How do you solve $x^{3} < x^{2}$ $x^3<x^2$ using a sign chart?

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How do you solve $x^{3} < x^{2}$ $x^3<x^2$ using a sign chart?

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Answer 1 · 2016-10-22T15:58:25

$x \in) - \infty, 0 (\cup) 0, 1 ($ $x ∈)-oo,0(∪)0,1 ($

Explanation:

Rewrite the equation as $x^{3} - x^{2} < 0$ $x^3-x^2<0$

on factorization, $x^{2} (x - 1) < 0$ $x^2(x-1)<0$

the points we have to look at are $x = 0$ $x=0$ and $x = 1$ $x=1$

here is the sign chart

$x$ $x$ $color(white(aaaaaa))$ $- \infty$ $-oo$ $a a a a$ $color(white)(aaaa)$ $0$ $0$ $a a a a a a a a a$ $color(white)(aaaaaaaaa)$ $1$ $1$ $a a a a a a$ $color(white)(aaaaaa)$ $+ \infty$ $+oo$
$x$ $x$ $a a a a a$ $color(white)(aaaaa)$ $-$ $-$ $a a$ $color(white)(aa)$ $0$ $0$ $a a a a$ $color(white)(aaaa)$ $+$ $+$ $a a a a a a a$ $color(white)(aaaaaaa)$ $+$ $+$
$x^{2}$ $x^2$ $a a a a$ $color(white)(aaaa)$ $+$ $+$ $a a$ $color(white)(aa)$ $0$ $0$ $a a a a$ $color(white)(aaaa)$ $+$ $+$ $a a a a a a a$ $color(white)(aaaaaaa)$ $+$ $+$
$x - 1$ $x-1$ $a a$ $color(white)(aa)$ $-$ $-$ $a a$ $color(white)(aa)$ $a a a a a$ $color(white)(aaaaa)$ $-$ $-$ $a a a$ $color(white)(aaa)$ $0$ $0$ $a a$ $color(white)(aa)$ $a a$ $color(white)(aa)$ $+$ $+$ $a a a$ $color(white)(aaa)$
$x^{2} (x - 1)$ $x^2(x-1)$ $-$ $-$ $a a a a a$ $color(white)(aaaaa)$ $-$ $-$ $a a a a a a a a a$ $color(white)(aaaaaaaaa)$ $+$ $+$

So the answer is $x \in) - \infty, 0 (\cup) 0, 1 ($ $x ∈)-oo,0(∪)0,1 ($ for $x < 0$ $x<0$

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