How do solve $2x^3+x^2>6x$ and write the answer as a inequality and interval notation?

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How do solve $2x^3+x^2>6x$ and write the answer as a inequality and interval notation?

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Answer 1 · 2017-02-23T10:54:46

The solutions are $-2 < x < 0$ and $x > 3/2$
Or in interval notation, $x in ]-2,0[uu]3/2,+oo[$

Explanation:

Let's rewrite and factorise the inequality

$2x^3+x^2>6x$

$2x^3+x^2-6x>0$

$x(2x^2+x-6)>0$

$x(2x-3)(x+2)>0$

Let $f(x)=x(2x-3)(x+2)$

We can build the sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaa)$ $-2$ $color(white)(aaaaa)$ $0$ $color(white)(aaaaa)$ $3/2$ $color(white)(aaaaaa)$ $+oo$

$color(white)(aaaa)$ $x+2$ $color(white)(aaaaaa)$ $-$ $color(white)(aaaa)$ $+$ $color(white)(aaaa)$ $+$ $color(white)(aaaaa)$ $+$

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

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

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

Therefore,

$f(x)>0$ when $-2 < x < 0$ and $x > 3/2$

Or in interval notation, $x in ]-2,0[uu]3/2,+oo[$

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How do solve $2x^3+x^2>6x$ and write the answer as a inequality and interval notation?

1 Answer

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How do solve 2x^3+x^2>6x2x^3+x^2>6x and write the answer as a inequality and interval notation?

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How do solve $2x^3+x^2>6x$ and write the answer as a inequality and interval notation?