Question

a grocery bag contains ( x ) apples, each weighing ( \frac{1}{3} ) of a pound, and ( y ) pounds of grapes. the total weight of the grocery bag is less than 5 pounds. which graph represents the possible numbers of apples and pounds of grapes that can be in the bag?

Explanation:

Step1: Formulate the inequality

The weight of the apples is $\frac{1}{3}x$ pounds (since each of the $x$ apples weighs $\frac{1}{3}$ pound) and the weight of the grapes is $y$ pounds. The total weight is less than 5 pounds, so the inequality is $\frac{1}{3}x + y < 5$. We can rewrite this in slope - intercept form ($y=mx + b$) as $y<-\frac{1}{3}x + 5$.

Step2: Analyze the y - intercept and slope

• The y - intercept of the line $y =-\frac{1}{3}x+5$ is 5. This means when $x = 0$ (no apples), $y=5$. So the line should cross the y - axis at $(0,5)$.
• The slope of the line is $-\frac{1}{3}$, which is a negative slope, meaning the line goes downwards from left to right.
• Since the inequality is $y<-\frac{1}{3}x + 5$, the region we are interested in is below the line, and the line should be dashed (because the inequality is strict, $y$ is less than, not less than or equal to, the expression).

Step3: Analyze the x - intercept

To find the x - intercept, set $y = 0$ in the equation $y=-\frac{1}{3}x + 5$. Then $0=-\frac{1}{3}x+5$. Solving for $x$:
\[

$$\begin{align*} \frac{1}{3}x&=5\\ x&=15 \end{align*}$$

\]
So the line should cross the x - axis at $(15,0)$.
Now let's analyze the graphs:

• The first two graphs have a y - intercept of 15, which is incorrect (our y - intercept should be 5).
• The third graph has a dashed line with y - intercept 5, x - intercept 15, and the shaded region below the line (since $y<-\frac{1}{3}x + 5$), which matches our inequality.
• The fourth graph has a solid line, but our inequality is strict ($y<\dots$), so the line should be dashed.

Answer:

The third graph (the one in the bottom - left with y - intercept 5, x - intercept 15, dashed line, and shaded region below the line)