Question

How do you solve `2.5x + 1.75y \leq 200`?

Answer 1

Cant be solved as there are 2 unknowns but only 1 equation.

Answer 2

Please see below.

Explanation:

Let us first consider the equality `2.5x+1.75y=200`. It iss apparent that it is the equation of a straight line. Hence, we need to to identify two or three (later will confirm that we are on right track) points that satisfy this equality.

Let `x=24` then we get `2.5xx24+1.75y=200` or `7/4y=140` or `y=80`. Hence line passes through `(24,80)`.

Similarly one can identity that line passes through `(-4,120)` and `(-32,160)` as well and hence graph appears as
graph{10x+7y=800 [-40, 40, 0, 200]}

Note that the graph is not drawn to scale but it wil still serve our purpose. This graph divides Cartesian plane in three parts,

• one the line `2.5x+1.75y=200` or `10x+7y=800` itself. On this point we get `2.5x+.175y=0` and hence, the line is also a solution to `2.5x+1.75y <= 200`

• two the area to the left of the line `2.5x+1.75y=200`. Observe that `(0,0)` lies on it. On this point we get `2.5x+1.75y=0` i.e. `2.5x+1.75y < 0`. This satisfies the inequality and hence, area to left of the line is also a solution.

• three the area to the right of the line `2.5x+1.75y=200`. Observe that `(20,100)` lies on it. On this point we get `2.5x+1.75y=2.5xx20+1.75xx100=50+175=225` i.e. `2.5x+1.75y > 200`. This area does not satisfy the inequality and hence, area to right of the line is not a solution.

Hence solution to `2.5x+1.75y > 200` is all the points shown in the graph below.
graph{2.5x+1.75y <= 200 [-40, 40, 0, 200]}

Note `-` In case we had the inequality as `2.5x+1.75y < 200`, points on the line woulld not have been included and to show this we would have dotted line indicating points on the line are not included. It would have appeared as shown below.

graph{2.5x+1.75y < 200 [-40, 40, 0, 200]}

Answer 3

One way to do inequalities is to first solve the expression as an equality, then select the range that also satisfies the inequality.

Explanation:

In this case we will have a range of values across x and y. We can rearrange the expression as a linear equality:
`1.75*y = -2.5*x + 200`; `y = -1.43*x + 114.3`

This equation describes a line that we can plot. From the plot we will select an (x,y) pair value on one side of the line. If it satisfies the inequality, that is the area of values that are the solution. If it does not, then the solution space is on the other side of the line.

Let’s try (0,40):

`2.5*x + 1.75*y ≤ 200`
`2.5*(0) + 1.75*40 ≤ 200`
`70 ≤ 200` TRUE!
So, our solution range is under (and including the line).

Check the other side, (25,140):

`2.5*(25) + 1.75*140 ≤ 200`
`62.5 + 245 ≤ 200` ; `307.5 ≤ 200`
NOT true, so it confirms our selected solution space.