SOLUTION: what is the solution to -1/4y - 1/3 is less than or equal to 2/3?

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Question 275185: what is the solution to -1/4y - 1/3 is less than or equal to 2/3?
Found 2 solutions by dabanfield, jsmallt9:
Answer by dabanfield(803) About Me  (Show Source):
You can put this solution on YOUR website!
what is the solution to -1/4y - 1/3 is less than or equal to 2/3?
-1/(4y) - 1/3 <= 2/3
-1/(4y) <= 2/3 + 1/3
-1(4y) <= 1
Multiply both sides by 4y:
-1 <= 4y
-1/4 <= y

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The way you posted your question is ambiguous. The problem is either
%28%28-1%29%2F4%29y+-+1%2F3+%3C=+2%2F3
or
%28-1%29%2F%284y%29+-+1%2F3+%3C=+2%2F3
I'll solve both but in the future please put parentheses around any numerator or denominator that is not just a single number or variable.

%28%28-1%29%2F4%29y+-+1%2F3+%3C=+2%2F3
Adding 1/3 to each side we get:
%28%28-1%29%2F4%29y+%3C=+1
Multiplying both sides by -4 (and remembering the rule about reversing the inequality whenever you multiply or divide both sides of an inequality by any negative number):
y+%3E=+-4
This is the solution to the first version of the problem.

%28-1%29%2F%284y%29+-+1%2F3+%3C=+2%2F3
Adding 1/3 to each side:
%28-1%29%2F%284y%29+%3C=+1
Next we will multiply both sides by 4y. Since we don't know what y is we don't know if 4y is negative or not. If 4y is negative then we should reverse the inequality. So, in order to multiply both sides by 4y we need to consider both possibilities: 4y is positive (and we don't reverse the inequality) or 4y is negative and we do reverse the inequality. The result of multiplying both side by 4y, taking into account both possibilities, can expressed as follows:
(4y+%3E+0 and -1+%3C=+4y) or (4y+%3C+0 and -1+%3E=+4y
Solving all four of these we get:
(y+%3E+0 and -1%2F4+%3C=+y) or (y+%3C+0 and -1%2F4+%3E=+y
Looking at the first two inequalities we see that y must be greater than 0 and greater than or equal to zero. Only numbers that are greater than zero fit both inequalities so these two inequalities simplify to:
y+%3E+0
The last two inequalities above say that y has to be less than 0 and less than or equal to -1/4. Only numbers that are less than or equal to -1/4 fit both inequalities so these two simplify to
y+%3C=+-1%2F4
So
(y+%3E+0 and -1%2F4+%3C=+y) or (y+%3C+0 and -1%2F4+%3E=+y)
simplifies to:
y+%3E+0 or -1%2F4+%3E=+y
This is the solution to the second version of the problem.

Note: Another tutor provided a solution to the second version of the problem, %28-1%29%2F%284y%29+-+1%2F3+%3C=+2%2F3. But he/she did not take into account whether 4y was positive or negative so that solution is incorrect.