Questions: Solve the inequality: 2(x-3)(x+9)(x-6)>0

Solution Steps

To solve the inequality \(2(x-3)(x+9)(x-6) > 0\), we need to determine the intervals where the product of these factors is positive. First, identify the critical points by setting each factor equal to zero: \(x-3=0\), \(x+9=0\), and \(x-6=0\). These give us the critical points \(x=3\), \(x=-9\), and \(x=6\). Next, test the sign of the product in the intervals determined by these critical points: \((-∞, -9)\), \((-9, 3)\), \((3, 6)\), and \((6, ∞)\). Finally, combine the intervals where the product is positive.

The inequality given is:

\[ 2(x-3)(x+9)(x-6) > 0 \]

First, identify the critical points by setting each factor equal to zero:

1. \(x - 3 = 0 \Rightarrow x = 3\)
2. \(x + 9 = 0 \Rightarrow x = -9\)
3. \(x - 6 = 0 \Rightarrow x = 6\)

The critical points are \(x = -9\), \(x = 3\), and \(x = 6\).

The critical points divide the number line into four intervals:

1. \((- \infty, -9)\)
2. \((-9, 3)\)
3. \((3, 6)\)
4. \((6, \infty)\)

Choose a test point from each interval and determine the sign of the expression \(2(x-3)(x+9)(x-6)\).

1. Interval \((- \infty, -9)\): Choose \(x = -10\) \[ 2(-10-3)(-10+9)(-10-6) = 2(-13)(-1)(-16) = -416 \] The expression is negative.

2. Interval \((-9, 3)\): Choose \(x = 0\) \[ 2(0-3)(0+9)(0-6) = 2(-3)(9)(-6) = 324 \] The expression is positive.

3. Interval \((3, 6)\): Choose \(x = 4\) \[ 2(4-3)(4+9)(4-6) = 2(1)(13)(-2) = -52 \] The expression is negative.

4. Interval \((6, \infty)\): Choose \(x = 7\) \[ 2(7-3)(7+9)(7-6) = 2(4)(16)(1) = 128 \] The expression is positive.

The inequality \(2(x-3)(x+9)(x-6) > 0\) is satisfied in the intervals where the expression is positive. From the test results, these intervals are:

• \((-9, 3)\)
• \((6, \infty)\)

Final Answer