How do you solve the polynomial inequality and state the answer in interval notation given #(x^3+2x^2)/2<x+2#?

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Answer 1 · 2017-02-16T05:20:36

#(-oo, -2) uuu (-sqrt(2), sqrt(2))#

Multiply both sides by #2#.

#x^3 + 2x^2 < 2x + 4#

#x^3 + 2x^2 - 2x - 4 < 0#

Factor by extracting a common factor twice.

#x^2(x + 2) - 2(x + 2) < 0#

#(x^2 - 2)(x + 2) < 0#

If you rewrite as an equation, you get zeroes of #x = +- sqrt(2) and -2#.

Now, select test points.

Test point 1: x = -3

#((-3)^3 + 2(-3)^2)/2 <^? -3 + 2#

#(-27 + 18)/2 < -1#

This is absolutely true.

Repeat this process in the following intervals:

#(-2, -sqrt(2)); (-sqrt(2), sqrt(2)); (sqrt(2), oo)#. You should get as a final result that

#x < -2# and #-sqrt(2) < x < sqrt(2)# both satisfy the inequality.

Hopefully this helps!