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

1 Answer
Douglas K.
May 13, 2017

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]}

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