Question

The solution set of \( x^2 + 6x < 91 \) is

• A. \( \{ x : -3 < x < 7 \} \)
• B. \( \{ x : -10 < x < 7 \} \)
• C. \( \{ x : -13 < x < 7 \} \)
• D. \( \{ x : -7 < x < 13 \} \)

Hint:

To solve quadratic inequalities, first solve the corresponding quadratic equation, then test the intervals formed by the roots.

Answer 1

The Correct Option is C

We are given the inequality: \[ x^2 + 6x < 9(1) \] Rearranging the inequality: \[ x^2 + 6x - 91 < 0. \] Now, solve the corresponding equation: \[ x^2 + 6x - 91 = 0. \] We can solve this using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \] where \( a = 1 \), \( b = 6 \), and \( c = -91 \). Substituting these values: \[ x = \frac{-6 \pm \sqrt{6^2 - 4(1)(-91)}}{2(1)} = \frac{-6 \pm \sqrt{36 + 364}}{2} = \frac{-6 \pm \sqrt{400}}{2} = \frac{-6 \pm 20}{2}. \] Thus, the two roots are: \[ x = \frac{-6 + 20}{2} = 7, \quad x = \frac{-6 - 20}{2} = -1(3) \] The inequality \( x^2 + 6x - 91 < 0 \) holds between the roots, so the solution set is: \[ \{ x : -13 < x < 7 \}. \]