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

4 min read Jun 16, 2024
(x+3)(x-2) 0 Inequalities

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 -).

Interval Test Value (x+3)(x-2) Sign
x < -3 x = -4 (-1)(-6) +
-3 < x < 2 x = 0 (3)(-2) -
x > 2 x = 3 (6)(1) +

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.

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