Since we can only spend $168 or less, the total cost of the purchased books can't exceed this amount of money, then, we can express the inequality:
money spent ≤ $168
Let's approach this problem with the two unknowns:
x -> number of hardback books at $14
y -> number of paperback books at $12
Then, the cost of the books would be the number of each type of book multiplied by its corresponding cost per book, then we can rewrite the above equation like this:
money spent ≤ 168
x*14 + y*12 ≤ 168
We can rearrange this equation in order to isolate the variable y on one side of the less or equal sign, like this:
14x + 12y ≤ 168
14x - 14x + 12y ≤ 168 - 14x
12y ≤ 168 - 14x
12y/12 ≤ (168 - 14x)/12
y ≤ (168 - 14x)/12
y ≤ 168/12 - 14x/12
y ≤ 14 - 7/6 x
To graph this inequality, we need to find two points of it, we can find these points by replacing some value into x and calculating the corresponding value of y, like this:
by taking x equals 0
y ≤ 14 - 7/6 *0
y ≤ 14 - 0
y ≤ 14, then we have the point (0,14)
by taking x equals 6:
y ≤ 14 - 7/6 *6
y ≤ 14 - 7
y ≤ 7, then we have the point (6,7)
The graph of the inequality looks is the region below the line that goes through the points (0,14) and (6,7).
The x-intercept is the point where the line intersects the x-axis, we can find it by replacing 0 for y into the inequality, like this:
x*14 +0*12 ≤ 168
x*14+0≤ 168
14x≤ 168
14x/14≤ 168/14
x≤ 12
Then, the x-intercept of the line is (12,0)
The y-intercept is the point where the line intersects the y-axis, in this case, we can see in the graph that this point is (0,14)
The slope of the line is the number multiplying the variable x, in this case -7/6
