How do solve $\frac{5 x - 8}{x - 5} \geq 2$ $(5x-8)/(x-5)>=2$ and write the answer as a inequality and interval notation?

Question

How do solve $\frac{5 x - 8}{x - 5} \geq 2$ $(5x-8)/(x-5)>=2$ and write the answer as a inequality and interval notation?

1 Answer

Impact of this question

2170 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-06-16T07:03:02

The solution is $x \in (- \infty, - \frac{2}{3}] \cup (5, + \infty)$ $x in (-oo,-2/3] uu(5, +oo)$

Explanation:

The inequality is

$\frac{5 x - 8}{x - 5} \geq 2$ $(5x-8)/(x-5)>=2$

We cannot do crossing over, so

$\frac{5 x - 8}{x - 5} - 2 \geq 0$ $(5x-8)/(x-5)-2>=0$

Putting on the same denominator

$\frac{5 x - 8 - 2 (x - 5)}{x - 5} \geq 0$ $(5x-8-2(x-5))/(x-5)>=0$

$\frac{5 x - 8 - 2 x + 10}{x - 5} \geq 0$ $(5x-8-2x+10)/(x-5)>=0$

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

Let $f (x) = \frac{3 x + 2}{x - 5}$ $f(x)=(3x+2)/(x-5)$

Build the sign chart

$a a a a$ $color(white)(aaaa)$ $x$ $x$ $a a a a$ $color(white)(aaaa)$ $- \infty$ $-oo$ $a a a a a$ $color(white)(aaaaa)$ $- \frac{2}{3}$ $-2/3$ $a a a a a a a$ $color(white)(aaaaaaa)$ $5$ $5$ $a a a a$ $color(white)(aaaa)$ $+ \infty$ $+oo$

$a a a a$ $color(white)(aaaa)$ $3 x + 2$ $3x+2$ $a a a a a$ $color(white)(aaaaa)$ $-$ $-$ $a a a$ $color(white)(aaa)$ $0$ $0$ $a a a$ $color(white)(aaa)$ $+$ $+$ $a a a a a$ $color(white)(aaaaa)$ $+$ $+$

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

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

Therefore,

$f (x) \geq 0$ $f(x)>=0$ when $x \in (- \infty, - \frac{2}{3}] \cup (5, + \infty)$ $x in (-oo,-2/3] uu(5, +oo)$ in interval notation

and $x \leq - \frac{2}{3}$ $x<=-2/3$ and $x > 5$ $x>5$ as an inequality

graph{(5x-8)/(x-5)-2 [-29.8, 35.14, -12.44, 20.05]}

Science

Math

Humanities

... and beyond

How do solve $\frac{5 x - 8}{x - 5} \geq 2$ $(5x-8)/(x-5)>=2$ and write the answer as a inequality and interval notation?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do solve 5x−8x−5≥2(5x-8)/(x-5)>=2 and write the answer as a inequality and interval notation?

1 Answer

Explanation:

Related questions

Impact of this question

How do solve $\frac{5 x - 8}{x - 5} \geq 2$ $(5x-8)/(x-5)>=2$ and write the answer as a inequality and interval notation?