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

Question

8x+5y≤100 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?

Answer 1

To graph the inequality \(8x + 5y \leq 100\), we first need to express it as an equation to find the boundary line. The equation is:

\[ 8x + 5y = 100 \]

Next, we'll find the intercepts:

1. Finding the x-intercept: Set \(y = 0\):

\[ 8x + 5(0) = 100 \implies 8x = 100 \implies x = 12.5 \]

So, the x-intercept is \( (12.5, 0) \).

2. Finding the y-intercept: Set \(x = 0\):

\[ 8(0) + 5y = 100 \implies 5y = 100 \implies y = 20 \]

So, the y-intercept is \( (0, 20) \).

Now we can plot these intercepts on a graph. The line connecting these points \( (12.5, 0) \) and \( (0, 20) \) will represent the boundary of the solution set.

Since the inequality is \( \leq \), the region below and including the line will be shaded to indicate that all points in this region satisfy the inequality.

The key features to include in the graph:

• Draw the line through the points \( (12.5, 0) \) and \( (0, 20) \).
• Shade the area below this line, including the line itself, as it represents all combinations \( (x, y) \) that satisfy the inequality \(8x + 5y \leq 100\).
• The axes should clearly indicate that \(x\) represents bath towels and \(y\) represents hand towels, both of which cannot be negative, meaning the graph should only represent the first quadrant where both \(x\) and \(y\) are non-negative.

In summary, the graph representing the inequality will show a straight line from \( (0, 20) \) to \( (12.5, 0) \), with the area below the line shaded.