Question

lesson 20: writing and solving inequalities in one variable
cool down: how many hours of work?
lins job pays $8.25 an hour plus $10 of transportation allowance each week. she has to work at least 5 hours a week to keep the job, and can earn up to $175 per week (including the allowance).

1. represent this situation mathematically. if you use variables, specify what each one means.
2. how many hours per week can lin work? explain or show your reasoning.

Explanation:

Step1: Define the variable

Let \(x\) be the number of hours Lin works per week.

Step2: Write the inequality for earnings

Lin's earnings per week are \(8.25x + 10\). She earns up to \(175\) per week, so \(8.25x+10\leqslant175\). Also, she has to work at least \(5\) hours, so \(x\geqslant5\).

Step3: Solve the inequality \(8.25x + 10\leqslant175\)

Subtract \(10\) from both sides: \(8.25x\leqslant175 - 10=165\).
Divide both sides by \(8.25\): \(x\leqslant\frac{165}{8.25}=20\).

Answer:

1. The mathematical representation is \(x\geqslant5\) (where \(x\) is the number of hours worked per week) and \(8.25x + 10\leqslant175\).
2. Lin can work \(5\leqslant x\leqslant20\) hours per week.