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 x(x+2)>0x(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)=0x(x + 2) = 0 to find the critical points (where the expression equals 0).

x(x+2)=0x(x + 2) = 0

This gives two solutions:

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

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

Step 2: Create Intervals

The critical points divide the real number line into three intervals:

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

Step 3: Test Each Interval

Now, we test a point from each interval in the expression x(x+2)x(x + 2) to determine the sign.

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

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

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

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

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

x(x+2)=(1)(1+2)=(1)(3)=3(positive)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:

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.

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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