How do you solve $\frac{2 x}{x - 2} \leq 3$ $(2x)/(x-2)<=3$ using a sign chart?

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How do you solve $\frac{2 x}{x - 2} \leq 3$ $(2x)/(x-2)<=3$ using a sign chart?

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Answer 1 · 2017-06-14T12:45:55

The solution is $x \in (- \infty, 2) \cup [6, + \infty)$ $x in (-oo,2)uu [6,+oo)$

Explanation:

We cannot do crossing over

Let's rearrange the inequality

$\frac{2 x}{x - 2} \leq 3$ $(2x)/(x-2)<=3$

$\frac{2 x}{x - 2} - 3 \leq 0$ $(2x)/(x-2)-3<=0$

$\frac{2 x - 3 (x - 2)}{x - 2} \leq 0$ $(2x-3(x-2))/(x-2)<=0$

$\frac{2 x - 3 x + 6}{x - 2} \leq 0$ $(2x-3x+6)/(x-2)<=0$

$\frac{6 - x}{x - 2} \leq 0$ $(6-x)/(x-2)<=0$

Let $f (x) = \frac{6 - x}{x - 2}$ $f(x)=(6-x)/(x-2)$

Let's build the sign chart

$a a a a$ $color(white)(aaaa)$ $x$ $x$ $a a a a a$ $color(white)(aaaaa)$ $- \infty$ $-oo$ $a a a a a a$ $color(white)(aaaaaa)$ $2$ $2$ $a a a a a a a$ $color(white)(aaaaaaa)$ $6$ $6$ $a a a a a a$ $color(white)(aaaaaa)$ $+ \infty$ $+oo$

$a a a a$ $color(white)(aaaa)$ $x - 2$ $x-2$ $a a a a a$ $color(white)(aaaaa)$ $-$ $-$ $a a a a$ $color(white)(aaaa)$ $∣ ∣$ $||$ $a a a$ $color(white)(aaa)$ $+$ $+$ $a$ $color(white)(a)$ $0$ $0$ $a a a a$ $color(white)(aaaa)$ $+$ $+$

$a a a a$ $color(white)(aaaa)$ $6 - x$ $6-x$ $a a a a a$ $color(white)(aaaaa)$ $+$ $+$ $a a a a$ $color(white)(aaaa)$ $∣ ∣$ $||$ $a a a$ $color(white)(aaa)$ $+$ $+$ $a$ $color(white)(a)$ $0$ $0$ $a a a a$ $color(white)(aaaa)$ $-$ $-$

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

Therefore,

$f (x) \leq 0$ $f(x)<=0$ when $x \in (- \infty, 2) \cup [6, + \infty)$ $x in (-oo,2)uu [6,+oo)$

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How do you solve $\frac{2 x}{x - 2} \leq 3$ $(2x)/(x-2)<=3$ using a sign chart?

1 Answer

Explanation:

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