To solve the inequality [tex]\( x^2 - 3x - 28 \geq 0 \)[/tex], let's follow these steps:
1. Factor the quadratic equation: First, we need to factor the quadratic to find its roots.
The quadratic equation is [tex]\( x^2 - 3x - 28 = 0 \)[/tex].
To factor this, we look for two numbers that multiply to [tex]\(-28\)[/tex] and add to [tex]\(-3\)[/tex]. These numbers are [tex]\( -7 \)[/tex] and [tex]\( 4 \)[/tex]. Therefore, we can factor the quadratic as:
[tex]\[
(x - 7)(x + 4) = 0
\][/tex]
2. Find the roots: Setting each factor equal to zero gives us the roots of the equation:
[tex]\[
x - 7 = 0 \quad \text{or} \quad x + 4 = 0
\][/tex]
Solving these, we get:
[tex]\[
x = 7 \quad \text{and} \quad x = -4
\][/tex]
3. Determine the intervals to test: The roots divide the number line into three intervals:
[tex]\[
(-\infty, -4), \quad (-4, 7), \quad (7, \infty)
\][/tex]
4. Test the intervals: We need to test each interval to determine where the inequality [tex]\( x^2 - 3x - 28 \geq 0 \)[/tex] holds true.
- For the interval [tex]\( (-\infty, -4) \)[/tex], choose a test point like [tex]\( x = -5 \)[/tex]:
[tex]\[
(-5)^2 - 3(-5) - 28 = 25 + 15 - 28 = 12 \quad (\text{which is} \geq 0)
\][/tex]
- For the interval [tex]\( (-4, 7) \)[/tex], choose a test point like [tex]\( x = 0 \)[/tex]:
[tex]\[
0^2 - 3(0) - 28 = -28 \quad (\text{which is} < 0)
\][/tex]
- For the interval [tex]\( (7, \infty) \)[/tex], choose a test point like [tex]\( x = 8 \)[/tex]:
[tex]\[
8^2 - 3(8) - 28 = 64 - 24 - 28 = 12 \quad (\text{which is} \geq 0)
\][/tex]
5. Combine the intervals: The inequality [tex]\( x^2 - 3x - 28 \geq 0 \)[/tex] holds true for the intervals where the test points satisfy the inequality. Based on the test results, these are:
[tex]\[
(-\infty, -4] \quad \text{and} \quad [7, \infty)
\][/tex]
Therefore, the solution to the inequality [tex]\( x^2 - 3x - 28 \geq 0 \)[/tex] is:
[tex]\[
(-\infty, -4] \cup [7, \infty)
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{(-\infty, -4] \; \text{or} \; [7, \infty)}
\][/tex]
