[tex] \large \bold{SOLUTION:} [/tex]
[tex] \begin{array}{l} \textsf{Let }x\textsf{ and }y\textsf{ be the number of hours required for} \\ \textsf{house cleaning and sales job, respectively.} \\ \\ \textsf{Work time per week: } \quad x + y \leq 10 \\ \textsf{“No more than” is translated as “less than or} \\ \:\:\textsf{equal to”.} \\ \\ \textsf{Earnings per week: } \quad 50x + 80y \geq 560 \\ \textsf{“At least” is translated as “greater than or} \\ \:\:\textsf{equal to”.} \\ \\ \\ \textsf{ For }Q1,\: \textsf{is }(4, 5)\: \textsf{ one of the solutions?} \\ \\ \textsf{Work time: } \begin{cases} x + y \negthickspace &\leq 10 \\ 4 + 5 &\leq 10 \\ 9 &\leq 10 \quad \textsf{(True)} \end{cases} \\ \\ \textsf{Earnings: } \begin{cases} 50x + 80y &\geq 560 \\ 50(4) + 80(5) &\geq 560 \\ 200 + 400 &\geq 560 \\ 600 &\geq 560\quad \textsf{(True)} \end{cases} \\ \\ \textit{Answer:}\:\textsf{Yes. You can still earn more than} \\ \qquad \!\!\qquad \textsf{Php 560 each week if you do 4 hours} \\ \qquad \!\!\qquad \textsf{for house cleaning and 5 hours for} \\ \qquad \!\!\qquad \textsf{sales jobs.} \\ \\ \\ \textsf{ For }Q2,\: \textsf{is }(1, 6)\: \textsf{ one of the solutions?} \\ \\ \textsf{Work time: } \begin{cases} x + y &\leq 10 \\ 1 + 6 &\leq 10 \\ 7 &\leq 10 \quad \textsf{(True)} \end{cases} \\ \\ \textsf{Earnings: } \begin{cases} 50x + 80y &\geq 560 \\ 50(1) + 80(6) &\geq 560 \\ 50 + 480 &\geq 560 \\ 530 &\geq 560\quad \textsf{(False)} \end{cases} \\ \\ \textit{Answer:}\:\textsf{No. You can't earn more than Php 560} \\ \qquad \!\!\qquad \textsf{each week if you do 1 hour for house} \\ \qquad \!\!\qquad \textsf{cleaning and 6 hours for sales jobs.} \\ \\ \\ \textsf{ For }Q3,\: \textsf{is }(3, 4)\: \textsf{ one of the solutions?} \\ \\ \textsf{Work time: }\begin{cases} x + y &\leq 10 \\ 3 + 4 &\leq 10 \\ 7 &\leq 10 \quad \textsf{(True)} \end{cases} \\ \\ \textsf{Earnings: }\begin{cases} 50x + 80y &\geq 560 \\ 50(3) + 80(4) &\geq 560 \\ 150 + 320 &\geq 560 \\ 470 &\geq 560\quad \textsf{(False)} \end{cases} \\ \\ \textit{Answer:}\:\textsf{No. You won't earn more than} \\ \qquad \!\!\qquad \textsf{Php 560 each week if you choose} \\ \qquad \!\!\qquad \textsf{to work 3 hours for house cleaning} \\ \qquad \!\!\qquad \textsf{and 4 hours for sales jobs.} \end{array} [/tex]
[tex] \begin{array}{l} \bold{GRAPH:} \\ \\\textsf{Transform each inequality into its slope - inter-}\\ \textsf{cept form then graph.} \\ \\ \underline{\textsf{For the first inequality,}}\\ \\ \red{\begin{aligned} \qquad \quad x + y &\leq 10 \\ \implies y &\leq -x + 10 \end{aligned}} \\ \\ \textsf{The inequality symbol next to }y\textsf{ is }\leq\textsf{ so use a} \\ \textsf{solid border line then }\red{\textsf{shade}}\textsf{ below the line.} \\ \\ \underline{\textsf{For the second inequality,}}\\ \\ \blue{\begin{aligned} \quad 50x + 80y &\geq 560 \\ 80y &\geq -50x + 560 \\ y &\geq -\frac{50}{80}x + \frac{560}{80} \\ \implies y &\geq -\frac{5}{8}x + 7 \end{aligned}} \\ \\ \textsf{The inequality symbol next to }y\textsf{ is }\geq\textsf{ so use a} \\ \textsf{solid border line then }\blue{\textsf{shade}}\textsf{ above the line.} \\ \\ \textsf{All the solutions of the system of linear inequali-} \\ \textsf{ties are located on the overlapped region or the} \\ \textsf{darker shaded region.} \end{array} [/tex]
[tex] \footnotesize \:\:\textsf{Refer to the attached picture for the graph.} [/tex]
[tex] \purple{\mathfrak{\#CarryOnLearning}} [/tex]
