Math Help for Section 3.2, Page 111

Example
2(c): Tip
This linear equation involves both brackets and parentheses.
Watch out for nested symbols of grouping such as these. The innermost
symbols of grouping should be removed first. 

Example
2: Check

$\eqalign{ {\textbf{a.}\quad\quad} 5\left( {x + 2}
\right) =& 2\left( {x – 1}
\right)&{\color{red}\small\quad\quad\text{Write original
equation.}} \cr 5\left( { \color{red}- 4\color{black} + 2} \right)
\overset{?}{=}&
2\left( {\color{red}- 4\color{black} –
1} \right)&{\color{red}\small\quad\quad\text{Substitute
}-4\text{ for }x.} \cr
5\left( {
– 2} \right) \overset{?}{=}& 2\left( { – 5}
\right)&{\color{red}\small\quad\quad\text{Simplify.}}
\cr – 10 =& –
10&{\color{red}\small\quad\quad\text{Solution
checks. }\checkmark}
\cr}$

$\eqalign{ {\textbf{b.}\quad\quad} 2\left( {x – 7}
\right) – 3\left( {x + 4} \right) &= 4 – \left( {5x – 2}
\right)&{\color{red}\small\quad\quad\text{Write original
equation.}}  \cr 2\left( {\color{red}8\color{black} – 7}
\right) – 3\left( {\color{red}8\color{black} + 4}
\right)&\overset{?}{=} 4 – \left[
{5\left(\color{red}8\color{black}
\right) – 2} \right]&{\color{red}\small\quad\quad\text{Substitute 8
for }x.} \cr 2\left( 1 \right) – 3\left( {12}
\right) &\overset{?}{=} 4 – \left( {40 – 2}
\right) &{\color{red}\small\quad\quad\text{Simplify.}}
\cr 2 –
36 &\overset{?}{=} 4-38
&{\color{red}\small\quad\quad\text{Simplify.}} \cr – 34
&=- 34
&{\color{red}\small\quad\quad\text{Solution 
checks. }\checkmark} \cr} $

$\eqalign{ {\textbf{c.}\quad\quad\quad\quad\quad\quad\quad\quad} 5x –
2\left[ {4x +
3\left( {x – 1} \right)}
\right] =& 8 –
3x&{\color{red}\small\quad\quad\text{Write original
equation.}}  \cr 5\left( {\color{red}-{1 \over
3}\color{black}} \right) – 2\left[
{4\left( {\color{red}-{1 \over 3}\color{black}} \right) +
3\left( {\color{red}-{1 \over 3}\color{black} – 1}
\right)} \right]\overset{?}{=} & 8 –
3\left({\color{red}-{1 \over 3}\color{black}}
\right)&{\color{red}\small\quad\quad\text{Substitute }-{1\over
3}\text{
for }x.}  \cr – {5 \over 3} – 2\left[ { – {4 \over 3}
+ 3\left( { – {4 \over 3}} \right)} \right]\overset{?}{=}
& 8 – \left( { – 1}
\right)&{\color{red}\small\quad\quad\text{Simplify.}} 
\cr – {5 \over 3} – 2\left[ { – {4 \over 3}
+ \left( { – {{12} \over 3}} \right)} \right]\overset{?}{=}
&
9&{\color{red}\small\quad\quad\text{Simplify.}}  \cr –
{5 \over 3} – 2\left( { – {{16} \over
3}} \right)\overset{?}{=} &
9&{\color{red}\small\quad\quad\text{Add.}}  \cr – {5
\over 3} – \left( { – {{32} \over
3}} \right)\overset{?}{=} &
9&{\color{red}\small\quad\quad\text{Multiply.}} 
\cr {{27} \over 3}\overset{?}{=} &
9&{\color{red}\small\quad\quad\text{Subtract.}}  \cr 9
= &
9&{\color{red}\small\quad\quad\text{Solution
checks. }\checkmark} \cr} $

Technology:
Tip
Here are the keystrokes to check the solution found in part (b) with a
scientific or graphing calculator. Note that these instructions may not
agree precisely with the steps required by your calculator.

Left Side of Equation: $2\left( {8 – 7} \right) – 3\left( {8 + 4} \right)$
Keystrokes	Display		Calculator
2  8  7  3 8 4	$-34$		Scientific
2 8 7 3 8 4	$-34$		Graphing

Right Side of Equation: $4 – \left[ {5\left( 8 \right) – 2}\right]$
Keystrokes	Display		Calculator
4   5  8 2	$-34$		Scientific
4   5  8 2	$-34$		Graphing

You might find it easier to evaluate expressions like this one step at
a time, starting with the innermost grouping symbols, rather than all
at once.