A wholesaler requires a minimum of 4 items in each order from its retail customers. The manager of one retail store is considering ordering a certain number of sofas, x, and a certain number of pillows that come in pairs, y. Which graph represents the overall equation represented by this scenario (all points may not apply to the scenario)?

On a coordinate plane, a dashed straight line has a negative slope and goes through (0, 2) and (4, 0). Everything below and to the left of the line is shaded.

On a coordinate plane, a solid straight line has a negative slope and goes through (0, 2) and (4, 0). Everything below and to the left of the line is shaded.

On a coordinate plane, a dashed straight line has a negative slope and goes through (0, 2) and (4, 0). Everything above and to the right of the line is shaded.

On a coordinate plane, a solid straight line has a negative slope and goes through (0, 2) and (4, 0). Everything above and to the right of the line is shaded.

1 answer

To model the wholesaler's ordering requirements, we assume the following:

  • The retail store is ordering sofas, represented by \(x\).
  • The retail store is ordering pillows, which come in pairs, represented by \(y\) (where \(y\) is the number of pairs of pillows).

The wholesaler requires a minimum of 4 items in total from the retail customer, which can be expressed as:

\[ x + y \geq 4 \]

Now, let's analyze the equation \(x + y = 4\) which corresponds to the boundary line:

  1. When \(x = 0\), \(y = 4\) (0 sofas and 4 pairs of pillows).
  2. When \(y = 0\), \(x = 4\) (4 sofas and 0 pairs of pillows).

This line has a negative slope and will pass through points (0, 4) and (4, 0).

In terms of the graph:

  • The region above this line is where the inequality \(x + y \geq 4\) is satisfied (where the total number of items is greater than or equal to 4). Therefore, the area of interest would be above and to the right of the line.

In the options you provided, we need to find the one that represents this condition:

  1. If the line is dashed, it suggests that points on the line do not include the boundary (not appropriate here since we want to include the values where \(x + y = 4\)).
  2. If the line is solid, it suggests that points on the line are included.

Since we need \(x + y \geq 4\), we want:

  • A solid line (indicating we include the boundary).
  • A shaded area above and to the right of the line.

Therefore, the correct answer would be:

On a coordinate plane, a solid straight line has a negative slope and goes through (0, 4) and (4, 0). Everything above and to the right of the line is shaded.

(Note: The specific points mentioned in options seem to indicate a mistake because the line should pass through (0, 4) rather than (0, 2). Just ensure to verify the line's equation in context of the required total.)