High School

Solve the inequality:

[tex] x^2 - 9x - 112 \geq 0 [/tex]

Options:
A. [tex] -7 \leq x \leq 16 [/tex]
B. [tex] -7 \ \textless \ x \ \textless \ 16 [/tex]
C. [tex] x \leq -7 [/tex] or [tex] x \geq 16 [/tex]
D. [tex] x \ \textless \ -7 [/tex] or [tex] x \ \textgreater \ 16 [/tex]

Answer :

To solve the inequality [tex]\(x^2 - 9x - 112 \geq 0\)[/tex], we need to find where the quadratic function is greater than or equal to zero.

1. Find the Roots of the Quadratic Equation:
We start by solving the equation [tex]\(x^2 - 9x - 112 = 0\)[/tex]. The solutions to this equation, or roots, are where the expression equals zero and likely where it changes sign.

The roots of this quadratic equation are:
- [tex]\(x = 16\)[/tex]
- [tex]\(x = -7\)[/tex]

2. Sign Chart or Analysis of the Quadratic:
The quadratic function [tex]\(x^2 - 9x - 112\)[/tex] is a parabola that opens upwards (since the coefficient of [tex]\(x^2\)[/tex] is positive). The parabola will cross the x-axis at the roots, [tex]\(x = -7\)[/tex] and [tex]\(x = 16\)[/tex].

3. Analyze the Intervals:

- For [tex]\(x < -7\)[/tex]:
The quadratic expression [tex]\(x^2 - 9x - 112\)[/tex] is positive. You can test a point in this interval, like [tex]\(x = -8\)[/tex], to confirm.

- For [tex]\(-7 \leq x \leq 16\)[/tex]:
The quadratic expression changes values, but since the roots are included in this interval, the expression equals zero at [tex]\(x = -7\)[/tex] and [tex]\(x = 16\)[/tex]. It is negative between these roots.

- For [tex]\(x > 16\)[/tex]:
The quadratic expression is positive again. You can test a point in this interval, like [tex]\(x = 17\)[/tex], to confirm.

4. Combine the Results:
From the analysis, the quadratic inequality [tex]\(x^2 - 9x - 112 \geq 0\)[/tex] is satisfied when:
- [tex]\(x \leq -7\)[/tex]
- [tex]\(x \geq 16\)[/tex]

Therefore, the solution to the inequality is that [tex]\(x\)[/tex] can be any value less than or equal to [tex]\(-7\)[/tex], or any value greater than or equal to [tex]\(16\)[/tex].