# Question Video: Creating and Solving Inequalities

Matthew needs to buy some clothes. The store’s parking lot has the shown sign outside. Parking: The first hour is free, \$1.50 per hour after that. Write an inequality for 𝑡, the time in hours, that Matthew can park if he only has \$8.25 in cash. Given that you must pay for whole hours of parking, use your inequality to find the maximum time that Matthew can park.

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### Video Transcript

Matthew needs to buy some clothes. The store’s parking lot has the shown sign outside. Parking: the first hour is free, one dollar 50 per hour after that. Write an inequality for 𝑡, the time in hours, that Matthew can park if he only has eight dollars 25 in cash. Given that you must pay for whole hours of parking, use your inequality to find the maximum time that Matthew can park.

Let’s consider the information given on the sign. We’re told that the first hour of parking is free, and the amount of time parked in hours is 𝑡. Every hour after that costs one dollar 50. So, we might’ve seen that we need to multiply one dollar 50 by 𝑡. However, as that first hour is free, we need to multiply one dollar 50 or 1.5 by 𝑡 minus one. We know that parking for two hours would only cost one dollar 50. And substituting two into this expression gives us one dollar 50. Likewise, three hours of parking would cost three dollars as the first hour is free. Substituting three into the expression gives us three minus one, which is two, and multiplying this by 1.5 gives us three dollars.

Matthew only has eight dollars and 25 cents in cash. Therefore, this expression needs to be less than or equal to 8.25. We could distribute the parentheses on the left-hand side. However, there is no need at this time. The inequality in terms of 𝑡 is 1.5 multiplies by 𝑡 minus one is less than or equal to 8.25. The second part of this question asks us to solve the inequality to find the maximum time that Matthew can park. We can solve the inequality using inverse operations.

Our first step is to divide both sides by 1.5. The left-hand side of the inequality becomes 𝑡 minus one. 8.25 divided by 1.5 is 5.5. Therefore, 𝑡 minus one is less than or equal to 5.5. Our second and final step is to add one to both sides of the new inequality. This gives us 𝑡 is less than or equal to 6.5. We’re told that you must pay for whole hours of parking. Therefore, 𝑡 must be an integer. As 𝑡 must be less than or equal to 6.5, the greatest integer value it can take is six. This means that the maximum time that Matthew can park for is six hours.