Question

solve for x.
-5(x + 4)(x + 6) ≥ 0
write the solution using interval notation. use the union symbol ∪ to express the solution as a union of disjoint intervals. finite endpoints of all intervals should be integers. if there are no solutions, use the symbol ∅ for the empty set. use the set notation \\{a\\} to represent an isolated solution a.

Explanation:

Step1: Simplify the inequality

Divide both sides by $-5$ (reverse inequality sign):
$$(x + 4)(x + 6) \leq 0$$

Step2: Find critical points

Set factors equal to 0:
$$x + 4 = 0 \implies x = -4$$
$$x + 6 = 0 \implies x = -6$$

Step3: Test interval sign

• For $x < -6$: $(x+4)(x+6) > 0$ (does not satisfy)
• For $-6 < x < -4$: $(x+4)(x+6) < 0$ (satisfies)
• For $x > -4$: $(x+4)(x+6) > 0$ (does not satisfy)

Step4: Include endpoints

Inequality is $\leq 0$, so include $x=-6$ and $x=-4$.

Answer:

$[-6, -4]$