Question

How do you solve the inequality `x^2+2x>=24`?

Answer 1

By trial and error. Your answer is with the exception of -6<x<4, all x values provide solution.

Explanation:

x must be a real number and let me try +3 provide the solution. There are two numbers provide the solution (one is positive and the other one is negative):

`3^2+2*3=15` therefore it does not provide solution.

Now let me try +4

`4^2+2*4=24` Yes. This is one answer. Therefore x must be greater than or equal to +4.

On the negative side, let me try -3 first:

`(-3)^2-2*3=2` does not satisfy the requested condition.

Try -4 now:

`(-4)^2-2*4=8` does not satisfy the requested condition.

Try -5 now:

`(-5)^2-2*5=15` does not satisfy the requested condition.

Try -6 now:

`(-6)^2-2*6=24` Yes it satisfies.

Any number less than -6 (including -6) will satisfy the given.

For instance (-8):

`(-8)^2-2*8=48` which is greater than 24.

Now your solution is x must be greater than or equal to 4 or x must be less than or equal to -6.

Answer 2

The solution is `x in (-oo,-6] uu[4,+oo)`

Explanation:

We solve this equation with a sign chart

Let's rearrange and factorise the inequality

`x^2+2x>=24`

`x^2+2x-24>=0`

`(x+6)(x-4)>=0`

Let `f(x)=(x+6)(x-4)`

Now, we construct the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-6` `color(white)(aaaa)` `4` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+6` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-4` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

Therefore,

`f(x)>=0`, when `x in (-oo,-6] uu[4,+oo)`