Question

The owners of the pet sitting business have set aside $48 to purchase chewy

toys for dogs, x, and collars for the cats, y, but do not want to use all of it. The
price of a chewy toy for dogs is $2 while the price of a cat collar is $6. Write and
graph an inequality in standard form to represent how many of each item can be
purchased.

Answer 1

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Let's use "d" to represent the number of chewy toys for dogs, and "c" to represent the number of collars for cats.

The total cost of the chewy toys and collars cannot exceed the $48 budget, so we can write the inequality:

2d + 6c < 48

This is the standard form of the inequality. To graph it, we can first rewrite it in slope-intercept form by solving for "c":

6c < -2d + 48

c < (-2/6)d + 8

c < (-1/3)d + 8

This inequality represents a line with a slope of -1/3 and a y-intercept of 8. We can graph this line by plotting the y-intercept at (0, 8) and then using the slope to find additional points.

To determine which side of the line to shade, we can test a point that is not on the line, such as (0, 0):

2d + 6c < 48

2(0) + 6(0) < 48

0 < 48

Since the inequality is true for (0, 0), we know that the region below the line is the solution. We can shade this region to show that any combination of d and c below the line will satisfy the inequality.