Solve the polynomial inequality: −(x−9)(x+1)(x+3)>0-(x-9)(x+1)(x+3)>0

Give your answer in interval notation. Enter DNE if there is no solution.

To solve the polynomial inequality \( -(x-9)(x+1)(x+3) > 0 \), let's proceed step by step:

### Step 1: Simplify the inequality

We can first distribute the negative sign to simplify the inequality:
\[
-(x-9)(x+1)(x+3) > 0 \quad \text{is the same as} \quad (x-9)(x+1)(x+3) < 0
\]
Now, we need to solve:
\[
(x-9)(x+1)(x+3) < 0
\]

### Step 2: Identify the critical points

The critical points are found by setting each factor equal to zero:
\[
x - 9 = 0 \quad \Rightarrow \quad x = 9
\]
\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\]
\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\]
So, the critical points are \( x = -3, -1, 9 \).

### Step 3: Analyze the sign of the expression

We will analyze the sign of the expression in each interval formed by the critical points. These intervals are:
- \( (-\infty, -3) \)
- \( (-3, -1) \)
- \( (-1, 9) \)
- \( (9, \infty) \)

Pick test points from each interval to determine the sign of the expression \( (x-9)(x+1)(x+3) \) in that interval:

1. **For \( x \in (-\infty, -3) \)**, choose \( x = -4 \):
   \[
   (-4 - 9)(-4 + 1)(-4 + 3) = (-13)(-3)(-1) = -39 \quad \Rightarrow \quad \text{negative}.
   \]

2. **For \( x \in (-3, -1) \)**, choose \( x = -2 \):
   \[
   (-2 - 9)(-2 + 1)(-2 + 3) = (-11)(-1)(1) = 11 \quad \Rightarrow \quad \text{positive}.
   \]

3. **For \( x \in (-1, 9) \)**, choose \( x = 0 \):
   \[
   (0 - 9)(0 + 1)(0 + 3) = (-9)(1)(3) = -27 \quad \Rightarrow \quad \text{negative}.
   \]

4. **For \( x \in (9, \infty) \)**, choose \( x = 10 \):
   \[
   (10 - 9)(10 + 1)(10 + 3) = (1)(11)(13) = 143 \quad \Rightarrow \quad \text{positive}.
   \]

### Step 4: Determine where the expression is negative

We need the intervals where the product is negative, as we are solving \( (x-9)(x+1)(x+3) < 0 \).

From the sign analysis, the expression is negative in the intervals:
\[
(-\infty, -3) \quad \text{and} \quad (-1, 9)
\]

### Step 5: Combine the solution

The solution in interval notation is:
\[
(-\infty, -3) \cup (-1, 9)
\]

### Final Answer:
\[
\boxed{(-\infty, -3) \cup (-1, 9)}
\]

Would you like more details or have any questions?

Here are some related questions for further practice:
1. Solve \( (x-4)(x+2)(x-7) > 0 \).
2. Solve \( (x-5)(x+3)(x-2) \geq 0 \).
3. Solve \( -2(x+1)(x-6)(x-2) > 0 \).
4. Solve \( (x-8)(x+4)^2(x-1) < 0 \).
5. Solve \( (x-1)(x+3) < 0 \).

**Tip:** Always check the sign of the expression between critical points to solve polynomial inequalities.

Solve the polynomial inequality: −(x−9)(x+1)(x+3) > 0. Give your answer in interval notation. Enter DNE if there is no solution.

Learn how to solve the polynomial inequality −(x−9)(x+1)(x+3) > 0 with detailed step-by-step instructions. This solution covers concepts such as factoring, sign analysis, critical points, and interval notation. Suitable for students in Grades 10-12.

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