To solve the quadratic inequality \( x^2 - 5x - 24 < 0 \), follow these steps:
1. Factor the quadratic expression if possible.
2. Find the roots of the corresponding quadratic equation by setting the expression equal to zero.
3. Determine the intervals where the inequality is satisfied.
4. Check the intervals against the inequality to find the solution set.
Let's solve it step-by-step:
**Step 1: Factor the quadratic expression.**
The expression \( x^2 - 5x - 24 \) factors into \( (x - 8)(x + 3) \).
**Step 2: Find the roots of the corresponding quadratic equation.**
Set the factored expression equal to zero:
\[ (x - 8)(x + 3) = 0 \]
Now, solve for x:
\[ x - 8 = 0 \Rightarrow x = 8 \]
\[ x + 3 = 0 \Rightarrow x = -3 \]
**Step 3: Determine the intervals.**
The roots divide the number line into three intervals: \( (-\infty, -3) \), \( (-3, 8) \), and \( (8, +\infty) \).
**Step 4: Check the intervals against the inequality.**
Using test points from each interval, plug them into the inequality to see where it holds true. We are looking for where the expression is less than zero.
For \( x \) in \( (-\infty, -3) \), choose a test point like \( x = -4 \):
\[ (-4 - 8)(-4 + 3) = (-12)(-1) = 12 > 0 \]
This interval does not satisfy the inequality since we are looking for less than zero.
For \( x \) in \( (-3, 8) \), choose a test point like \( x = 0 \):
\[ (0 - 8)(0 + 3) = (-8)(3) = -24 < 0 \]
This interval does satisfy the inequality.
For \( x \) in \( (8, +\infty) \), choose a test point like \( x = 9 \):
\[ (9 - 8)(9 + 3) = (1)(12) = 12 > 0 \]
This interval does not satisfy the inequality.
The solution set for the inequality \( x^2 - 5x - 24 < 0 \) is \( x \) in the interval \( (-3, 8) \). This can be written in interval notation as:
\[ (-3, 8) \]
This means that for all \( x \) values between -3 and 8 (but not including -3 and 8), the inequality \( x^2 - 5x - 24 < 0 \) holds true.
