Solving the Inequality (x+3)(x2) > 0
This article explores how to solve the inequality (x+3)(x2) > 0. We'll break down the process stepbystep, 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)(x2) > 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)(x2) to determine its sign (+ or ).
Interval  Test Value  (x+3)(x2)  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)(x2) 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)(x2) > 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)(x2) > 0. This process helps us understand the behavior of the expression and identify the values of x that satisfy the given condition.