boomtangboy wrote:
If y + | y | = 0, which of the following must be true?
A. y > 0
B. y≥0
C. y < 0
D. y≤0
E. y = 0
Why is just E incorrect?
\(y + |y| = 0\)
\(|y| = 0 - y\)
\(|y| = -y\)
The last expression means that \(y\leq{0}\). That rule can seem odd or counterintuitive.
The variable has a "hidden" negative sign. With the variable, it's hard to remember that there ARE two negative signs on RHS. We just do not (cannot) write the minus sign twice with the variable. These equations are equivalent, where y = -2:
|-2| = -(-2) = 2
|y| = -(y) = -y
So if \(y + |y| = 0\), then \(|y| = -y\) and
\(y\leq{0}\)
Answer D
If none of the above occurs to you or if it makes no sense, pick and list three numbers: negative, 0, and positive.
Use them to try to DISPROVE the answers. Even one example that defies the rule being tested makes "must be true" false.
-2, 0, and 2
A. y > 0
\(y + |y| = 0\). Try y = 0
\(0 + |0| = 0\). That works. \(y\) does not have to be positive. REJECT
B. y≥0. Use -2
\(y + |y| = 0\)
\(-2 + |-2| = 0\). That works. \(y\) can be negative. REJECT
C. y < 0. We know from (A) that \(y\) CAN equal 0. REJECT
D. y≤0. Try 2
\(y + |y| = 0\)
\(2 + |2| \neq{0}\)
We know from (A) that \(y\) can equal 0.
We know from (B) that \(y\) can be negative.
And having tested +2, we know that \(y\) CANNOT be positive.
This expression MUST be true. KEEP
E. y = 0
We know from (B) that \(y\) can be negative. Yes, \(y\)
can also be 0. But it does not
have to be 0 -- it can be negative, e.g. -2. REJECT
Answer D