Question

the solution of which quadratic inequality is shown on the nu
-3 -2 -1 0 1 2
) $(x + 7)(x + 3) \leq 0$
) $(x + 7)(x - 3) < 0$
) $(x - 7)(x - 3) < 0$
) $(x - 7)(x + 3) \leq 0$

Explanation:

Step1: Identify the interval from the graph

The number line shows a closed dot at $x=-3$ and the line extends to the right, so the solution is $x \geq -3$.

Step2: Analyze Option A

Find roots: $x=-7, x=-3$. Test intervals:

• For $x < -7$: $(-)(-)=+>0$, does not satisfy $(x+7)(x+3)\leq0$
• For $-7 < x < -3$: $(+)(-)=-<0$, satisfies
• For $x > -3$: $(+)(+)=+>0$, does not satisfy

Solution: $-7 \leq x \leq -3$, does not match.

Step3: Analyze Option B

Find roots: $x=-7, x=3$. Test intervals:

• For $x < -7$: $(-)(-)=+>0$, does not satisfy $(x+7)(x-3)<0$
• For $-7 < x < 3$: $(+)(-)=-<0$, satisfies
• For $x > 3$: $(+)(+)=+>0$, does not satisfy

Solution: $-7 < x < 3$, does not match.

Step4: Analyze Option C

Find roots: $x=7, x=3$. Test intervals:

• For $x < 3$: $(-)(-)=+>0$, does not satisfy $(x-7)(x-3)<0$
• For $3 < x < 7$: $(-)(+)=-<0$, satisfies
• For $x > 7$: $(+)(+)=+>0$, does not satisfy

Solution: $3 < x < 7$, does not match.

Step5: Analyze Option D

Find roots: $x=7, x=-3$. Test intervals:

• For $x < -3$: $(-)(-)=+>0$, does not satisfy $(x-7)(x+3)\leq0$
• For $-3 < x < 7$: $(-)(+)=-<0$, satisfies
• For $x > 7$: $(+)(+)=+>0$, does not satisfy
• At $x=-3$ and $x=7$, the expression equals 0, so included.

Solution: $x \geq -3$ and $x \leq 7$, which matches the shown interval $x \geq -3$.

Answer:

D. $(x - 7)(x + 3) \leq 0$