(x-5)(x+4) 0 Number Line

4 min read Jun 17, 2024

Solving (x-5)(x+4) = 0 on a Number Line

This article will guide you through solving the equation (x-5)(x+4) = 0 using a number line.

Understanding the Equation

The equation (x-5)(x+4) = 0 represents a product of two factors equaling zero. The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero.

Finding the Roots

To find the solutions (roots) of the equation, we need to determine the values of x that make each factor equal to zero.

Factor 1: (x-5) = 0
- Solving for x, we get x = 5.
Factor 2: (x+4) = 0
- Solving for x, we get x = -4.

Therefore, the solutions to the equation (x-5)(x+4) = 0 are x = 5 and x = -4.

Visualizing on a Number Line

Draw a number line: Mark the points -4 and 5 on the number line. These points represent the solutions (roots) we found.
Divide the number line: The two points divide the number line into three intervals:
- Interval 1: x < -4
- Interval 2: -4 < x < 5
- Interval 3: x > 5
Test a value in each interval: To determine the sign of the expression (x-5)(x+4) in each interval, we can choose a test value within each interval and evaluate the expression.
- Interval 1 (x < -4): Let's choose x = -5.
  - (x-5)(x+4) = (-5-5)(-5+4) = (-10)(-1) = 10 (positive)
- Interval 2 (-4 < x < 5): Let's choose x = 0.
  - (x-5)(x+4) = (0-5)(0+4) = (-5)(4) = -20 (negative)
- Interval 3 (x > 5): Let's choose x = 6.
  - (x-5)(x+4) = (6-5)(6+4) = (1)(10) = 10 (positive)
Mark the sign: Above each interval on the number line, indicate whether the expression (x-5)(x+4) is positive (+) or negative (-).
Solution: The solution to the equation (x-5)(x+4) = 0 is represented by the points where the expression changes sign from positive to negative or vice versa. This occurs at x = -4 and x = 5.

Conclusion

By visualizing the solution on a number line, we can clearly see that the equation (x-5)(x+4) = 0 has two distinct solutions: x = -4 and x = 5. This method provides a visual representation of the solution and helps to understand the behavior of the expression (x-5)(x+4) across different intervals.