Question

How do you solve `\frac { x + 4} { x - 2} < x ^ { 2}`?

Answer 1

Please see the explanation.

Explanation:

Given: `(x+4)/(x-2) < x^2`

Restrict the domain to prevent division by 0:

`(x+4)/(x-2) < x^2; x !=2`

Subtract `x^2` from both sides:

`(x+4)/(x-2)-x^2 < x^2-x^2;x !=2`

Combine like terms:

`(x+4)/(x-2)-x^2 < 0;x !=2`

Make a common denominator:

`(x+4)/(x-2)-(x^2(x-2))/(x-2) < 0;x !=2`

Distribute the `-x^2` factor:

`(x+4)/(x-2)+(-x^3+2x^2)/(x-2) < 0;x !=2`

Combine the two fractions:

`(-x^3+2x^2+x+4)/(x-2) < 0;x !=2`

Multiply both sides by -1:

`(x^3-2x^2-x-4)/(x-2) > 0;x !=2`

I used WolframAlpha to solve the above. The cubic in the numerator has a zero at `x~~ 2.84547`

This gives the interval `(2.84547, oo)`

The expression becomes less than `oo` at `x < 2` but never becomes negative as `x to -oo`; this gives the interval `(-oo,2)`

The answer is `(-oo,2)` and `(2.84547, oo)`

Here is a graph of `y = (x^3-2x^2-x-4)/(x-2)`

graph{(x^3-2x^2-x-4)/(x-2) [-10, 10, -5, 5]}