How do you solve x^2+16x+24>6xx2+16x+24>6x?

2 Answers
Sonnhard
May 25, 2018

x<-6x<−6or x>-4x≻4

Explanation:

simplifying to x^2+10x+24>0x2+10x+24>0

solving x^2+10x+24=0x2+10x+24=0 we get
x_1=-4x1=−4
x_2=-6x2=−6
so our inequality is equivalent to
(x+4)(x+6)>0(x+4)(x+6)>0
this gives us
x>-4x≻4 or x<-6x<−6

George C.
Aug 9, 2018

x in (-oo, -6) uu (-4, oo)x∈(−∞,−6)∪(−4,∞)

Explanation:

Given:

x^2+16x+24 > 6xx2+16x+24>6x

Subtract 6x6x from both sides to get:

x^2+10x+24 > 0x2+10x+24>0

We can make the left hand side into a perfect square trinomial by adding 11, so let us add 11 to both sides to get:

(x+5)^2 = x^2+10x+25 > 1(x+5)2=x2+10x+25>1

Note that this would give equality when (x+5)^2 = 1(x+5)2=1, i.e. when x+5 = +-1x+5=±1, i.e. when x=-6x=−6 or x=-4x=−4.

Hence the inequality is achieved when:

x < -6" "x<−6 or " "x > -4 x>−4

In interval notation, when:

x in (-oo, -6) uu (-4, oo)x∈(−∞,−6)∪(−4,∞)

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