(x+4)(x-7) 0 Number Line

4 min read Jun 16, 2024

Solving (x+4)(x-7) = 0 using a Number Line

This article will guide you through solving the equation (x+4)(x-7) = 0 using a number line. This method is visually intuitive and helps understand the solution process clearly.

Understanding the Equation

The equation (x+4)(x-7) = 0 represents a product of two factors that equals zero. According to the Zero Product Property, if the product of two factors is zero, at least one of the factors must be zero.

Therefore, we can find the solutions to the equation by setting each factor equal to zero:

x + 4 = 0
x - 7 = 0

Solving for x

Solving these simple equations gives us:

x = -4
x = 7

These are our two solutions to the equation (x+4)(x-7) = 0.

Visualizing with a Number Line

Now, let's use a number line to visualize these solutions:

Mark the Solutions: Draw a number line and mark the points -4 and 7.
Divide the Line: These points divide the number line into three intervals:
- Interval 1: x < -4
- Interval 2: -4 < x < 7
- Interval 3: x > 7
Test Points: Choose a test point within each interval and substitute it into the original equation (x+4)(x-7) = 0.
- Interval 1: Let's use x = -5. (-5 + 4)(-5 - 7) = (-1)(-12) = 12. The result is positive.
- Interval 2: Let's use x = 0. (0 + 4)(0 - 7) = (4)(-7) = -28. The result is negative.
- Interval 3: Let's use x = 8. (8 + 4)(8 - 7) = (12)(1) = 12. The result is positive.
Interpret the Results:
- The intervals where the result is positive represent values of x that make the expression (x+4)(x-7) greater than zero.
- The interval where the result is negative represents values of x that make the expression (x+4)(x-7) less than zero.
Solution: We are looking for values of x that make the expression equal to zero. This occurs precisely at the points x = -4 and x = 7.

Conclusion

The number line provides a clear visual representation of the solution process. We see that the equation (x+4)(x-7) = 0 has two solutions, x = -4 and x = 7, which are represented by the points where the number line changes from positive to negative or vice versa. This method offers a valuable tool for understanding and solving similar equations involving factored expressions.