Make a sign chart that represents the expression x, left parenthesis, x, plus, 2, right parenthesis, is greater than, 0x(x+2)>0 and use your sign chart to solve the inequality.

To solve the inequality \( x(x + 2) > 0 \), we will create a sign chart to determine the intervals where the expression is positive. Here's the step-by-step process:

### Step 1: Find Critical Points
Set \( x(x + 2) = 0 \) to find the critical points (where the expression equals 0).

\[
x(x + 2) = 0
\]

This gives two solutions:

\[
x = 0 \quad \text{or} \quad x + 2 = 0 \quad \Rightarrow \quad x = -2
\]

So, the critical points are \( x = -2 \) and \( x = 0 \).

### Step 2: Create Intervals
The critical points divide the real number line into three intervals:

1. \( (-\infty, -2) \)
2. \( (-2, 0) \)
3. \( (0, \infty) \)

### Step 3: Test Each Interval
Now, we test a point from each interval in the expression \( x(x + 2) \) to determine the sign.

- **Interval \( (-\infty, -2) \):** Choose \( x = -3 \)

\[
x(x + 2) = (-3)((-3) + 2) = (-3)(-1) = 3 \quad (\text{positive})
\]

- **Interval \( (-2, 0) \):** Choose \( x = -1 \)

\[
x(x + 2) = (-1)((-1) + 2) = (-1)(1) = -1 \quad (\text{negative})
\]

- **Interval \( (0, \infty) \):** Choose \( x = 1 \)

\[
x(x + 2) = (1)(1 + 2) = (1)(3) = 3 \quad (\text{positive})
\]

### Step 4: Create the Sign Chart
We now summarize the signs in each interval:

\[
\begin{array}{c|c}
x & x(x + 2) \\
\hline
(-\infty, -2) & + \\
(-2, 0) & - \\
(0, \infty) & +
\end{array}
\]

### Step 5: Solve the Inequality \( x(x + 2) > 0 \)
We are interested in where \( x(x + 2) \) is positive, which occurs in the intervals where the sign is "+". From the sign chart:

\[
x(x + 2) > 0 \quad \text{for} \quad x \in (-\infty, -2) \cup (0, \infty)
\]

### Solution:
\[
x \in (-\infty, -2) \cup (0, \infty)
\]

### Final Answer:
The solution to the inequality is \( x \in (-\infty, -2) \cup (0, \infty) \).

---

Would you like any further details or clarifications on this solution?

Here are 5 related questions to explore:
1. How do you solve \( x(x - 3) > 0 \)?
2. Can the sign chart method be used for quadratic equations?
3. What happens when the inequality is \( x(x + 2) < 0 \)?
4. How do we handle inequalities with higher-degree polynomials?
5. How do sign charts work with rational expressions?

**Tip**: Always test points within each interval of the sign chart to confirm the sign of the expression.

Make a sign chart that represents the expression x(x + 2) > 0 and use your sign chart to solve the inequality.

Learn how to solve the inequality x(x + 2) > 0 using a sign chart. This detailed step-by-step guide covers critical points, interval testing, and solving quadratic inequalities. Ideal for students in Grades 9-11.

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