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# Quant Fundamentals: Absolute Value

What is the value of \$\$|x| - x\$\$?

(1) \$\$x \ge 0\$\$

(2) \$\$|x| > 0\$\$

Incorrect.

[[snippet]] Stat.(2): According to the definition of the absolute value, \$\$|x|>0\$\$ is true for any \$\$x\$\$ except for \$\$x=0\$\$. Plug in a positive number, such as \$\$\color{blue}{x=3}\$\$ and answer the stem. The value of \$\$\color{blue}{|x|-x}\$\$ is \$\$\color{blue}{|3|-3=0}\$\$. Now plug in a negative number. For example, \$\$\color{orange}{x=-3}\$\$. The value of \$\$|x|-x\$\$ is then \$\$\color{orange}{|{-3}|-(-3)=6}\$\$. No single value can be determined for \$\$|x|-x\$\$; therefore, Stat.(2) → IS → ACE. So Stat.(2) is insufficient, but what about Stat.(1)?

Incorrect.

[[snippet]] Stat.(1): Plug in a number that fits the statement, such as \$\$\color{blue}{x=3}\$\$, and answer the stem. The value of \$\$\color{blue}{|x|-x}\$\$ is \$\$\color{blue}{|3|-3=0}\$\$. Now plug in another number that fits the statement. For example, \$\$\color{purple}{x=4}\$\$. The value of \$\$\color{purple}{|x|-x}\$\$ is \$\$\color{purple}{|4|-4=0}\$\$ again. Obviously, also for \$\$x=0\$\$, the value of \$\$|x|-x=0\$\$. Ask yourself, "Is it _always_ \$\$0\$\$?" Under test conditions, you may want to guess that it is _always_ the same (i.e., that Statement (1) is sufficient). Alternatively, take a closer look at the two checks you have so neatly written on your note board. Try to find a pattern. If \$\$x \ge 0\$\$, then \$\$|x|=x\$\$, and therefore \$\$|x|-x=0\$\$. Therefore Stat.(1) → S → AD.

Incorrect.

[[snippet]] Stat.(2): According to the definition of the absolute value, \$\$|x|>0\$\$ is true for any \$\$x\$\$ except for \$\$x=0\$\$. Plug in a positive number, such as \$\$\color{blue}{x=3}\$\$ and answer the stem. The value of \$\$\color{blue}{|x|-x}\$\$ is \$\$\color{blue}{|3|-3=0}\$\$. Now plug in a negative number. For example, \$\$\color{orange}{x=-3}\$\$. The value of \$\$|x|-x\$\$ is then \$\$\color{orange}{|{-3}|-(-3)=6}\$\$. No single value can be determined for \$\$|x|-x\$\$, so Stat.(2) → IS → ACE.

Stat.(2): According to the definition of the absolute value, \$\$|x|>0\$\$ is true for any \$\$x\$\$ except for \$\$x=0\$\$. From Stat.(1), we already know that the value of \$\$|x|-x\$\$ is \$\$0\$\$ for positive values of \$\$x\$\$. But is it always true for _any_ number? Plug in negative numbers that do not fit Stat.(1), such as \$\$\color{orange}{x= -3}\$\$. The value of \$\$|x|-x\$\$ is then \$\$\color{orange}{|{-3}| - (-3)=6}\$\$. Plug in \$\$\color{green}{x=-4}\$\$. Now the value of \$\$|x|-x\$\$ is \$\$\color{green}{|{-4}|-(-4)=8}\$\$. Since no single value can be determined for \$\$|x|-x\$\$, Stat.(2) → IS → A.

Incorrect.

[[snippet]] Be sure to consider each statement separately before you consider them together. Stat.(1): Plug in a number that fits the statement, such as \$\$\color{blue}{x=3}\$\$, and answer the stem. The value of \$\$\color{blue}{|x|-x}\$\$ is \$\$\color{blue}{|3|-3=0}\$\$. Now plug in another number that fits the statement. For example, \$\$\color{purple}{x=4}\$\$. The value of \$\$\color{purple}{|x|-x}\$\$ is \$\$\color{purple}{|4|-4=0}\$\$ again. Obviously, also for \$\$x=0\$\$, the value of \$\$|x|-x=0\$\$. Ask yourself, "Is it _always_ \$\$0\$\$?" Under test conditions, you may want to guess that it is _always_ the same (i.e., that Statement (1) is sufficient). Alternatively, take a closer look at the two checks you have so neatly written on your note board. Try to find a pattern. If \$\$x \ge 0\$\$, then \$\$|x|=x\$\$, and therefore \$\$|x|-x=0\$\$. Therefore Stat.(1) → S → AD.

Correct.

[[snippet]] Stat.(1): Plug in a number that fits the statement, such as \$\$\color{blue}{x=3}\$\$, and answer the stem. The value of \$\$\color{blue}{|x|-x}\$\$ is \$\$\color{blue}{|3|-3=0}\$\$. Now plug in another number that fits the statement. For example, \$\$\color{purple}{x=4}\$\$. The value of \$\$\color{purple}{|x|-x ~\mbox{is} ~|4|-4=0}\$\$ again. Obviously, also for \$\$x=0\$\$, the value of \$\$|x|-x=0\$\$. Ask yourself, "Is it _always_ \$\$0\$\$?" Under test conditions, you may want to guess that it is _always_ the same (i.e., that Statement (1) is sufficient). Alternatively, take a closer look at the two checks you have so neatly written on your note board. Try to find a pattern. If \$\$x \ge 0\$\$, then \$\$|x|=x\$\$, and therefore \$\$|x|-x=0\$\$. Therefore Stat.(1) → S → AD.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.