Answer
$(3,15]$
Work Step by Step
$4x/(x-3) \geq 5$
$4x/(x-3)-5 \geq 5-5$
$4x/(x-3)-5 \geq 0$
$4x/(x-3)-5*(x-3)/(x-3) \geq 0$
$4x/(x-3) -5x+15/(x-3) \geq 0$
$(4x-5x+15)/(x-3) \geq 0$
$(-x+15)/(x-3) \geq 0$
The denominator is zero when $x=3$, and the numerator is zero when $x=15$.
We have three sections: $(−∞,3)$, $(3,15)$, and $(15,∞)$. 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 include the end points and use brackets instead of parentheses.
Let $x=0$, $x=5$, and $x=20$
$x=0$
$4x/(x-3) \geq 5$
$4*0/(0-3) \geq 5$
$0/-3 \geq 5$
$0 \geq 5$ (false)
$x=5$
$4x/(x-3) \geq 5$
$4*5/(5-3) \geq 5$
$20/2 \geq 5$
$10 \geq 5$ (true)
$x=20$
$4x/(x-3) \geq 5$
$4*20/(20-3) \geq 5$
$80/17 \geq 5$ (false)
