# Video: SAT Practice Test 1 β’ Section 4 β’ Question 12

Let π and π₯ be numbers such that βπ < π₯ < π. Which of the following must be true? [A] |π₯| < π [B] π₯ > 0 [C] π > 0

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### Video Transcript

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

Well, weβre gonna start by looking at the first possible scenario. And that is that the modulus or absolute value of π₯ is less than π. Well, the modulus of π₯ or absolute value of π₯ is the first thing to consider, because what does this mean?

Well, the modulus or absolute value of π₯ tells us that weβre only interested in the positive values of π₯, because the absolute value of a number is just the distance. So we consider it as a distance away from a point. So therefore, itβs not going to include the negative values.

So therefore, if this is the case, this first situation is going to be true. And thatβs because weβre told in our inequality that π₯ is less than π. But weβre also told that π₯ is greater than negative π. But therefore, if weβre only looking at the positive π₯-values, then this must be less than π. And this is because there are no values of π₯ that would satisfy the inequality π₯ is less than π but greater than negative π if the modulus or absolute value of π₯ wasnβt less than π.

Okay, letβs move on to number ii π₯ is greater than zero. Well, if we look at π₯ is greater than zero, then this isnβt necessarily true. And letβs take this example. If π₯ was equal to negative one, π could be two. And thatβs because negative one would be less than two. So thatβd be correct. But negative one is greater than negative two. So it would fulfill our inequality. So in that case, number ii π₯ is greater than zero is not necessarily true.

So now letβs take a look at π is greater than zero, number iii. Well, letβs consider this. Letβs consider if π was equal to negative one. So π wasnβt greater than zero. Then weβd have a scenario where π₯ will be greater than negative negative one but less than negative one. So therefore, π₯ would have to be greater than one β and thatβs because negative negative one is positive one β but less than negative one. Well, this would be impossible because π₯ couldnβ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 π is greater than zero must also be true.

So therefore, if π and π₯ are numbers such that π₯ is greater than negative π but less than π, the following must be true. The modulus or absolute value of π₯ must be less than π. So thatβs answer i. And π must be greater than zero. So thatβs answer iii. So we can say that i and iii must be true.