$, $[5, ∞)$
Let $x=-10$, $x=-3$, $x=0$, $x=3$, $x=10$
$x=-10$
$(x^2-4)(y^2-25) <=0$
$((-10)^2-4)((-10) ^2-25) <=0$
$(100-4)(100-25)<=0$
$96*75 <=0$
$7200 <=0$ (false)
$x=-3$
$(x^2-4)(y^2-25) <=0$
$((-3)^2-4)((-3)^2-25) <=0$
$(9-4)(9-25)<=0$
$5*-16 <=0$
$-80 <=0$ (true)
$x=0$
$(x^2-4)(y^2-25) <=0$
$(0^2-4)(0^2-25) <=0$
$(0-4)(0-25) <=0$
$-4*-25 <=0$
$100 <=0$ (false)
$x=3$
$(x^2-4)(y^2-25) <=0$
$((3)^2-4)((3)^2-25) <=0$
$(9-4)(9-25)<=0$
$5*-16 <=0$
$-80 <=0$ (true)
$x=10$
$(x^2-4)(y^2-25) <=0$
$((10)^2-4)((10)^2-25) <=0$
$(100-4)(100-25)<=0$
$96*75 <=0$
$7200 <=0$ (false)
