(x-7)(x+1) 0 Number Line

Jasonbradley

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Jun 17, 2024

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Solving (x-7)(x+1) = 0 and Representing it on a Number Line

This article will guide you through solving the equation (x-7)(x+1) = 0 and visualizing the solution on a number line.

Understanding the Equation

The equation (x-7)(x+1) = 0 represents a product of two factors equaling zero. This principle, known as the Zero Product Property, states that if the product of two or more factors is zero, at least one of the factors must be zero.

Solving the Equation

To solve the equation, we need to find the values of x that make each factor equal to zero:

• x - 7 = 0 => x = 7
• x + 1 = 0 => x = -1

Therefore, the solutions to the equation (x-7)(x+1) = 0 are x = 7 and x = -1.

Representing the Solutions on a Number Line

1. Draw a number line: Start by drawing a horizontal line with an arrow at each end, indicating that the line extends infinitely in both directions.

2. Mark the solutions: Locate the points -1 and 7 on the number line and mark them with dots or small circles.

3. Divide the number line: The solutions divide the number line into three distinct sections:

   • x < -1: This section represents all values of x that are less than -1.
   • -1 < x < 7: This section represents all values of x between -1 and 7.
   • x > 7: This section represents all values of x that are greater than 7.

4. Test each section: Choose a test value within each section and substitute it into the original equation:

   • x < -1: Choose x = -2. (-2 - 7)(-2 + 1) = 9 > 0. This means the expression is positive in this section.
   • -1 < x < 7: Choose x = 0. (0 - 7)(0 + 1) = -7 < 0. This means the expression is negative in this section.
   • x > 7: Choose x = 8. (8 - 7)(8 + 1) = 9 > 0. This means the expression is positive in this section.

5. Label the sections: Label each section of the number line based on the sign of the expression (positive or negative) within that section.

Visualizing the Solution

The number line should now visually represent the solution to the equation (x-7)(x+1) = 0:

• The points -1 and 7 are the solutions where the expression equals zero.
• The sections between the solutions indicate where the expression is positive or negative.

This visualization helps understand how the equation behaves for different values of x and provides a clear representation of the solution set.