Can someone help me through the steps of how to do this problem? I find it quite difficult.

Question

This is a pretty hard inequality word problem for me. Help regarding what steps I need to take to solve this problem is greatly appreciated.

Roberto plans to start a new job. In preparation, he decides that he should spend no more than $30$ hours per week on the job and homework combined. If Roberto wants to have at least $2$ homework hours for every $1$ hour at his job, what is the maximum number of hours that he should spend at his job each week?

Answer 1

Let $x$ be the number of hours on the job. How many hours does he spend on homework? What is the total time spent on the job and homework? That has to be less than or equal to $30$ and as you are looking for the maximum it should be equal.

Answer 2

Let $h$ be the homework hours and $j$ be the job hours.

We are given that $h+j \le 30$ and $h \ge 2 j$.

Solve $\max \{ j | h+j \le 30, h \ge 2 j \}$.

If either constraint is not active, then we can increase $j$, so both constraints must be active, that is, $h+j=30, h = 2j$ from which we get $j=10$.

Alternatively, note that we always have $3 j \le h+j \le 30$, so we must have $j \le 10$.

If we choose $j=10$ and $h=20$, then $j$ attains the upper bound and the constraints are met.

Answer 3

Let $x$ and $y$ be the number of hours spent on job and homework respectively. Now we have, $x+y\le 30$ and $2x\le y$ adding both we get $3x+y\le 30+y\implies x\le 10$. Hence maximum hours he should spend at his job is $10$.