Math Help for Section 3.6, Page 147

Example
4: Tip
The solution set in set notation is $\{x|x\ge -4\}$.

Example
5: Tip
The solution set in set notation
is $\{x|x>{3\over2}\}$.

Study
Tip
Checking the solution set of an
inequality is not as simple as
checking the solution set of an
equation. (There are usually too
many x-values
to substitute back
into the original inequality.) You
can, however, get an indication of
the validity of a solution set by
substituting a few convenient values
of x. For
instance, in Example 4 on page 81, try
checking that $x=0$ and $x=-4$ satisfy the
original inequality, whereas $x=-5$
does not.

Example
4: Check
The solution was found to be $x\ge -4$. As stated above, check that
$x=0$ and
$x=-4$
satisfy the inequality and that $x=-5$ does not. 

$x=0$:
    $\eqalign{8-3x\le & 20
&{\small\color{red}\quad\quad \text{Write original
inequality.}} \cr 8-3({\color{red}0})
\overset{?}{\le}& 20 &{\small\color{red}\quad\quad
\text{Substitute 0 for }x.}
\cr 8-0\overset{?}{\le}& 20
&{\small\color{red}\quad\quad \text{Multiply.}} \cr 8 \le& 20 &{\small\color{red}\quad\quad
\text{Solution
checks. }\checkmark} } $

$x=-4$:
    $\eqalign{8-3x\le& 20
&{\small\color{red}\quad\quad \text{Write original
inequality.}} \cr 8-3({\color{red}-4})
\overset{?}{\le}& 20 &{\small\color{red}\quad\quad
\text{Substitute }-4\text{ for }x.}
\cr 8+12\overset{?}{\le}& 20
&{\small\color{red}\quad\quad \text{Multiply.}} \cr 20 \le& 20 &{\small\color{red}\quad\quad
\text{Solution
checks. }\checkmark} } $

$x=-5$:
    $\eqalign{8-3x\le&
20 &{\small\color{red}\quad\quad \text{Write original
inequality.}} \cr 8-3({\color{red}-5})
\overset{?}{\le}& 20 &{\small\color{red}\quad\quad
\text{Substitute }-5\text{ for }x.}
\cr 8+15\overset{?}{\le}& 20
&{\small\color{red}\quad\quad \text{Multiply.}} \cr 23
\not\le& 20 &{\small\color{red}\quad\quad
\text{Solution does not check. ✗}} } $

Example
5: Check
The solution was found to be $x>{3\over2}$, so check that
$x=2$
satisfies the inequality and that $x=1$ does not. 

$x=2$:
    $\eqalign{7x-3>& 3(x+1)
&{\small\color{red}\quad\quad \text{Write original
inequality.}} \cr 7({\color{red}2})-3
\overset{?}{>}& 3({\color{red}2}+1)
&{\small\color{red}\quad\quad
\text{Substitute 2 for }x.}
\cr 14-3\overset{?}{>}& 3(3)
&{\small\color{red}\quad\quad \text{Simplify.}}
\cr 11>& 9 
&{\small\color{red}\quad\quad
\text{Solution
checks. }\checkmark} } $

$x=1$:
    $\eqalign{7x-3>&
3(x+1)
&{\small\color{red}\quad\quad \text{Write original
inequality.}} \cr 7({\color{red}1})-3
\overset{?}{>}& 3({\color{red}1}+1)
&{\small\color{red}\quad\quad
\text{Substitute 1 for }x.}
\cr 7-3\overset{?}{>}& 3(2)
&{\small\color{red}\quad\quad \text{Simplify.}} \cr
4\not>& 6 &{\small\color{red}\quad\quad
\text{Solution does not check. ✗}} } $