How do you solve the inequality $(x^2-2x-24)/(x^2-8x-20)>=0$ ?

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How do you solve the inequality $(x^2-2x-24)/(x^2-8x-20)>=0$ ?

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Answer 1 · 2018-04-01T12:52:49

The solution is $x in (-oo,-4] uu (-2, 6] uu (10, +oo)$

Explanation:

Factorise the inequality

$(x^2-2x-24)/(x^2-8x-20)=((x+4)(x-6))/((x+2)(x-10))>=0$

Let $f(x)=((x+4)(x-6))/((x+2)(x-10))$

Perform a sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaa)$ $-4$ $color(white)(aaaa)$ $-2$ $color(white)(aaaaa)$ $6$ $color(white)(aaaa)$ $10$ $color(white)(aaaa)$ $+oo$

$color(white)(aaaa)$ $x+4$ $color(white)(aaaaa)$ $-$ $color(white)(aa)$ $0$ $color(white)(aaa)$ $+$ $color(white)(aaa)$ $+$ $color(white)(aaa)$ $+$ $color(white)(aaa)$ $+$

$color(white)(aaaa)$ $x+2$ $color(white)(aaaaa)$ $-$ $color(white)(aa)$ #color(white)(aaaa)-# $color(white)(a)$ $||$ $color(white)(a)$ $+$ $color(white)(aaa)$ $+$ $color(white)(aaa)$ $+$

$color(white)(aaaa)$ $x-6$ $color(white)(aaaaa)$ $-$ $color(white)(aa)$ #color(white)(aaaa)-# $color(white)(a)$ #color(white)(aa)-# $color(white)(a)$ $0$ $color(white)(a)$ $+$ $color(white)(aaa)$ $+$

$color(white)(aaaa)$ $x-10$ $color(white)(aaaa)$ $-$ $color(white)(aa)$ #color(white)(aaaa)-# $color(white)(a)$ #color(white)(aa)-# $color(white)(a)$ #color(white)(aa)-# $color(white)(a)$ $||$ $color(white)(a)$ $+$

$color(white)(aaaa)$ $f(x)$ $color(white)(aaaaaa)$ $+$ $color(white)(aa)$ $0$ $color(white)(aaa)$ $-$ $color(white)(a)$ $||$ $color(white)(a)$ $+$ $color(white)(a)$ $0$ $color(white)(a)$ $-$ $color(white)(a)$ $||$ $color(white)(a)$ $+$

Therefore,

$f(x)>=0$ when $x in (-oo,-4] uu (-2, 6] uu (10, +oo)$

graph{(x^2-2x-24)/(x^2-8x-20) [-25.65, 25.66, -12.83, 12.84]}

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How do you solve the inequality $(x^2-2x-24)/(x^2-8x-20)>=0$ ?

1 Answer

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How do you solve the inequality (x^2-2x-24)/(x^2-8x-20)>=0(x^2-2x-24)/(x^2-8x-20)>=0?

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Explanation:

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How do you solve the inequality $(x^2-2x-24)/(x^2-8x-20)>=0$ ?