A hook in an office storage closet can hold no more than 6 pounds. An order of jumbo paperclips weighs 2 pounds and an order of packing tape weighs 3 pounds. If x is the number of orders of paperclips and y is the number of orders of packing tape, which graph represents how many of each order could be put in a bag hanging from the hook?

To determine which graph represents how many of each order could be put in a bag hanging from the hook given the weight restrictions, you would start by writing an inequality that models the situation.

Given: - A hook can hold no more than 6 pounds. - An order of jumbo paperclips weighs 2 pounds. - An order of packing tape weighs 3 pounds.

Let x be the number of orders of paperclips, and y be the number of orders of packing tape.

The weight of the x orders of paperclips is 2x pounds and the weight of the y orders of packing tape is 3y pounds. The total weight of the bag must not exceed 6 pounds. Thus, the inequality that represents this scenario is:

2x + 3y ≤ 6

To graph this inequality, you create a coordinate system with x on the horizontal axis and y on the vertical axis. The next steps would be as follows:

Step 1: Determine the boundary line by considering the equation 2x + 3y = 6 which represents the maximum weight that the hook can hold. Step 2: Find the intercepts to plot the boundary line. If x = 0, then 3y = 6 and y = 6/3, so y = 2. If y = 0, then 2x = 6 and x = 6/2, so x = 3. Step 3: Plot these intercepts. You will plot (0, 2) and (3, 0). Step 4: Draw the line that connects these two points. This line represents the weight limit for the hook. Step 5: Determine the appropriate side of the line that represents the solutions to the inequality. Since we want the weight to be less than or equal to 6 pounds, you would shade the area below or to the left of the line, depending on the direction of the slope. Step 6: The shaded area represents all possible combinations of orders of paperclips and packing tape that can be safely hung from the hook without exceeding the 6-pound weight limit.

Extra: The inequality we've used is a linear inequality because when it is graphed, it forms a straight line. Linear inequalities like this one often represent constraints or limitations in real-world scenarios, such as weight limits in this example.

When working with such problems, understanding that the equations or inequalities represent lines or curves on a graph helps visualize the possible solutions. In the context of this problem, the line 2x + 3y = 6 divides the coordinate plane into two halves, and the inequality 2x + 3y ≤ 6 tells us that the solution set includes the line itself (because of the 'less than or equal to' sign) and the half-plane that is below the line (if the slope is positive) or above the line (if the slope is negative).

When you are graphing inequalities, you often use a dashed line to represent the boundary when the inequality does not include the boundary (using a 'less than' or 'greater than' sign), and a solid line when it does include the boundary (using a 'less than or equal to' or 'greater than or equal to' sign).

Understanding how to graph inequalities and interpret these graphs is an important math skill and is particularly relevant in fields that involve optimization and planning, such as operations research, economics, and management science.