Suppose you need $2.40 in postage to mail a package to a friend. You have 9 stamps, some $0.20 and some $0.34. How many of each do you need to mail the package?
So to solve this problem, I came up with two inequalities:
1. a + b is less than or equal to 9
2. 20a + 34b is greater than or equal to 240
Now I'm supposed to graph both inequalities, but how? I don't know how to find the intercepts for the inequalities or even which interecepts I'm supposed to find..
let's switch your a and b to x and y to make it more obvious for the x-y plane
x + y ≤ 9 (#1)
20x + 34y ≤ 240 or
10x + 17y ≤ 120 (#2)
consider the "equation" for each
remember to find the x-intercept you let the y=0 and
to find the y-intercept you let x=0
so for #1, the x and y intercepts are both 9, so... easy to graph
do the same for 10x + 17y = 120
the y-intercept is (0,12) and the x-intercept is (appr.7, 0)
now shade in the region which falls below the first line AND the second line in the first quadrant.
Notice they intersect at a point, call that point P
solve the two "equations" to find P.
I used substitution and found x = 33/7
but x is the number of 20 cent stamps, and must be a whole number.
so we have to go either to the next higher x or the next lower x
if x = 5 then y = 9
if x = 4, y = 9
test: if x=5, y=4, cost = 20(5)+34(4)= 236 , not enough
if x=4, y = 5, cost = 20(4) + 5(34) = 250 , which is more than what is needed but is the best we can do.
thank you!
two of my statements were not complete
in <<if x = 5 then y = 9
if x = 4, y = 9 >>
it should have said:
if x = 5 then y = 9-5 = 4
if x = 4, y = 9-4 = 5
by the way, which way do you think is easier: Finding the intercepts(like above), or rearranging the inequality and graph it by knowing the slope and y-intercept?
To graph the inequalities, you first need to find the intercepts of each inequality. The intercepts are the points where the line crosses the x-axis (for the x-intercept) or the y-axis (for the y-intercept).
For the first inequality, a + b ≤ 9:
To find the intercepts, set one variable equal to zero and solve for the other variable:
- Setting a = 0, we have 0 + b ≤ 9, which simplifies to b ≤ 9.
- Setting b = 0, we have a + 0 ≤ 9, which simplifies to a ≤ 9.
So the intercepts for the first inequality are (0, 9) and (9, 0).
For the second inequality, 20a + 34b ≥ 240:
Similarly, set one variable equal to zero and solve for the other variable:
- Setting a = 0, we have 0 + 34b ≥ 240, which simplifies to b ≥ 240/34 or b ≥ 7.06 (approximately).
- Setting b = 0, we have 20a + 0 ≥ 240, which simplifies to a ≥ 240/20 or a ≥ 12.
So the intercepts for the second inequality are (0, 7.06) and (12, 0).
Now, you can plot these intercepts on a graph. The x-axis represents the number of $0.20 stamps (a), and the y-axis represents the number of $0.34 stamps (b). Plot each intercept as a point on the graph.
Next, you will shade the area that satisfies each inequality:
- For a + b ≤ 9, shade the region below the line connecting the intercepts (0, 9) and (9, 0). This indicates that the sum of a and b should be less than or equal to 9.
- For 20a + 34b ≥ 240, shade the region above the line connecting the intercepts (0, 7.06) and (12, 0). This indicates that the total value of the stamps should be greater than or equal to $2.40.
The shaded region where the two regions overlap represents the solutions that satisfy both inequalities. Thus, find the area where the two shaded regions intersect.
Identify the points within this region to determine the specific combination of $0.20 and $0.34 stamps needed to equal $2.40 in postage.
Note: Due to the decimals in the intercepts, it may be challenging to precisely pinpoint the solution on the graph. However, you can use these regions as a general guide to determine the possible combinations of stamps.