To solve the inequality x3(−3x−8)4≤0, we first find the critical points where the expression equals zero. The expression is zero at:
x=0andx=−38
Step 2: Analyze the Sign of the Expression
Next, we analyze the sign of the expression x3(−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 x3. The expression x3≤0 holds true when:
x≤0
Step 3: Determine the Solution Set
Combining the critical points and the sign analysis, we find that the inequality x3(−3x−8)4≤0 is satisfied for:
−∞<x≤0