(12-4x)(x-6)>= 0
To solve the inequality \((12 - 4x)(x - 6) \geq 0\), we need to find the values of \(x\) that make the expression non-negative.
Step 1: Find the critical points
The critical points occur where the expression equals zero:
\(12 - 4x = 0\) \[ 4x = 12 \] \[ x = 3 \]
\(x - 6 = 0\) \[ x = 6 \]
So, the critical points are \(x = 3\) and \(x = 6\).
Step 2: Test intervals
The critical points divide the number line into three intervals:
- \((-∞, 3)\)
- \((3, 6)\)
- \((6, ∞)\)
We will test a point from each interval to determine whether the product \((12 - 4x)(x - 6)\) is positive or negative in each interval.
Interval 1: \(x < 3\) (Test with \(x = 0\)) \[ (12 - 4(0))(0 - 6) = 12 \cdot (-6) = -72 \quad (\text{negative}) \]
Interval 2: \(3 < x < 6\) (Test with \(x = 4\)) \[ (12 - 4(4))(4 - 6) = (12 - 16)(-2) = (-4) \cdot (-2) = 8 \quad (\text{positive}) \]
Interval 3: \(x > 6\) (Test with \(x = 7\)) \[ (12 - 4(7))(7 - 6) = (12 - 28)(1) = -16 \cdot 1 = -16 \quad (\text{negative}) \]
Step 3: Determine the intervals for the inequality
From our testing:
- The product is non-negative in the interval \((3, 6)\), inclusive of the endpoints since the equality condition (≥ 0) allows for the expression to equal zero.
- At \(x = 3\) and \(x = 6\), the product equals zero.
Conclusion
The solution to the inequality \((12 - 4x)(x - 6) \geq 0\) is: \[ x \in [3, 6] \]
Thus, the final result is: \[ \boxed{[3, 6]} \]
