Math Help for Section 3.7, Page 153

Solving
Equations Involving Absolute Value
When solving absolute value equations, remember that it is possible
that they
have no solution. For instance, the equation $|3x+4|=-10$ has no
solution
because the absolute value of a real number cannot be negative. Do not
make the
mistake of trying to solve such an equation by writing the “equivalent”
linear
equations as $3x+4=-10$ and $3x+4=10$. These equations have solutions,
but they are both extraneous.

Example
3: Tip
The equation in the Example 3
on page 87 is not given in the standard
form

    $|ax+b|=c,\quad c\ge0$.

Notice that the first step in solving such an equation is to
write it in standard form.

Example
3: Check

$x=-2$:

   
$\eqalign{|2x-1|+3=&8
&{\small\color{red}\quad\quad\text{Write
original equation.}} \cr |2({\color{red}-2})-1|+3=&8
&{\small\color{red}\quad\quad\text{Substitute }-2\text{ for
}x.}
\cr |-4-1|+3=&8
&{\small\color{red}\quad\quad\text{Multiply.}}
\cr |-5|+3=&8
&{\small\color{red}\quad\quad\text{Subtract.}} \cr
5+3=&8
&{\small\color{red}\quad\quad\text{The absolute value
of }-5\text{ is 5}} \cr 8=&8
&{\small\color{red}\quad\quad\text{Solution checks.
}\checkmark} }$

$x=3$:

    $\eqalign{|2x-1|+3=&8
&{\small\color{red}\quad\quad\text{Write
original equation.}} \cr |2({\color{red}3})-1|+3=&8
&{\small\color{red}\quad\quad\text{Substitute 3 for }x.}
\cr |6-1|+3=&8
&{\small\color{red}\quad\quad\text{Multiply.}}
\cr |5|+3=&8
&{\small\color{red}\quad\quad\text{Subtract.}}
\cr 5+3=&8 &{\small\color{red}\quad\quad\text{The
absolute
value of 5 is 5.}} \cr 8=&8
&{\small\color{red}\quad\quad\text{Solution checks.
}\checkmark} }$