How do you solve $x^2<=8x$ using a sign chart?

Question

1912 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-12-06T09:50:13

The answer is $x in [0,8]$

Let's rearrange the equation

$x^2<=8x$

$x^2-8x<=0$

$x(x-8)<=0$

Let $f(x)=x(x-8)$

Let's do the sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaa)$ $0$ $color(white)(aaaa)$ $8$ $color(white)(aaaa)$ $+oo$

$color(white)(aaaa)$ $x$ $color(white)(aaaaaaa)$ $-$ $color(white)(aaaa)$ $+$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $x-8$ $color(white)(aaaa)$ $-$ $color(white)(aaaa)$ $-$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $f(x)$ $color(white)(aaaaa)$ $+$ $color(white)(aaaa)$ $-$ $color(white)(aaaa)$ $+$

So, $f(x)<=0$ , when $x in [0,8]$

graph{x(x-8) [-41.1, 41.1, -20.56, 20.56]}

How do you solve x^2<=8xx^2<=8x using a sign chart?