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Mark each equation as True or False.

$\abs{10}=10\text{.}$

$\abs{-3}$ is both $3$ and $-3\text{.}$

$\abs{x+4}=x+4\text{.}$

$\abs{-6}=\abs{6}\text{.}$

$\abs{x-3}=\abs{x+3}\text{.}$

Solution

Remember that to be “false” when there is a variable in the equation, all that has to occur is a single input number that makes the equation false.

True: $\abs{10}=10\text{.}$

False: $\abs{-3}$ is only $3\text{.}$

False: $\abs{x+4}\neq x+4$ for many values of $x\text{.}$ When you have to decide whether or not an equation is true, one good method to help you decide is to plug in a few numbers to see if each number makes the equation true or false. Be sure to pick a variety and input at least two numbers, if not three. In this case, we will choose $10$ and $-20\text{.}$
When $x=\substitute{10}\text{:}$
$\begin{aligned} \abs{x+4}\amp\stackrel{?}{=}x+4\\ \abs{\substitute{10}+4}\amp\stackrel{?}{=}\substitute{10}+4\\ \abs{14}\amp\stackrel{\checkmark}{=}14 \end{aligned}$
When $x=\substitute{-20}\text{:}$
$\begin{aligned} \abs{x+4}\amp\stackrel{?}{=}x+4\\ \abs{\substitute{-20}+4}\amp\stackrel{?}{=}\substitute{-20}+4\\ \abs{-16}\amp\stackrel{\text{no}}{=}-16 \end{aligned}$
When we input $-20\text{,}$ the equation was false, which indicates that the equation $\abs{x+4}=x+4$ is false for general $x\text{.}$

True: $\abs{-6}=\abs{6}\text{.}$ Both $\abs{-6}$ and $\abs{6}$ are equal to $6\text{.}$

False: $\abs{x-3}\neq\abs{x+3}\text{.}$ Again we should choose some numbers to check the validity of the equation. We will choose $-12$ and $15\text{.}$
When $x=\highlight{-12}\text{:}$
$\begin{aligned} \abs{x-3}\amp\stackrel{?}{=}\abs{x+3}\\ \abs{\highlight{-12}-3}\amp\stackrel{?}{=}\abs{\highlight{-12}+3}\\ \abs{-15}\amp\stackrel{?}{=}\abs{-9}\\ 15\amp\stackrel{\text{no}}{=}9 \end{aligned}$
Since we had a false equation for our first value, we don’t need to check the second input. The original equation is simply false.