A customer is buying bath towels and hand towels and can spend no more than $100. Each bath towel costs $8 and each hand towel costs $5. The inequality 8x+ 5y is less than or equal to 100 represents all possible combinations of x, the number of bath towels, and y, the number of hand towels the customer can buy.

Question

Which graph best represents the solution set for this inequality?

Answer 1

To represent the inequality \( 8x + 5y \leq 100 \) graphically, you will follow these steps:

1. Identify the Boundary Line: First, convert the inequality into an equation: \[ 8x + 5y = 100 \]

2. Find Intercepts: To graph this line, determine the x-intercept and y-intercept:

   • When \( y = 0 \): \[ 8x = 100 \implies x = 12.5 \]
   • When \( x = 0 \): \[ 5y = 100 \implies y = 20 \]

3. Plot the Intercepts: Plot the points \( (12.5, 0) \) and \( (0, 20) \) on a graph.

4. Draw the Boundary Line: Draw a solid line through these two points because the inequality includes \( \leq \) (indicating that the boundary itself is included in the solution set).

5. Shade the Region: Since the inequality is \( 8x + 5y \leq 100 \), shade the region below the line, which represents all the combinations of towels that satisfy the inequality.

6. Consider Non-Negativity: Since \( x \) and \( y \) (the number of bath towels and hand towels) cannot be negative, you will shade only in the first quadrant of the graph.

7. Final Graph: The resulting graph will have a solid line from (12.5, 0) to (0, 20), with shading below this line and only in the first quadrant.

By following these steps, you can create the appropriate graph for the solution set of the inequality \( 8x + 5y \leq 100 \). If you are presented with multiple graph options, look for the one that displays a solid line for the boundary and shading below that line within the first quadrant.