You have $47 to spend on music and movie downloads. Each album download cost $7 and each movie download costs $8. Write and graph a linear inequality that represents this situation. Let x represent the number of albums and y the number of movies.
how about
7x + 8y ≤ 47
To write a linear inequality that represents this situation, we need to consider the total cost of the downloads and the available budget.
Let's start by defining our variables:
x = the number of albums
y = the number of movies
The cost of downloading albums is $7 each, so the total cost of the albums will be 7x.
The cost of downloading movies is $8 each, so the total cost of the movies will be 8y.
Considering that the total budget is $47, we can write the inequality as follows:
7x + 8y ≤ 47
Graphically, we can plot this inequality on a coordinate plane by first converting it into slope-intercept form y ≤ mx + b, where m is the slope and b is the y-intercept.
Rearranging the inequality, we get:
8y ≤ -7x + 47
Dividing both sides of the inequality by 8, we have:
y ≤ (-7/8)x + 47/8
Now, we can graph the line y = (-7/8)x + 47/8 as a boundary line. Since the inequality includes "less than or equal to," the boundary line should be a solid line.
Next, choose a point on one side of the line (usually the origin, or (0,0)), and test it in the original inequality. If the point satisfies the inequality, shade the region containing that point; otherwise, shade the opposite region.
In this case, (0,0) can be used as a test point. Plugging it into the inequality, we get:
7(0) + 8(0) ≤ 47
0 ≤ 47
Since this statement is true, we will shade the region below the boundary line.
The resulting graph should show a filled area below the line y = (-7/8)x + 47/8.