Must Be True Questions | Algebra

Often, Quant questions are of two types: Problem Solving and Data Sufficiency.

But, if we look carefully, we will see a third type of question – combination of the two. There are a few statements given in them (like in Data Sufficiency questions) and five options to choose from (like in Problem Solving questions). But since we know how to solve both these question types, we shouldn’t really have a problem in solving this third type, or so one would think!

In any Quant question, it is very important to know two things:

1. What is given

2. What is asked

Now, one might think that it is a very obvious distinction and why are we even trying to discuss it in a post. In this third question type, this exact distinction is far harder to explain because here the statements do NOT represent the data given. Here the statements actually ask “Is this true?” and many test-takers find it hard to make that switch. To clarify, let’s discuss the structure of the three question types.

Type I – Problem Solving Question:

Question: $A$ and $B$ are given, what is $X$ ?

(A) $X$ is …

(B) $X$ is …

(D) $X$ is …

(E) $X$ is …

Type II – Data Sufficiency Question:

Question: $A$ and $B$ are given, what is $X$ ?

I. We are given that $X$ and $Y$ are related.

II. We are given that $X$ and $Z$ are related.

Type III – “Which of the following must be true?” Question:

Question: $A$ and $B$ are given, which of the following must be true about $X$ ?

I. Is this true about $X$ ?

II. Is this true about $X$ ?

III. Is this true about $X$ ?

(A) I is true

(B) I and II are true

and so on…

We hope you see that the statements in a Data Sufficiency question are different from the statements in this third type of question.
We will elaborate with the help of an example now:

Question: If ${|x| > 3}$ , which of the following must be true?

I. ${x > 3}$

II. ${x^2 > 9}$

III. ${|x - 1| > 2}$

(A) I only

(B) II only

(D) II and III only

(E) I, II, and III

Solution:

We are given that $|x| > 3$

This implies that $x$ is a point at a distance of more than 3 from 0. So $x$ could be greater than 3 or less than -3. Before we go any further, let’s think about the values $x$ can take: 3.00001, 3.5, 4.2, 5.7, 67, 1000, -3.45, -4, -8, -100 etc. The only values it cannot take are $-3 \le x \le 3$

Which of the following must be true?

I. $x > 3$

This is a question even though it looks like a statement.

Is it necessary that $x > 3$ ?

For every value that $x$ can take, must $x$ be greater than 3? No. As discussed above, $x$ could take values such as 3.00001, 3.5, 4.2, 5.7, 67, 1000 but it could also take values such as -3.45, -4, -8, -100.

So this is not necessarily true.

II. $x^2 > 9$

Again, this is a question even though it looks like a statement.

Taking square root on both sides since they are positive, we get

$\sqrt{x^2} > \sqrt9$

$|x| > 3$

This is what we are given, hence it certainly is true.

III. $|x-1|>2$

Yet again, we are asked: Is $|x - 1| > 2$ ?

What does $|x - 1|> 2$ imply?

The distance of $x$ from 1 must be greater than 2. So $x$ is either greater than 3 or less than -1. Now, recall all the values that $x$ can take.

So this is the question now: Is every value that $x$ can take greater than 3 or less than -1?

Recall the values that $x$ can take (discussed above)

3.00001 : $x$ is greater than 3

3.5 : $x$ is greater than 3

4.2 : $x$ is greater than 3

5.7 : $x$ is greater than 3

67 : $x$ is greater than 3

1000 : $x$ is greater than 3

-3.45 : $x$ is less than -1

-4 : $x$ is less than -1

-8 : $x$ is less than -1

-100 : $x$ is less than -1

For every value that $x$ can take, $x$ will be either greater than 3 or less than -1. Note that we are not saying that every value less than -1 must be valid for $x$ . We are saying that every value that is valid for $x$ (found by using $|x| > 3$ ) will be either greater than 3 or less than -1 since any value less than -3 is obviously less than -1 too. Hence $|x-1|>2$ must be true for every value that $x$ can take.

Answer (D).

We hope you are quite clear about how to handle this third question type now!

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