BTGModeratorVI wrote: ↑Wed Oct 07, 2020 7:18 am
What is the value of y?
(1) 3|x^2 – 4| = y – 2
(2) |3 – y| = 11
Answer:
C
Source: Manhattan prep
Target question: What is the value of y?
Statement 1: 3|x² – 4| = y – 2
Notice that there are INFINITELY MANY solutions to this equation. Just choose any value of x, and you will find a corresponding value of y.
Consider these two possible cases:
Case a: x = 1, which means y = 11 (once you plug x = 1 into the equation and then solve for y). In this case, the answer to the target question is
y = 3
Case b: x = 2, which means y = 2. In this case, the answer to the target question is
y = 2
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: |3 – y| = 11
Useful property:
If |x| = k, then x = k or x = -k
So, EITHER 3 – y = 11 OR 3 – y = -11
Let's examine each case:
Case a: If 3 – y = 11, then y = -8. In this case, the answer to the target question is
y = -8
Case b: If 3 – y = -11, then y = 14. In this case, the answer to the target question is
y = 14
Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
Statement 2 tells us that EITHER y = -8 OR y = 14
Great! There are only TWO possible values of y.
At this point, we need only determine whether each of these y-values will yield an actual solution for the statement 1 equation (3|x² – 4| = y – 2 )
Case a: y = -8, which means we get: 3|x² – 4| = (-8) – 2
Simplify: 3|x² – 4| = -10
So we get: |x² – 4| = -10/3
Since the absolute value of an expression is always
greater than or equal to 0, we can conclude that this equation has ZERO solutions.
At this point, we can see that y = -8 is NOT a proper solution, which means it MUST be the case that
y = 14, which means
the COMBINED statements are sufficient.
However, let's see why by examining what happens when we test y = 14...
Case b: y = 14, which means we get: 3|x² – 4| = 14 – 2
Simplify: 3|x² – 4| = 12
Divide both sides by 3 to get: |x² – 4| = 4
From the
above property, we can conclude that EITHER x² – 4 = 4 OR x² – 4 = -4
If x² – 4 = 4, then x² = 8, so x = √8 OR x = -√8
In other words, one possible solution is
x = √8 and y = 12
Another possible solution is
x = -√8 and y = 12
Likewise, if x² – 4 = -4, then x² = 0, which means x = 0
So another possible solution is
x = 0 and y = 12
Notice that, for ALL THREE possible solutions, the answer to the
target question is always the same:
y = 12
So, it MUST be the case that
y = 12
Since we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent
