Math Help for Section 3.6, Page 150

Study
Tip
Do not forget to reverse the inequality symbol when multiplying or
dividing by a negative number. For example,

  
 $\eqalign{ -x\ge& 8&{\small\color{red}\quad\quad
\text{Original inequality}} \cr
{\color{red}\left(-1\right)}\left({-x}\right) \le&
{\color{red}\left(-1\right)}\left({8}\right)
&{\small\color{red}\quad\quad \text{Multiply each
side by }-1.\text{ Reverse the
inequality.}} \cr x\le& -8.&{\small\color{red}\quad\quad
\text{Simplify.}} } $

Checking
a Solution
After solving an inequality for x,
you can get an indication of the validity
of a solution set by checking a few convenient values of x. Check some
numbers that are
in the solution set and some numbers that are not. Confirm that numbers
in the solution set satisfy the inequality and that numbers not in the
solution set do not satisfy the inequality. For instance, the solution of the
problem in the Concept Summary was found to be $x<-2.$ Try checking that $x = -3$
satisfies the original inequality, while $x=-1$ does not. 

$x=-3$:
    $\eqalign{-2x+4 >&
8&{\small\color{red}\quad\quad\text{Write original
inequality.}} \cr -\!2({\color{red}-3})+4
\overset{?}{>}&
8&{\small\color{red}\quad\quad\text{Substitute -3
for }x.} \cr 6+4 \overset{?}{>}&
8&{\small\color{red}\quad\quad\text{Multiply.}} \cr 10
>& 8&{\small\color{red}\quad\quad\text{Solution
checks. }\checkmark}} $ 

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

    For $\le$ or $\ge$
inequalities,
also check that the endpoint of
the solution interval satisfies the inequality. For instance, if the
solution set is $y\ge 6$, you could check that $y=6$ and $y=7$
satisfy the inequality, while $y=5$ does not.