(x+3)(x-2) 0 Inequalities

Jasonbradley

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Jun 16, 2024

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Solving the Inequality (x+3)(x-2) > 0

This article explores how to solve the inequality (x+3)(x-2) > 0. We'll break down the process step-by-step, focusing on understanding the concept of inequalities and applying it to this specific example.

Understanding Inequalities

An inequality is a mathematical statement that compares two expressions using symbols like:

• > (greater than)
• < (less than)
• ≥ (greater than or equal to)
• ≤ (less than or equal to)

Solving an inequality means finding all the values of the variable that make the inequality true.

Solving (x+3)(x-2) > 0

1. Finding the Critical Points

• The critical points are the values of x that make the expression equal to zero. In our case, we set each factor to zero:
  • x + 3 = 0 => x = -3
  • x - 2 = 0 => x = 2

2. Creating a Sign Chart

• We use the critical points to divide the number line into three intervals:

  • x < -3
  • -3 < x < 2
  • x > 2

• We choose a test value within each interval and evaluate the expression (x+3)(x-2) to determine its sign (+ or -).

3. Interpreting the Results

• We are looking for the intervals where (x+3)(x-2) is greater than zero (positive). From the sign chart, we see that this happens in the intervals:
  • x < -3
  • x > 2

4. Expressing the Solution

The solution to the inequality (x+3)(x-2) > 0 is:

x < -3 or x > 2

This can be represented graphically on a number line, with open circles at -3 and 2 to indicate that these values are not included in the solution.

Conclusion

By using critical points, sign charts, and test values, we successfully solved the inequality (x+3)(x-2) > 0. This process helps us understand the behavior of the expression and identify the values of x that satisfy the given condition.