A customer is buying bath towels and hand towels and can spend no more than $100. Each bath towel cost $8, and each hand towel costs $5. The inequality `8x+5y\le100` represents all possible combinations of `x`, the number of bath towels, and `y`, the number of hand towels the customer can buy. Which graph best represents the solution set for this inequality?

Question

A
B
C
D

Answer 1

To determine which graph represents the solution set for the inequality \(8x + 5y \leq 100\), we need to interpret the inequality and find its boundary line.

1. Find the boundary line: We rewrite the inequality as an equation: \[ 8x + 5y = 100 \]

   To find the intercepts:

   • x-intercept: Set \(y = 0\): \[ 8x = 100 \implies x = \frac{100}{8} = 12.5 \]
   • y-intercept: Set \(x = 0\): \[ 5y = 100 \implies y = \frac{100}{5} = 20 \]

   So, the boundary line intersects the x-axis at \( (12.5, 0) \) and the y-axis at \( (0, 20) \).

2. Graph the inequality:

   • The line \(8x + 5y = 100\) will be drawn as a solid line (since the inequality is \(\leq\), we include the line itself).
   • The solution set for \(8x + 5y \leq 100\) will be the area below this line, including the line itself.

3. Check the axes: Both \(x\) and \(y\) must be non-negative since they represent quantities of towels. Thus, we focus on the first quadrant (where \(x \geq 0\) and \(y \geq 0\)).

4. Look for options: From the described information, we need to identify the graph where:

   • It shows a solid line through points \( (12.5, 0) \) and \( (0, 20) \).
   • The shaded area is below this line, in the first quadrant.

Once you have these criteria, you can compare the options (graphs A, B, C, D) provided to find the one that matches the description of the solution set for the inequality \(8x + 5y \leq 100\).

If you can describe the details of each option briefly, I can help you analyze them further to find the correct answer.