Math Problem Statement
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.
Solution
To solve the inequality , 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 to find the critical points (where the expression equals 0).
This gives two solutions:
So, the critical points are and .
Step 2: Create Intervals
The critical points divide the real number line into three intervals:
Step 3: Test Each Interval
Now, we test a point from each interval in the expression to determine the sign.
- Interval : Choose
- Interval : Choose
- Interval : Choose
Step 4: Create the Sign Chart
We now summarize the signs in each interval:
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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Inequalities
Sign Chart
Quadratic Expressions
Formulas
x(x + 2) = 0 (for finding critical points)
Testing intervals to determine sign
Theorems
Sign chart method for inequalities
Properties of quadratic expressions
Suitable Grade Level
Grades 9-11