Question

Solve the inequality x^2-4x-5 greater than 0

Answer 1

To solve the inequality \( x^2 - 4x - 5 > 0 \), we first need to find the roots of the corresponding equation \( x^2 - 4x - 5 = 0 \).

We can factor the quadratic expression:

\[
x^2 - 4x - 5 = (x - 5)(x + 1)
\]

Now we set each factor equal to zero to find the roots:

1. \( x - 5 = 0 \) → \( x = 5 \)
2. \( x + 1 = 0 \) → \( x = -1 \)

Now that we have the roots \( x = -1 \) and \( x = 5 \), we can use these points to determine the intervals where the inequality \( (x - 5)(x + 1) > 0 \) holds true.

The roots divide the real number line into three intervals:
1. \( (-\infty, -1) \)
2. \( (-1, 5) \)
3. \( (5, \infty) \)

Next, we will test a point from each interval to determine where the expression is positive.

1. For the interval \( (-\infty, -1) \), let's test \( x = -2 \):
\[
(-2 - 5)(-2 + 1) = (-7)(-1) = 7 > 0
\]

2. For the interval \( (-1, 5) \), let's test \( x = 0 \):
\[
(0 - 5)(0 + 1) = (-5)(1) = -5 < 0
\]

3. For the interval \( (5, \infty) \), let's test \( x = 6 \):
\[
(6 - 5)(6 + 1) = (1)(7) = 7 > 0
\]

Now we summarize the findings:
- The expression is positive in the intervals \( (-\infty, -1) \) and \( (5, \infty) \).
- The expression is negative in the interval \( (-1, 5) \).

Since we are looking for where \( x^2 - 4x - 5 > 0 \), the solution to the inequality is:

\[
(-\infty, -1) \cup (5, \infty)
\]

Thus, the solution in interval notation is:

\[
\boxed{(-\infty, -1) \cup (5, \infty)}
\]