Question

the graph below represents the solution set of which inequality?
←
-5 -4 -3 -2 -1 0 1 2 3 4 5 x
○ ( x^2 - 2x - 8 < 0 )
○ ( x^2 + 2x - 8 < 0 )
○ ( x^2 - 2x - 8 > 0 )
○ ( x^2 + 2x - 8 > 0 )

Explanation:

Step1: Analyze the number line graph

The number line has open circles at \( x = -4 \) and \( x = 2 \), and the shaded region is between them, so the solution set is \( -4 < x < 2 \).

Step2: Factor each quadratic inequality

• For \( x^{2}-2x - 8\): Factor as \( (x - 4)(x+ 2) \). The roots are \( x = 4 \) and \( x=-2 \), not matching the graph's roots.
• For \( x^{2}+2x - 8\): Factor as \( (x + 4)(x - 2) \). The roots are \( x=-4 \) and \( x = 2 \), matching the graph's roots.

Step3: Determine the inequality direction

Since the solution is between the roots (\( -4 < x < 2 \)) and the parabola \( y=x^{2}+2x - 8 \) opens upwards (coefficient of \( x^{2} \) is positive), the inequality \( x^{2}+2x - 8<0 \) holds between the roots.

Answer:

\( \boldsymbol{x^{2}+2x - 8 < 0} \) (the second option)