How do you solve $\frac{2 x}{x - 5} \geq 3$ $(2x)/(x-5)>=3$ ?

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How do you solve $\frac{2 x}{x - 5} \geq 3$ $(2x)/(x-5)>=3$ ?

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Answer 1 · 2018-08-04T14:02:40

The solution is $x \in (5, 15]$ $x in (5,15]$

Explanation:

You cannot do crossing over in inequalities

$\frac{2 x}{x - 5} \geq 3$ $(2x)/(x-5)>=3$

$\Leftrightarrow$ $<=>$ , $\frac{2 x}{x - 5} - 3 \geq 0$ $(2x)/(x-5)-3>=0$

$\Leftrightarrow$ $<=>$ , $\frac{2 x - 3 (x - 5)}{x - 5} \geq 0$ $(2x-3(x-5))/(x-5)>=0$

$\Leftrightarrow$ $<=>$ , $\frac{2 x - 3 x + 15}{x - 5} \geq 0$ $(2x-3x+15)/(x-5)>=0$

$\Leftrightarrow$ $<=>$ , $\frac{- x + 15}{x - 5} \geq 0$ $(-x+15)/(x-5)>=0$

Let $f (x) = \frac{- x + 15}{x - 5}$ $f(x)=(-x+15)/(x-5)$

Build a sign chart

$a a a a$ $color(white)(aaaa)$ $x$ $x$ $a a a a$ $color(white)(aaaa)$ $- \infty$ $-oo$ $a a a a a a a$ $color(white)(aaaaaaa)$ $5$ $5$ $a a a a a a a$ $color(white)(aaaaaaa)$ $15$ $15$ $a a a a$ $color(white)(aaaa)$ $+ \infty$ $+oo$

$a a a a$ $color(white)(aaaa)$ $x - 5$ $x-5$ $a a a a a$ $color(white)(aaaaa)$ $-$ $-$ $a a a a$ $color(white)(aaaa)$ $∣ ∣$ $||$ $a a$ $color(white)(aa)$ $+$ $+$ $a a a a a a$ $color(white)(aaaaaa)$ $+$ $+$

$a a a a$ $color(white)(aaaa)$ $- x + 15$ $-x+15$ $a a a$ $color(white)(aaa)$ $+$ $+$ $a a a a a$ $color(white)(aaaaa)$ #color(white)(aa)+# $a a a$ $color(white)(aaa)$ $0$ $0$ $a a$ $color(white)(aa)$ $-$ $-$

$a a a a$ $color(white)(aaaa)$ $f (x)$ $f(x)$ $a a a a a a a$ $color(white)(aaaaaaa)$ $-$ $-$ $a a a$ $color(white)(aaa)$ $∣ ∣$ $||$ $a a a$ $color(white)(aaa)$ $+$ $+$ $a a a$ $color(white)(aaa)$ $0$ $0$ $a a$ $color(white)(aa)$ $-$ $-$

Therefore,

$f (x) \geq 0$ $f(x)>=0$ when $x \in (5, 15]$ $x in (5,15]$

graph{(2x)/(x-5)-3 [-25.56, 32.17, -14.65, 14.22]}

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How do you solve $\frac{2 x}{x - 5} \geq 3$ $(2x)/(x-5)>=3$ ?

1 Answer

Explanation:

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