Absolutely, let's solve this quadratic inequality step-by-step.
We're given the inequality:
[tex]\[ x^2 - 13x - 30 > 200 \][/tex]
First, let's rewrite this inequality in a standard format by moving all terms to one side:
[tex]\[ x^2 - 13x - 30 - 200 > 0 \][/tex]
This simplifies to:
[tex]\[ x^2 - 13x - 230 > 0 \][/tex]
Next, let's solve the corresponding quadratic equation to find the critical points (where the inequality equals zero):
[tex]\[ x^2 - 13x - 230 = 0 \][/tex]
The solutions to this quadratic equation are the points where the expression changes sign. Solving the quadratic equation, we find the critical points to be:
[tex]\[ x = -10 \][/tex] and [tex]\[ x = 23 \][/tex]
Now we need to determine the intervals where the inequality [tex]\( x^2 - 13x - 230 > 0 \)[/tex] holds true. We'll test intervals determined by the critical points [tex]\(-10\)[/tex] and [tex]\(23\)[/tex]:
1. Interval [tex]\( (-\infty, -10) \)[/tex]:
Choose a test point, say [tex]\( x = -11 \)[/tex].
Plugging in [tex]\( x = -11 \)[/tex]:
[tex]\[ (-11)^2 - 13(-11) - 230 = 121 + 143 - 230 = 34 \][/tex]
Since 34 is greater than 0, the inequality holds in this interval.
2. Interval [tex]\((-10, 23)\)[/tex]:
Choose a test point, say [tex]\( x = 0 \)[/tex].
Plugging in [tex]\( x = 0 \)[/tex]:
[tex]\[ 0^2 - 13(0) - 230 = -230 \][/tex]
Since -230 is less than 0, the inequality does not hold in this interval.
3. Interval [tex]\( (23, \infty) \)[/tex]:
Choose a test point, say [tex]\( x = 24 \)[/tex].
Plugging in [tex]\( x = 24 \)[/tex]:
[tex]\[ 24^2 - 13(24) - 230 = 576 - 312 - 230 = 34 \][/tex]
Since 34 is greater than 0, the inequality holds in this interval.
Therefore, combining these intervals, the solution set for the inequality [tex]\( x^2 - 13x - 230 > 0 \)[/tex] is:
[tex]\[ x \in (-\infty, -10) \cup (23, \infty) \][/tex]
This matches the second option provided:
[tex]\[ x <-10 \text{ and } x > 23 \][/tex]
So, the correct answer is:
[tex]\[ x <-10 \text{ and } x > 23 \][/tex]
This means the quadratic inequality holds true for values of [tex]\( x \)[/tex] that either are less than [tex]\(-10\)[/tex] or greater than [tex]\(23\)[/tex].
