# Video: SAT Practice Test 1 • Section 4 • Question 12

Let 𝑎 and 𝑥 be numbers such that −𝑎 < 𝑥 < 𝑎. Which of the following must be true? [A] |𝑥| < 𝑎 [B] 𝑥 > 0 [C] 𝑎 > 0

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### Video Transcript

Let 𝑎 and 𝑥 be numbers such that 𝑥 is greater than negative 𝑎 but less than 𝑎. Which of the following must be true? The options are i) the modulus or absolute value of 𝑥 is less than 𝑎, ii) 𝑥 is greater than zero, or iii) 𝑎 is greater than zero.

Well, we’re gonna start by looking at the first possible scenario. And that is that the modulus or absolute value of 𝑥 is less than 𝑎. Well, the modulus of 𝑥 or absolute value of 𝑥 is the first thing to consider, because what does this mean?

Well, the modulus or absolute value of 𝑥 tells us that we’re only interested in the positive values of 𝑥, because the absolute value of a number is just the distance. So we consider it as a distance away from a point. So therefore, it’s not going to include the negative values.

So therefore, if this is the case, this first situation is going to be true. And that’s because we’re told in our inequality that 𝑥 is less than 𝑎. But we’re also told that 𝑥 is greater than negative 𝑎. But therefore, if we’re only looking at the positive 𝑥-values, then this must be less than 𝑎. And this is because there are no values of 𝑥 that would satisfy the inequality 𝑥 is less than 𝑎 but greater than negative 𝑎 if the modulus or absolute value of 𝑥 wasn’t less than 𝑎.

Okay, let’s move on to number ii 𝑥 is greater than zero. Well, if we look at 𝑥 is greater than zero, then this isn’t necessarily true. And let’s take this example. If 𝑥 was equal to negative one, 𝑎 could be two. And that’s because negative one would be less than two. So that’d be correct. But negative one is greater than negative two. So it would fulfill our inequality. So in that case, number ii 𝑥 is greater than zero is not necessarily true.

So now let’s take a look at 𝑎 is greater than zero, number iii. Well, let’s consider this. Let’s consider if 𝑎 was equal to negative one. So 𝑎 wasn’t greater than zero. Then we’d have a scenario where 𝑥 will be greater than negative negative one but less than negative one. So therefore, 𝑥 would have to be greater than one — and that’s because negative negative one is positive one — but less than negative one. Well, this would be impossible because 𝑥 couldn’t be greater than a positive number at the same time being less than a negative number. So therefore, we can say that scenario iii 𝑎 is greater than zero must also be true.

So therefore, if 𝑎 and 𝑥 are numbers such that 𝑥 is greater than negative 𝑎 but less than 𝑎, the following must be true. The modulus or absolute value of 𝑥 must be less than 𝑎. So that’s answer i. And 𝑎 must be greater than zero. So that’s answer iii. So we can say that i and iii must be true.