x^2 -90 > x

Answer 1

To solve this inequality, we can rearrange it as follows:

x^2 - x - 90 > 0

Next, we can factor the left side of the inequality:

(x - 10)(x + 9) > 0

Now we can find the critical points by setting each factor equal to zero:

x - 10 = 0
x = 10

x + 9 = 0
x = -9

So the critical points are x = 10 and x = -9. We can then test intervals created by the critical points to determine where the inequality holds true:

For x < -9, both factors are negative, so the inequality is not true.
For -9 < x < 10, the factor (x + 9) is positive while (x - 10) is negative, so the inequality is true.
For x > 10, both factors are positive and the inequality is true.

Therefore, the solution to the inequality x^2 - 90 > x is:

x < -9 or x > 10