Questions: Solve the following inequality. [ (x+2)^2(x-9)>0 ] What is the solution? (Type your answer in interval notation. Simplify your answer. Use integers

Solution Steps

To solve the inequality \((x+2)^{2}(x-9)>0\), we need to determine the intervals where the product of these factors is positive. We will:

1. Identify the critical points where each factor is zero.
2. Determine the sign of the expression in each interval defined by these critical points.
3. Combine the intervals where the expression is positive.

To solve the inequality \((x + 2)^{2}(x - 9) > 0\), we first find the critical points by setting each factor to zero:

• \((x + 2)^{2} = 0\) gives \(x = -2\) (with multiplicity 2).
• \((x - 9) = 0\) gives \(x = 9\).

Thus, the critical points are \(x = -2\) and \(x = 9\).

The critical points divide the number line into the following intervals:

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

We will test a point from each interval to determine where the product \((x + 2)^{2}(x - 9)\) is positive:

• For \(x < -2\) (e.g., \(x = -3\)): \(((-3 + 2)^{2}(-3 - 9) = (1)(-12) < 0\)
• For \(-2 < x < 9\) (e.g., \(x = 0\)): \(((0 + 2)^{2}(0 - 9) = (4)(-9) < 0\)
• For \(x > 9\) (e.g., \(x = 10\)): \(((10 + 2)^{2}(10 - 9) = (144)(1) > 0\)

The expression is positive only in the interval \((9, \infty)\).

Final Answer