Answer: {2, 3}
Explanation:
Define Variables:
May choose any letters.
\text{Let }b=
Let b=
\,\,\text{the number of hours babysitting}
the number of hours babysitting
\text{Let }c=
Let c=
\,\,\text{the number of hours clearing tables}
the number of hours clearing tables
\text{\textquotedblleft a maximum of 17 hours"}\rightarrow \text{17 or fewer hours}
“a maximum of 17 hours"→17 or fewer hours
Use a \le≤ symbol
Therefore the total number of hours worked in both jobs, b+cb+c, must be less than or equal to 17:17:
b+c\le 17
b+c≤17
\text{\textquotedblleft a minimum of \$180"}\rightarrow \text{\$180 or more}
“a minimum of $180"→$180 or more
Use a \ge≥ symbol
Sofia makes $12 per hour babysitting, so in bb hours she will make 12b12b dollars. Sofia makes $8 per hour clearing tables, so in cc hours she will make 8c8c dollars. The total amount earned 12b+8c12b+8c must be greater than or equal to \$180:$180:
12b+8c\ge 180
12b+8c≥180
\text{Plug in }\color{green}{14}\text{ for }b\text{ and solve each inequality:}
Plug in 14 for b and solve each inequality:
Sofia worked 14 hours babysitting
\begin{aligned}b+c\le 17\hspace{10px}\text{and}\hspace{10px}&12b+8c\ge 180 \\ \color{green}{14}+c\le 17\hspace{10px}\text{and}\hspace{10px}&12\left(\color{green}{14}\right)+8c\ge 180 \\ c\le 3\hspace{10px}\text{and}\hspace{10px}&168+8c\ge 180 \\ \hspace{10px}&8c\ge 12 \\ \hspace{10px}&c\ge 1.50 \\ \end{aligned}
b+c≤17and
14+c≤17and
c≤3and
12b+8c≥180
12(14)+8c≥180
168+8c≥180
8c≥12
c≥1.50
\text{The values of }c\text{ that make BOTH inequalities true are:}
The values of c that make BOTH inequalities true are:
\{2,\ 3\}
{2, 3}