To solve the inequality [tex]\( x^2 - 18x < 77 \)[/tex], we can follow these steps:
1. Rewrite the inequality in standard form:
Start by moving 77 to the left side of the inequality:
[tex]\[
x^2 - 18x - 77 < 0
\][/tex]
2. Factor the quadratic expression:
We look for two numbers that multiply to [tex]\(-77\)[/tex] and add up to [tex]\(-18\)[/tex]. These numbers are [tex]\(-7\)[/tex] and [tex]\(-11\)[/tex]. Therefore, we can factor the quadratic as:
[tex]\[
(x - 7)(x - 11) < 0
\][/tex]
3. Determine the critical points:
The critical points are the roots of the quadratic equation [tex]\( x^2 - 18x + 77 = 0 \)[/tex]. Solving [tex]\( (x - 7)(x - 11) = 0 \)[/tex] gives us the points [tex]\( x = 7 \)[/tex] and [tex]\( x = 11 \)[/tex].
4. Analyze the intervals:
The critical points divide the x-axis into three intervals:
- Interval [tex]\( (-\infty, 7) \)[/tex]
- Interval [tex]\( (7, 11) \)[/tex]
- Interval [tex]\( (11, \infty) \)[/tex]
5. Determine where the inequality holds true:
We need to test each interval to see where the product [tex]\( (x - 7)(x - 11) \)[/tex] is less than 0 (i.e., negative).
- For [tex]\( x \in (-\infty, 7) \)[/tex]:
Choose a point like [tex]\( x = 0 \)[/tex]:
[tex]\[
(0 - 7)(0 - 11) = 77 > 0
\][/tex]
This interval does not satisfy the inequality.
- For [tex]\( x \in (7, 11) \)[/tex]:
Choose a point like [tex]\( x = 8 \)[/tex]:
[tex]\[
(8 - 7)(8 - 11) = 1 \cdot (-3) = -3 < 0
\][/tex]
This interval satisfies the inequality.
- For [tex]\( x \in (11, \infty) \)[/tex]:
Choose a point like [tex]\( x = 12 \)[/tex]:
[tex]\[
(12 - 7)(12 - 11) = 5 \cdot 1 = 5 > 0
\][/tex]
This interval does not satisfy the inequality.
6. Conclusion:
The inequality [tex]\( (x - 7)(x - 11) < 0 \)[/tex] holds true in the interval [tex]\( (7, 11) \)[/tex].
Therefore, the solution to the inequality [tex]\( x^2 - 18x < 77 \)[/tex] is:
[tex]\[
7 < x < 11
\][/tex]
Among the given options, the correct answer is:
[tex]\((7 < x < 11) \)[/tex]
