Beginning Algebra With Applications, Chapter 1, 1.2, Section 1.2, Problem 172

Determine which statement is true for all real numbers.

a.) $||x| - |y|| \leq |x| - |y|$

b.) $||x| - |y|| = |x| - |y|$

c.) $||x| - |y|| \geq |x| - |y|$


If we let $x = -2$ and $y = 3$, then we substitute this to the given statement. We have

a.)


$
\begin{equation}
\begin{aligned}

||-2| - |3|| & \leq |-2|-|3|
\\
|2-3| & \leq 2-3
\\
|-1| & \leq -1
\\
1 & \leq -1

\end{aligned}
\end{equation}
$


The statement is false.

b.)


$
\begin{equation}
\begin{aligned}

||-2| - |3|| =& |-2| - |3|
\\
|2-3| =& 2-3
\\
|-1| =& -1
\\
1 =& -1

\end{aligned}
\end{equation}
$


The statement is false.

c.)


$
\begin{equation}
\begin{aligned}

||-2| - |3|| \geq & |-2| - |3|
\\
|2 -3| =& 2-3
\\
|-1| =& -1
\\
1 \geq & -1

\end{aligned}
\end{equation}
$


The statement is true.

So the statement that is true for all real numbers is $||x| - |y|| \geq |x| - |y|$