How do you solve $7-x^2<=0$ ?

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Answer 1 · 2017-02-15T09:02:48

The answer is $x in ]-oo,-sqrt7]uu[sqrt7,+oo[$

Let's factorise the inequality

$7-x^2<=0$

$(sqrt7+x)(sqrt7-x)<=0$

Let $f(x)=(sqrt7+x)(sqrt7-x)$

Now, we build the sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaaaa)$ $-oo$ $color(white)(aaaa)$ $-sqrt7$ $color(white)(aaaa)$ $sqrt7$ $color(white)(aaaaa)$ $+oo$

$color(white)(aaaa)$ $sqrt7+x$ $color(white)(aaaaaa)$ $-$ $color(white)(aaaaa)$ $+$ $color(white)(aaaaa)$ $+$

$color(white)(aaaa)$ $sqrt7-x$ $color(white)(aaaaaa)$ $+$ $color(white)(aaaaa)$ $+$ $color(white)(aaaaa)$ $-$

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

Therefore,

$f(x)<=0$ when $x in ]-oo,-sqrt7]uu[sqrt7,+oo[$

How do you solve 7-x^2<=07-x^2<=0?