# Word problem with systems of inequations that involves a mixture

I've been struggling to convert this world problem into mathematical expressions:

In an oil mill, they decide to make a mixture from two types of oil: the extra virgin whose price is \$4 per liter and the virgin \$3 per liter. They have 200 L of extra virgin olive oil and 500 L of virgin olive oil and they also want the cost of the mixed oil to be at least 1500 dollars.

The extra virgin olive oil will be $$e$$, while the virgin olive oil will be $$v$$.

a) Set up a system of inequalities that contains the restrictions set out in the problem.

I'm mainly struggling to transform the problem into a system. I've done the following:

The total number of mixed oil is:$$e+v=m$$$$200+500=700$$

Considering $$4e+3v$$, the maximum total price of the mixed oil is 2300 :$$4(200)+3(500)=2300$$

Then, I've formed a system of inequalities $$\left\{ \begin{array}e e+v \leq 700 \\ 4e+3v \geq 1500 \end{array} \right.$$

I'm not sure if I've transformed the problem into a system correctly.

b) Is it possible to meet the conditions if the mixture contains the same amount of the two oils?

Since I wasn't sure about my system of equations, I've done the last two questions using $$4e+3v$$ I'm taking the max amount of extra virgin oil, 200L, and this is what we get: $$4(200)+3(200)=800+600=\1400\,$$ So, the answer would be $$1400\lt 1500$$

So, no, it would meet the conditions.

c) If they decide to use all the extra virgin oil they have in the mixture, what will be the amount of virgin oil that they should use if they want the cost of the mixture to be \\$1700?

$$1700=4(200)+3v$$$$\frac{1700-800}{3}=v$$$$300L=v$$

If you could help me identify my errors I would greatly appreciate it. Thank you in advance.

Your first inequality $$e+v\leq 700$$ is not written correctly, since it would allow the possibility of $$(e, v) = (600, 100)$$ which is not possible in this problem.
$$e \leq 200$$ $$v \leq 500$$
Your cost inequality is correct: $$4e+3v \geq 1500$$