If x and y are nonnegative integers and x+y=25 what is x?

YES, you can subtract with confidence, because this operation is equivalent to subtracting the same quantity from both sides, operation allowed when working with inequalities. In the above case, you subtract 25, only that on the left you express it as x + y, and on the right you just simply write it as 25.
Therefore, 2x + y - (x + y) < 30 - 25 or x < 5.
If you use substitution, replacing x + y in the inequality by 25, you obtain x + 25 < 30, from which by now subtracting 25 from both sides, you get the same conclusion x < 5.
So, you do the same thing, just in slightly different forms.

No, there should be no problem substituting for x and then getting y. There is no one way to get the answer. You must have made some calculation mistakes. Though I would strongly recommend sticking with the variable you need so that extra steps are not required.
When you solve in inequalities for y (by putting x = 25 - y), you get y > 20 and y < 22. So y must be 21 and hence x must be 4.

ooh that makes sense!

not that important but what about for :

a) they say the maximum for 20x + 10y is 250 when x=0 and y=25, why didn't they make the maximum value using x=25 , y=0 since that would give you a larger number?

Hey Bunuel,

You taught me one hard rule, I'll never forget with inequilities. You can only add two inequalities in the same direction, never subtract!!

Ex
a<b
x<y

a+x<b+y is ok

but a-x<b-y is WRONG!!

Now, does this rule differ if I compare an inequality to an equation? Can I subtract an inequality and an equation in the same direction?

(1) 2x+y<30
(2) x+y=25

Can I simply subtract with confidence to get: x<5? I know that I can solve for y in equation (2) and substitute in equation (1). I would appreciate if you could confirm

Many thanks!!

Awesome question..
Inequality mixed with word problems
here x=4 and y=21 is the only value value that will work
SO C

for me ,the right answer is E because the question says that x and y are nonnegative integer means that x and y can be negative fraction and could be positive integer and fraction Therefore when we combine statments we find that x is between 3 and 4 means that x can be 3.5 3.6 3.2 so is insufficient can you please tell me where my reasoning is false ?
Thankyou

x and y are nonnegative integers means that x and y are integers greater than or equal to 0: 0, 1, 2, 3, ... This is the only reading of this statement.

I hope, I can think the way Banuel does.
To solve this simple question I used substitution method.

Yes, only one value of X can satisfy the equations, x=4 and y=21.

Answer C.

x = 25 - y

(1) 20x + 10y < 300
20(25 - y) + 10y < 300
y > 20

INSUFFICIENT.

(2) 20x + 10y > 280
20(25-y) + 10y > 280
y < 22

INSUFFICIENT.

(1&2) 20 < y < 22
y = 21; x = 4.

SUFFICIENT.

Answer is C.

KarishmaB

I would be so appreciative for your insights on this problem.

I originally solved for y in the inequalities.
For statement 1, I simplified to y > 200/19. For statement 2, I got y <38.

So, if I go back to the original equation, y can be really any number greater than 11 and less than 24 because x must be an integer. Then depending on what value y is, x can take on an array of values.

I see how if you solved for the inequalities to tell you what x was greater than and less than by combining the two statements, you arrive at the correct answer based on the restrictions. However, why did solving for y lead me down a rabbit hole? Where am I wrong in my math? Or is the takeaway to just focus on the variable needed when simplifying inequalities?

Thank you thank you thank you again