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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].
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].
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