Math Help for Section 3.7, Page 158

Study
Tip
Remember that an absolute
value inequality of the form $|x|<a$  (or $|x|\le a$)
can be
solved with the double inequality

    $-a<x<a$,

but an inequality of the
form $|x|>a$  (or $|x|\ge a$) must be solved using the
two separate
inequalities

    $x<-a$  
 or    $x>a$.

Checking
a Solution
After solving an absolute value equation, you should check your
answer(s) in the original equation. For instance, you can check the
solution of the equation in the Concept Summary as follows.

$x=11$:

   
$\eqalign{|x+1|=&12
&{\small\color{red}\quad\quad\text{Write
original equation.}} \cr |{\color{red}11}+1|=&12
&{\small\color{red}\quad\quad\text{Substitute }11\text{ for
}x.}
\cr |12|=&12
&{\small\color{red}\quad\quad\text{Add.}}
\cr 12=&12
&{\small\color{red}\quad\quad\text{Solution checks.
}\checkmark} }$

$x=-13$:

    $\eqalign{|x+1|=&12
&{\small\color{red}\quad\quad\text{Write
original equation.}}
\cr |{\color{red}-13}+1|=&8 &{\small\color{red}\quad\quad\text{Substitute
}-13\text{ for
}x.}
\cr |-12|=&12
&{\small\color{red}\quad\quad\text{Add.}}
\cr 12=&12
&{\small\color{red}\quad\quad\text{Solution checks.
}\checkmark} }$

    For absolute value
inequalities,
you should check some values in the solution set and some
values outside the solution set to get an indication of the validity of
the solution. For instance, to check the solution $4\le x\le 10$, you
could check that $x=4$ and $x=10$ satisfy the original inequality and
that $x=7$ does not.