Question

to help pay for his tuition, a college student plans to work in the evenings and on weekends. he has been offered two part - time jobs: working in the guest services department at a hotel and waiting tables at a popular restaurant.
the job at the hotel pays $18 an hour and offers $33 in transportation allowance per month.
the job at the restaurant pays $7.50 an hour plus tips. the entire wait staff typically collects about $50 in tips each hour. tips are divided equally among the four wait staff members who share a shift.
1 the equation ( 7.50h+\frac{50}{4}h = 18h + 33 ) represents a possible constraint about the situation.
a. solve the equation and check your solution.
b. here is a graph on a number line.
put a scale on the number line so that the point marked with a circle represents the solution to the equation.
2 does one job pay better if:
a. the student works fewer hours than the solution you found earlier? if so, which job?
b. the student works more hours than the solution you found earlier? if so, which job?
be prepared to explain or show how you know

Explanation:

Step1: Simplify the equation

$$\begin{align*} 7.50h+\frac{50}{4}h&=18h + 33\\ 7.50h + 12.5h&=18h+33\\ 20h&=18h + 33 \end{align*}$$

Step2: Solve for \(h\)

Subtract \(18h\) from both sides:

$$\begin{align*} 20h-18h&=18h + 33-18h\\ 2h&=33\\ h&=\frac{33}{2}=16.5 \end{align*}$$

Step3: Check the solution

Left - hand side: \(7.50\times16.5+\frac{50}{4}\times16.5=7.50\times16.5 + 12.5\times16.5=(7.50 + 12.5)\times16.5=20\times16.5 = 330\)
Right - hand side: \(18\times16.5+33=297+33=330\)

Answer:

The solution of the equation \(7.50h+\frac{50}{4}h = 18h+33\) is \(h = 16.5\)

For part 2a:
Let \(y_1\) be the income from the hotel job \(y_1=18h + 33\) and \(y_2\) be the income from the restaurant job \(y_2=7.5h+\frac{50}{4}h=20h\)
If \(h<16.5\), then \(y_1=18h + 33\) and \(y_2=20h\)
\(y_1-y_2=18h + 33-20h=33 - 2h\)
When \(h<16.5\), \(33-2h>0\) (since if \(h = 16\), \(33-2h=33-32 = 1>0\)), so the hotel job pays better when \(h<16.5\)

For part 2b:
If \(h>16.5\), then \(y_1 - y_2=33-2h<0\) (since if \(h = 17\), \(33-2h=33 - 34=-1<0\)), so the restaurant job pays better when \(h>16.5\)