Video: US-SAT02S4-Q21-693137126863v2

John would like to buy a used car. He is looking at cars priced in a range from a maximum of $4000 to a minimum of $2500. Which inequality expresses the complete range of car prices, 𝑥, in dollars that he is looking at? [A] |𝑥 − 3250| ≤ 750 [B] |𝑥 − 3250| ≥ 750 [C] |𝑥 − 3500| ≤ 700 [D] |𝑥 − 3500| ≤ 700

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Video Transcript

John would like to buy a used car. He is looking at cars priced in a range from a maximum of 4000 dollars to a minimum of 2500 dollars. Which inequality expresses the complete range of car prices, 𝑥, in dollars that he is looking at? A) The absolute value of 𝑥 minus 3250 is less than or equal to 750, B) the absolute value of 𝑥 minus 3250 is greater than or equal to 750, C) the absolute value of 𝑥 minus 3500 is less than or equal to 700, or D) the absolute value of 𝑥 minus 3500 is greater than or equal to 700.

First, let’s consider what we know. We have a price range with a minimum of 2500 and a maximum of 4000. And we use the variable 𝑥 to represent the car prices in dollars. So we’ll say that 𝑥, the price in dollars, can be greater than or equal to 2500, but must be less than or equal to 4000. And while this is a correct inequality for the range of prices, none of our answer choices are in this format. That means we need to consider which of our answer choices is equivalent to this range. Let’s start by considering answer choice A.

When we consider an inequality with absolute value like this one, the absolute value of 𝑥 minus 3250 is less than or equal to 750, we’ll have two cases to consider: the positive case 𝑥 minus 3250 is less than or equal to 750 and the negative case which would say the negative of 𝑥 minus 3250 is less than or equal to 750. Solving for 𝑥 on the positive side, we’ll add 3250 to both sides of the equation. On the left, we’ll be left with 𝑥 is less than or equal to 750 plus 3250 which is 4000. 𝑥 is less than or equal to 4000. And based on the information we’re given, that is true.

To solve for the negative side takes an additional step. We first need to distribute this negative. And we’ll now have negative 𝑥 plus 3250 is less than or equal to 750. From there, we’ll subtract 3250 from both sides of the equation. On the left, we have negative 𝑥 and on the right, we’ll have negative 2500. We want to consider our positive 𝑥 values. So we’ll multiply through by negative one. We change all the signs and we make sure that we flip our inequality when we’re multiplying by negative one. And our statement would say that 𝑥 is greater than or equal to 2500, which is also true based on the information we were given.

That means the first inequality does express the complete range of car prices. But let’s go ahead and consider why the other three wouldn’t work. Looking at option B, we see that we’re dealing with all the same values and our inequality symbol is flipped. This means when we solve, we would have gotten 𝑥 is greater than or equal to 4000 and 𝑥 is less than or equal to 2500, which means we can eliminate option B.

If we tried to solve option C, we would have 𝑥 minus 3500 is less than or equal to 700. Then, we would add 3500 to both sides which would be 𝑥 is less than or equal to 4200. 4200 is outside of the maximum we were given, which means C is not an option.

And finally, with option D, we again see this greater than sign in addition to having the incorrect values. So we can say that option D would not yield the correct range of car prices.

Option A the absolute value of 𝑥 minus 3250 is less than or equal to 750 correctly expresses the car price range that John has.

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