Questions: Make a sign chart that represents the expression -6 x^4(2 x-6)^2<0 and use your sign chart to solve the inequality.

Transcript text: Make a sign chart that represents the expression $-6 x^{4}(2 x-6)^{2}<0$ and use your sign chart to solve the inequality.

Solution

Solution Steps

Step 1: Find the roots of the polynomial.

The roots are the values of x that make the expression equal to zero. -6x⁴(2x-6)² = 0

This occurs when x⁴=0 or (2x-6)² = 0 So, x=0 or 2x-6=0 => 2x=6 => x=3.

The roots are x=0 and x=3.

Step 2: Create a sign chart.

A sign chart helps determine the sign of the expression in the intervals determined by the roots. We have the roots 0 and 3, which divide the number line into three intervals: (-∞, 0), (0, 3), and (3, ∞).

| Interval | Test Value | Sign of -6x⁴ | Sign of (2x-6)² | Sign of -6x⁴(2x-6)² | |---|---|---|---|---| | (-∞, 0) | -1 | - | + | - | | (0, 3) | 1 | - | + | - | | (3, ∞) | 4 | - | + | - |

Step 3: Solve the inequality.

We are looking for where -6x⁴(2x-6)² < 0. From the sign chart, we can see that the expression is negative in the intervals (-∞, 0) and (0,3) and (3, ∞). Since x=0 and x=3 make the expression equal to zero, they are not part of the solution.

Final Answer

(-∞, 0) U (0, 3) U (3, ∞) or simply x≠0 and x≠3.

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