Questions: x^3(-3 x-8)^4 leq 0

Transcript text: $x^{3}(-3 x-8)^{4} \leq 0$

Solution

Solution Steps

Step 1: Identify Critical Points

To solve the inequality $x^{3}(-3x-8)^{4} \leq 0$, we first find the critical points where the expression equals zero. The expression is zero at: \[ x = 0 \quad \text{and} \quad x = -\frac{8}{3} \]

Step 2: Analyze the Sign of the Expression

Next, we analyze the sign of the expression $x^{3}(-3x-8)^{4}$. The term $(-3x-8)^{4}$ is always non-negative since it is raised to an even power. Therefore, the sign of the entire expression depends solely on $x^{3}$. The expression $x^{3} \leq 0$ holds true when: \[ x \leq 0 \]

Step 3: Determine the Solution Set

Combining the critical points and the sign analysis, we find that the inequality $x^{3}(-3x-8)^{4} \leq 0$ is satisfied for: \[ -\infty < x \leq 0 \]