Math Help for Section 3.7, Page 157

Example
9: Tip
The solution set in set notation is $\{x|\,175,000\le x\le225,000\}$.

Example
9: Check
The solution was found to be $175,000\le x\le225,000$, so check that
the original statement of the problem is satisfied when production is
175,000 barrels and 225,000 barrels, but not when production
is 170,000 barrels and 230,000 barrels.

175,000 barrels:

   
$\eqalign{ |x-200,000|\le & 25,000
&{\small\color{red}\quad\quad\text{Write original inequality.}}
\cr  \left|{\color{red}175,000}-200,000\right|\overset{?}{\le}
&25,000 &{\small\color{red}\quad\quad\text{Substitute
175,000 for }x.} \cr |-25,000| \overset{?}{\le}&
25,000 &{\small\color{red}\quad\quad\text{Subtract.}}
\cr 25,000\le&
25,000 &{\small\color{red}\quad\quad\text{Solution checks.
}\checkmark}  }$

225,000 barrels:

   
$\eqalign{ |x-200,000|\le & 25,000
&{\small\color{red}\quad\quad\text{Write original inequality.}}
\cr  \left|{\color{red}225,000}-200,000\right|\overset{?}{\le}
&25,000 &{\small\color{red}\quad\quad\text{Substitute
225,000
for }x.} \cr |25,000| \overset{?}{\le}&
25,000 &{\small\color{red}\quad\quad\text{Subtract.}}
\cr 25,000\le&
25,000 &{\small\color{red}\quad\quad\text{Solution checks.
}\checkmark}  }$  

170,000 barrels:

   
$\eqalign{ |x-200,000|\le & 25,000
&{\small\color{red}\quad\quad\text{Write original inequality.}}
\cr  \left|{\color{red}170,000}-200,000\right|\overset{?}{\le}
&25,000 &{\small\color{red}\quad\quad\text{Substitute
170,000
for }x.} \cr |-30,000| \overset{?}{\le}&
25,000 &{\small\color{red}\quad\quad\text{Subtract.}}
\cr 30,000\not\le&
25,000 &{\small\color{red}\quad\quad\text{Solution does not
check.
}✗}  }$

230,000 barrels:

   
$\eqalign{ |x-200,000|\le & 25,000
&{\small\color{red}\quad\quad\text{Write original inequality.}}
\cr  \left|{\color{red}230,000}-200,000\right|\overset{?}{\le}
&25,000 &{\small\color{red}\quad\quad\text{Substitute 230,000
for }x.} \cr |30,000| \overset{?}{\le}&
25,000 &{\small\color{red}\quad\quad\text{Subtract.}}
\cr 30,000\not\le&
25,000 &{\small\color{red}\quad\quad\text{Solution does not
check.
}✗}  }$ 

Example
10: Tip
The solution set in set notation is $\{A|-15\le A\le 15\}$.