Video: Writing and Solving Linear Inequalities with Unknowns on Both Sides in Word Problems

You are choosing between two different window-washing companies. The first charges $5 per window, while the second charges a base fee of $40 plus $3 per window. How many windows would you need to have for the second company to be preferable?

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Video Transcript

You are choosing between two different window-washing companies. The first charges five dollars per window, while the second charges a base fee of 40 dollars plus three dollars per window. How many windows would you need to have for the second company to be preferable?

We could set up a table to compare the cost of one window, two windows, three windows, and so on for company one and company two. As company one charges five dollars per window, for one window it would cost us five dollars; for two windows, 10 dollars; for three windows, 15 dollars; increasing in the five times table.

Company two on the other hand charges a base fee of 40 dollars plus three dollars per window. Therefore, for one window it would cost 43 dollars. For two windows, it would cost 46 dollars as it would cost us another three dollars. For three windows, 49 dollars, increasing by three dollars for each extra window.

Whilst we could find the correct answer using this process, it would take a long time. Therefore, a more sensible way to approach this problem would be to use inequalities. If we let 𝑥 be the number of windows, we can find an expression for company one and also an expression for company two.

Company one charges five dollars per window. Therefore, the total cost would be five multiplied by 𝑥 or five 𝑥. Company two on the other hand charges three dollars per window, which gives us three 𝑥. But we also need to add the base fee. Therefore the expression for company two would be three 𝑥 plus 40.

The question wants us to work out when the second company would be preferable. Therefore, it would need to be cheaper. This would occur when five 𝑥 is greater than three 𝑥 plus 40. In order to solve this inequality, we can subtract three 𝑥 from both sides.

This leaves us with two 𝑥 is greater than 40. Dividing both sides by two tells us that 𝑥 must be greater than 20. This means that in order for the second company to be preferable, we need to have more than 20 windows.

If you have less than 20 windows, you should choose company one. But if you have more than 20 windows, you should choose company two. If you happen to have exactly 20 windows, the cost of both companies would be equal.

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