1963 TMTA High School Algebra I Contest Problem 19

Problem

If the symbol $|x|$ is read "the absolute value of $x$ " and is equal to $x$ when $x\ge 0$ and is equal to $-x$ when $x \le 0,$ which of the following is always true?

$\text{(A)} \quad |-8|<|5| \quad \text{(B)} \quad |a+b|=|a-b| \quad \text{(C)} \quad |4-5|<5-4 \quad \text{(D)} \quad |7|=|-5-2| \quad \text{(E)} |+a|>|-a|$

Solution

Examine the validity of the statements one by one. $\text{A}$ simplifies to $8<5,$ which is false.

$\text{B}$ is not true in all cases (specifically, one case where it would be true is $a>0$ and $b=0.$

$\text{C}$ simplifies to $1<1,$ which is false.

$\text{D}$ simplifies to $7=7,$ which is true.

Finally, $\text{E}$ simplifies to $a>a,$ which is false.

The only true statement is $\boxed{\text{(D)} \quad |7|=|-5-2|.}$

1963 TMTA High School Mathematics Contests (Problems)
Preceded by Problem 18	TMTA High School Mathematics Contest Past Problems/Solutions	Followed by Problem 20

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1963 TMTA High School Algebra I Contest Problem 19

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