Question

a craft vendor must sell at least $300 worth of merchandise to make a profit. scarves sell for $10 each and hats sell for $20 each. she expects to sell at least 20 items. the lines on the graph represent the situation in which she makes a profit. in which section of the graph does the actual solution lie? 1 2 3 4 graph: x-axis: number of scarves sold (0 - 20), y-axis: number of hats sold (0 - 20). two lines, sections labeled 1,2,3,4

Explanation:

Step1: Define Variables

Let \( x \) be the number of scarves sold and \( y \) be the number of hats sold.

Step2: Write Inequalities

1. Profit Requirement: The vendor must sell at least \$300 worth of merchandise. Since scarves sell for \$10 each and hats for \$20 each, we have \( 10x + 20y \geq 300 \), which simplifies to \( x + 2y \geq 30 \) (dividing by 10).
2. Item Requirement: She expects to sell at least 20 items, so \( x + y \geq 20 \).

Step3: Analyze the Graph

• The line for \( x + y = 20 \) (orange line) and \( x + 2y = 30 \) (blue line) divide the graph into sections.
• We need to find the region that satisfies both \( x + y \geq 20 \) and \( x + 2y \geq 30 \).
• Let's test a point in each section. For example, take the intersection of the two lines: solve \( x + y = 20 \) and \( x + 2y = 30 \). Subtracting the first equation from the second gives \( y = 10 \), then \( x = 10 \). So the intersection point is (10, 10).
• Now, test a point in section 1, say (15, 15):
• For \( x + y \geq 20 \): \( 15 + 15 = 30 \geq 20 \) (satisfied).
• For \( x + 2y \geq 30 \): \( 15 + 2(15) = 45 \geq 30 \) (satisfied).
• Test a point in section 2, say (5, 15):
• \( x + y = 20 \), but \( x + 2y = 5 + 30 = 35 \geq 30 \), but \( x + y = 20 \) (just meets), but let's check another point. Wait, actually, section 1 is above both lines, section 2 is between the two lines, section 3 is below both, and section 4 is below the orange line.
• Since we need \( x + y \geq 20 \) (above or on the orange line) and \( x + 2y \geq 30 \) (above or on the blue line), the region that satisfies both is section 1.

Answer:

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