Question

How do you solve the inequality `3 <= x^2 - 8x + 15`?

Answer 1

Because the coefficient of the `x^2` term is greater than 0, we know that the domain between the roots will cause the quadratic to be less than 0. We shall include all values of x except this region.

Explanation:

Given: `3 <= x^2 - 8x + 15`

Subtract 3 from both sides:

`0 <= x^2 - 8x + 12`

Flip the inequality:

`x^2 - 8x + 12 >= 0`

Because the coefficient of the `x^2` term is greater than 0, we know that the quadratic represents a parabola that opens upward. Therefore, the quadratic will be less than 0 between the two roots and greater than or equal to 0 elsewhere.

Let's find the roots by factoring:

`(x-2)(x-6) = 0`

`x = 2 and x = 6`

Therefore, we know that the two domains that will cause the quadratic to be greater than or equal to 0 are:

`x <= 2` and `x >=6`

Here is a graph to prove it:

graph{x^2-8x+12 [-10, 10, -5, 5]}