jjjinapinch wrote:On the number line, point R has coordinate r and point T has coordinate t. Is t < 0?
(1) -1 < r < 0
(2) The distance between R and T is equal to r²
Official Guide question
Answer: C
Target question: Is t NEGATIVE?
Statement 1: -1 < r < 0
No information about t
So, statement 1 is NOT SUFFICIENT
Statement 2: The distance between R and T is equal to r²
There are several values of r and t that satisfy statement 2. Here are two:
Case a: r = -1 and t = -2. The distance between r and t is 1 (aka r²). So, these values of r and t satisfy statement 2. In this case,
t IS negative
Case b: r = -1 and t = 0. The distance between r and t is 1 (aka r²). So, these values of r and t satisfy statement 2. In this case,
t is NOT negative
Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
Statement 1 tells us that -1 < r < 0
ASIDE: If j and k are on the number line, then |j - k| = the distance between j and k
So, from statement 2, we can write:
|t - r| = r²
-----------------ASIDE-------------------------------------
There are 3 steps to solving equations involving ABSOLUTE VALUE:
1. Apply the rule that says:
If |x| = k, then x = k and/or x = -k
2. Solve the resulting equations
3. Plug solutions into original equation to check for extraneous roots
--------BACK TO THE QUESTION---------------------------
Since
|t - r| = r², we'll examine two possible cases:
t - r = r² and t - r = -(r²)
case a: t - r = r²
Rearrange to get: t = r + r²
Factor: t = r(1 + r)
Since -1 < r < 0, we can conclude that (1 + r) is POSITIVE
So, t = r(1 + r) = (NEGATIVE)(POSITIVE) = NEGATIVE
So,
t is negative
case b: t - r = -(r²)
Rearrange to get: t = r - r²
Factor: t = r(1 - r)
Since -1 < r < 0, we can conclude that (1 - r) is POSITIVE
So, t = r(1 - r) = (NEGATIVE)(POSITIVE) = NEGATIVE
So,
t is negative
In both of the two possible cases,
t is negative
Since we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent
