Answer
$(-∞, -3)$ U $[2, 3)$
Work Step by Step
$(7x-14) / (x^2-9) \le 0$
$x^2-9 =0$
$x^2-9+9=0+9$
$x^2=9$
$\sqrt {x^2} =\sqrt 9$
$x = ±3$
The denominator is zero when $x=-3$ and $x=3$.
$(7x-14) / (x^2-9) = 0$
$(7x-14)* (x^2-9) / (x^2-9) = 0*(x^2-9)$
$7x-14 =0$
$7x-14+14=0+14$
$7x=14$
$7x/7 =14/7$
$x=2$
Four regions to test: $(-∞, -3)$, $(-3, 2]$, $[2, 3)$, $(3, ∞)$
Let $x=-4$, $x=0$, $x=2.5$, $x=4$
$x=-4$
$(7x-14) / (x^2-9) \le 0$
$(7*-4-14) / ((-4)^2-9) \le 0$
$(-28-14)/(16-9) \le 0$
$-42/ 7 \le 0$
$-6 \le 0$ (true)
$x=0$
$(7x-14) / (x^2-9) \le 0$
$(7*0-14) / (0^2-9) \le 0$
$(0-14) /(0-9) \le 0$
$-14/ -9 \le 0$
$14/9 \le 0$ (false)
$x=2.5$
$(7x-14) / (x^2-9) \le 0$
$(7*2.5-14) / (2.5^2-9) \le 0$
$(17.5-14) / (6.25-9) \le 0$
$3.5/-2.75 \le 0$
$14/4 / -11/4 \le 0$
$14/-11 \le 0$
$-14/11 \le 0$ (true)
$x=4$
$(7x-14) / (x^2-9) \le 0$
$(7*4-14) / (4^2-9) \le 0$
$(28-14) /(16-9) \le 0$
$14/ 7 \le 0$
$2 \le 0$ (false)
$(-∞, -3)$ U $[2, 3)$
