How do you solve $x^2<=9$ using a sign chart?

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Answer 1 · 2017-02-08T07:33:57

The answer is $x in [-3,3]$

Let's rewrite the equation

$x^2<=9$

$x^2-9<=0$

$(x+3)(x-3)<=0$

Let $f(x)=(x+3)(x-3)$

Now, we can build the sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaa)$ $-3$ $color(white)(aaaa)$ $3$ $color(white)(aaaaa)$ $+oo$

$color(white)(aaaa)$ $x+3$ $color(white)(aaaaa)$ $-$ $color(white)(aaaa)$ $+$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $x-3$ $color(white)(aaaaa)$ $-$ $color(white)(aaaa)$ $-$ $color(white)(aaaa)$ $+$

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

Therefore,

$f(x)<=0$ when $x in [-3,3]$

How do you solve x^2<=9x^2<=9 using a sign chart?