How do you solve $p/(p-16)+2/(p-6)<=0$ using a sign chart?

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How do you solve $p/(p-16)+2/(p-6)<=0$ using a sign chart?

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Answer 1 · 2017-05-07T14:05:14

The solution is $p in [-4,6) uu [8, 16)$

Explanation:

Let 's simplify the $LHS$ of the inequality

$p/(p-16)+2/(p-6)<=0$

$(p(p-6)+2(p-16))/((p-16)(p-6))<=0$

$(p^2-6p+2p-32)/((p-16)(p-6))<=0$

$(p^2-4p-32)/((p-16)(p-6))<=0$

$((p+4)(p-8))/((p-16)(p-6))<=0$

Let $f(p)=((p+4)(p-8))/((p-16)(p-6))$

We can build the sign chart

$color(white)(aaaa)$ $p$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaa)$ $-4$ $color(white)(aaaaaaa)$ $6$ $color(white)(aaaaaa)$ $8$ $color(white)(aaaaaaa)$ $16$ $color(white)(aaaa)$ $+oo$

$color(white)(aaaa)$ $p+4$ $color(white)(aaaa)$ $-$ $color(white)(aaaaaa)$ $+$ $color(white)(aaa)$ $||$ $color(white)(aaa)$ $+$ $color(white)(aaa)$ $+$ $color(white)(aaa)$ $||$ $color(white)(aa)$ $+$

$color(white)(aaaa)$ $p-6$ $color(white)(aaaa)$ $-$ $color(white)(aaaaaa)$ $-$ $color(white)(aaa)$ $||$ $color(white)(aaa)$ $+$ $color(white)(aaa)$ $+$ $color(white)(aaa)$ $||$ $color(white)(aa)$ $+$

$color(white)(aaaa)$ $p-8$ $color(white)(aaaa)$ $-$ $color(white)(aaaaaa)$ $-$ $color(white)(aaa)$ $||$ $color(white)(aaa)$ $-$ $color(white)(aaa)$ $+$ $color(white)(aaa)$ $||$ $color(white)(aa)$ $+$

$color(white)(aaaa)$ $p-16$ $color(white)(aaa)$ $-$ $color(white)(aaaaaa)$ $-$ $color(white)(aaa)$ $||$ $color(white)(aaa)$ $-$ $color(white)(aaa)$ $-$ $color(white)(aaa)$ $||$ $color(white)(aa)$ $+$

$color(white)(aaaa)$ $f(p)$ $color(white)(aaaaa)$ $+$ $color(white)(aaaaaa)$ $-$ $color(white)(aaa)$ $||$ $color(white)(aaa)$ $+$ $color(white)(aaa)$ $-$ $color(white)(aaa)$ $||$ $color(white)(aa)$ $+$

Therefore,

$f(p)<=0$ when $p in [-4,6) uu [8, 16)$

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How do you solve $p/(p-16)+2/(p-6)<=0$ using a sign chart?

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Explanation:

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How do you solve p/(p-16)+2/(p-6)<=0p/(p-16)+2/(p-6)<=0 using a sign chart?

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Explanation:

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How do you solve $p/(p-16)+2/(p-6)<=0$ using a sign chart?