Question

alejandra is packing a bag for a flight. the airline has a baggage weight limit, and alejandra has already packed her bag with some essentials. all she has left is her outfits and shoes, which must weigh less than 30 pounds total. if alejandra estimates that one outfit weighs two pounds and one pair of shoes weighs three pounds, which graph represents the number of outfits and shoes that alejandra can pack while staying under the weight limit? option a’s graph (with y - axis labeled “number of shoes”, x - axis (outfits), a line, and shaded region) is shown

Explanation:

To solve this problem, we first define the variables: let \( x \) be the number of outfits and \( y \) be the number of pairs of shoes.

Step 1: Formulate the inequality

Each outfit weighs 2 pounds, so the total weight of outfits is \( 2x \) pounds. Each pair of shoes weighs 3 pounds, so the total weight of shoes is \( 3y \) pounds. The total weight of outfits and shoes must be less than 30 pounds. Thus, the inequality is:
\[
2x + 3y < 30
\]

Step 2: Analyze the boundary line

Rewrite the inequality in slope - intercept form (\( y=mx + b \)) to understand the graph:

1. Start with \( 2x+3y < 30 \).
2. Subtract \( 2x \) from both sides: \( 3y<-2x + 30 \).
3. Divide both sides by 3: \( y<-\frac{2}{3}x + 10 \).

The boundary line is \( y =-\frac{2}{3}x + 10 \), which has a slope of \( -\frac{2}{3} \) and a \( y \) - intercept of 10. Since the inequality is \( y<-\frac{2}{3}x + 10 \) (not \( \leq \)), the boundary line should be dashed. Also, we shade the region below the line because \( y \) is less than \( -\frac{2}{3}x + 10 \).

Step 3: Analyze the intercepts (optional, but helpful)

• \( x \) - intercept: Set \( y = 0 \) in \( 2x+3y=30 \) (the boundary line equation). Then \( 2x=30\Rightarrow x = 15 \). So the \( x \) - intercept is \( (15,0) \).
• \( y \) - intercept: Set \( x = 0 \) in \( 2x + 3y=30 \). Then \( 3y=30\Rightarrow y = 10 \). So the \( y \) - intercept is \( (0,10) \).

Now, looking at the given graph (Option A), we can see that:

• The boundary line has a negative slope (consistent with \( y=-\frac{2}{3}x + 10 \)).
• The \( y \) - intercept is around 10 (consistent with our calculation).
• The region below the line is shaded (consistent with \( y<-\frac{2}{3}x + 10 \)). Also, since the number of outfits (\( x \)) and shoes (\( y \)) cannot be negative, we are only interested in the first quadrant (where \( x\geq0 \) and \( y\geq0 \)), which matches the graph's context (number of items can't be negative).

Answer:

A (the graph with a dashed line \( y =-\frac{2}{3}x + 10 \) (or equivalent slope - intercept form) and shading below the line in the first quadrant)