For what values of x is it true that x^2 - 5x - 4 < 10? Express your answer in interval notation.

Thank you!

x^2 - 5x - 4 < 10

x^2 - 5x - 14 < 0
(x-2)(x+7) < 0

if y = (x-2)(x+7)
we have a parabola opening upwards with x-intercepts of 2 and -7

the parabola would be below the x-axis, that is y < 0 ,
for -7 < x < 2

To determine the values of x for which the inequality x^2 - 5x - 4 < 10 is true, we can solve the inequality by bringing all terms to one side and factoring the quadratic expression.

First, let's rewrite the inequality:
x^2 - 5x - 4 - 10 < 0
x^2 - 5x - 14 < 0

Now, we can factor the quadratic expression:
(x - 7)(x + 2) < 0

To determine the sign of this inequality, we need to examine the signs of each factor. The sign of the expression depends on when the factors change their signs.

Considering the factor (x - 7):
When x < 7, (x - 7) is negative.
When x > 7, (x - 7) is positive.

Considering the factor (x + 2):
When x < -2, (x + 2) is negative.
When x > -2, (x + 2) is positive.

To determine when the overall inequality is negative (less than zero), we need one factor to be negative and the other to be positive. From the signs of the factors, we can identify the intervals where this is true.

There are three intervals that we need to consider:
1. x < -2
2. -2 < x < 7
3. x > 7

Now, we can express the solution in interval notation:
(-∞, -2) ∪ (-2, 7) ∪ (7, +∞)

Therefore, for any value of x in the intervals (-∞, -2), (-2, 7), and (7, +∞), the inequality x^2 - 5x - 4 < 10 is true.