(x-3)(x+4) 0 Quadratic Inequality

4 min read Jun 17, 2024

Solving the Quadratic Inequality: (x - 3)(x + 4) < 0

This article will guide you through the steps of solving the quadratic inequality (x - 3)(x + 4) < 0.

Understanding the Problem

We are looking for the values of 'x' that make the expression (x - 3)(x + 4) less than zero. This means we are looking for the intervals where the product of the two factors is negative.

Finding the Critical Points

Critical points are the values of 'x' where the expression equals zero. To find these points, set each factor equal to zero and solve:

x - 3 = 0 => x = 3
x + 4 = 0 => x = -4

These critical points divide the number line into three intervals:

Interval 1: x < -4
Interval 2: -4 < x < 3
Interval 3: x > 3

Testing the Intervals

We need to determine the sign of the expression (x - 3)(x + 4) in each interval:

Interval 1 (x < -4): Choose a value less than -4, for example, x = -5. Substitute it into the expression:
- (-5 - 3)(-5 + 4) = (-8)(-1) = 8 > 0. The expression is positive in this interval.
Interval 2 (-4 < x < 3): Choose a value between -4 and 3, for example, x = 0. Substitute it into the expression:
- (0 - 3)(0 + 4) = (-3)(4) = -12 < 0. The expression is negative in this interval.
Interval 3 (x > 3): Choose a value greater than 3, for example, x = 4. Substitute it into the expression:
- (4 - 3)(4 + 4) = (1)(8) = 8 > 0. The expression is positive in this interval.

Solution

We are looking for the intervals where the expression is less than zero. Therefore, the solution to the inequality (x - 3)(x + 4) < 0 is:

-4 < x < 3

This means all values of 'x' between -4 and 3, excluding -4 and 3, satisfy the inequality.

Graphing the Solution

You can visualize the solution by plotting the critical points on a number line and shading the interval where the expression is negative:

     <------|-----|----->
     -4    3 
     ------- shaded region -------

This shaded region represents the solution set for the inequality.

Remember: Always check the endpoints of the intervals (the critical points) to determine if they are included in the solution set. In this case, they are not included because the inequality is strictly less than zero (<).