Question

each month, a shopkeeper spends 7x + 19 dollars on rent and electricity. if he spends 2x - 7 dollars on rent, how much does he spend on electricity? use pencil and paper. for which value(s) of x is the amount the shopkeeper spends on electricity less than $100? explain how you found the value(s).

Explanation:

Step1: Set up the inequality

We want to find when the amount spent on electricity (\(2x - 7\)) is less than \(100\). So we set up the inequality \(2x - 7<100\).

Step2: Solve the inequality

First, add \(7\) to both sides of the inequality: \(2x - 7+7<100 + 7\), which simplifies to \(2x<107\). Then, divide both sides by \(2\): \(x<\frac{107}{2}=53.5\).

Step3: Interpret the solution

Since \(x\) is the amount the shopkeeper spends on rent, paper, etc., and we found that \(x < 53.5\) when the electricity cost (\(2x - 7\)) is less than \(100\). We can also check by plugging a value less than \(53.5\), say \(x = 50\). Then electricity cost is \(2(50)-7=100 - 7 = 93\), which is less than \(100\). If \(x = 54\), electricity cost is \(2(54)-7 = 108 - 7=101\), which is more than \(100\), so the solution makes sense.

Answer:

The shopkeeper spends less than \(100\) dollars on electricity when \(x<53.5\) (where \(x\) is the amount spent on rent, paper, etc.). To find the electricity cost when it's less than \(100\), we solve \(2x - 7<100\) to get \(x<53.5\).