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

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

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Solution Steps

Step 1: Identify Critical Points

To solve the inequality x3(3x8)40x^{3}(-3x-8)^{4} \leq 0, we first find the critical points where the expression equals zero. The expression is zero at: x=0andx=83 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 x3(3x8)4x^{3}(-3x-8)^{4}. The term (3x8)4(-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 x3x^{3}. The expression x30x^{3} \leq 0 holds true when: x0 x \leq 0

Step 3: Determine the Solution Set

Combining the critical points and the sign analysis, we find that the inequality x3(3x8)40x^{3}(-3x-8)^{4} \leq 0 is satisfied for: <x0 -\infty < x \leq 0

Final Answer

The solution to the inequality is: (,0] \boxed{(-\infty, 0]}

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