GMAT QUANTITATIVE REASONING

INEQUALITIES &

NUMBER PROPERTIES

DATA SUFFICIENCY

Diagnostic Test

Question

Is x > y ?

Statement 1: x + y > x – y

Statement 2: x + y < -(x + y)

Step 1

Jot down answers to these 3 questions

before looking at the statements

Is x > y?We will not even look at the statements while answering the following questions

Is x > y?We will not even look at the statements while answering the following questions

When is the data sufficient?

Is x > y?We will not even look at the statements while answering the following questions

When is the data sufficient?

It is an “is” question.

Is x > y?We will not even look at the statements while answering the following questions

When is the data sufficient?

For any “is” question, the data is sufficient when we can answer the question with a definite yes or a definite no.

It is an “is” question.

Is x > y?We will not even look at the statements while answering the following questions

When is the data sufficient? When is it an yes and when no?

For any “is” question, the data is sufficient when we can answer the question with a definite yes or a definite no.

It is an “is” question.

Is x > y?We will not even look at the statements while answering the following questions

When is the data sufficient? When is it an yes and when no?

For any “is” question, the data is sufficient when we can answer the question with a definite yes or a definite no.

It is an “is” question. In this question, the answer is yes when

x > y

Is x > y?We will not even look at the statements while answering the following questions

When is the data sufficient? When is it an yes and when no?

For any “is” question, the data is sufficient when we can answer the question with a definite yes or a definite no.

It is an “is” question. In this question, the answer is yes when

x > y

In this question, the answer is no when

a. x < y or when b. x = y

Is x > y?We will not even look at the statements while answering the following questions

When is the data sufficient? When is it an yes and when no? What do we know about x & y?

For any “is” question, the data is sufficient when we can answer the question with a definite yes or a definite no.

It is an “is” question. In this question, the answer is yes when

x > y

In this question, the answer is no when

a. x < y or when b. x = y

Is x > y?We will not even look at the statements while answering the following questions

When is the data sufficient? When is it an yes and when no? What do we know about x & y?

For any “is” question, the data is sufficient when we can answer the question with a definite yes or a definite no.

It is an “is” question. In this question, the answer is yes when

x > y

In this question, the answer is no when

a. x < y or when b. x = y

No additional information is available about x and y.

Is x > y?We will not even look at the statements while answering the following questions

When is the data sufficient? When is it an yes and when no? What do we know about x & y?

For any “is” question, the data is sufficient when we can answer the question with a definite yes or a definite no.

It is an “is” question. In this question, the answer is yes when

x > y

In this question, the answer is no when

a. x < y or when b. x = y

No additional information is available about x and y.

So, x and y belong to the set of Real numbers.

They could both be positive, negative, integers, fractions, irrational.

Step 2

Let’s evaluate statement 1 alone

Is x > y?Statement 1: x + y > x – y

Is x > y?Statement 1: x + y > x – y

x + y > x – y

Is x > y?Statement 1: x + y > x – y

x + y > x – y

i.e., 2y > 0 or y > 0

Is x > y?Statement 1: x + y > x – y

x + y > x – y

i.e., 2y > 0 or y > 0

From the statement we can deduce that y is positive.

Is x > y?Statement 1: x + y > x – y

x + y > x – y

i.e., 2y > 0 or y > 0

From the statement we can deduce that y is positive.

However, no information is available about x and its relation to y.

Is x > y?Statement 1: x + y > x – y

x + y > x – y

i.e., 2y > 0 or y > 0

From the statement we can deduce that y is positive.

However, no information is available about x and its relation to y.

Hence, we will not be able to determine whether x > y.

Is x > y?Statement 1: x + y > x – y

Statement 1 alone is NOT sufficient

x + y > x – y

i.e., 2y > 0 or y > 0

From the statement we can deduce that y is positive.

However, no information is available about x and its relation to y.

Hence, we will not be able to determine whether x > y.

Is x > y?Statement 1: x + y > x – y

Eliminate choices A and DStatement 1 alone is NOT sufficient

x + y > x – y

i.e., 2y > 0 or y > 0

From the statement we can deduce that y is positive.

However, no information is available about x and its relation to y.

Hence, we will not be able to determine whether x > y.

Is x > y?Statement 1: x + y > x – y

Choices narrow down to B, C or E.

Eliminate choices A and DStatement 1 alone is NOT sufficient

x + y > x – y

i.e., 2y > 0 or y > 0

From the statement we can deduce that y is positive.

However, no information is available about x and its relation to y.

Hence, we will not be able to determine whether x > y.

Step 3

Let’s evaluate statement 2 alone.

Is x > y?Statement 2 : x + y < -(x + y)

Is x > y?Statement 2 : x + y < -(x + y)

x + y < -(x + y)

Is x > y?Statement 2 : x + y < -(x + y)

x + y < -(x + y)

i.e., 2(x + y) < 0 or x + y < 0

Is x > y?Statement 2 : x + y < -(x + y)

x + y < -(x + y)

i.e., 2(x + y) < 0 or x + y < 0

If x + y < 0, what can we infer about x and y?

Is x > y?Statement 2 : x + y < -(x + y)

x + y < -(x + y)

i.e., 2(x + y) < 0 or x + y < 0

If x + y < 0, what can we infer about x and y?

Possibility 1: Both x and y are negative

Is x > y?Statement 2 : x + y < -(x + y)

x + y < -(x + y)

i.e., 2(x + y) < 0 or x + y < 0

If x + y < 0, what can we infer about x and y?

Possibility 1: Both x and y are negative

Possibility 2: One of x or y is positive and the other is negative

Is x > y?Statement 2 : x + y < -(x + y)

x + y < -(x + y)

i.e., 2(x + y) < 0 or x + y < 0

If x + y < 0, what can we infer about x and y?

Possibility 1: Both x and y are negative

Possibility 2: One of x or y is positive and the other is negative

What is the approach? Look for a counter example: Pick two sets of values satisfying the condition in statement 2. If 1 set provides an answer yes and the other set provides a no, the data is insufficient.

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Both x and y are negative

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Both x and y are negative

Set 1: x = -2, y = -3. x + y = -5

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Both x and y are negative

Set 1: x = -2, y = -3. x + y = -5

Satisfies statement 2.

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Both x and y are negative

Set 1: x = -2, y = -3. x + y = -5

Satisfies statement 2.

Yes

x > y

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Both x and y are negative

Set 1: x = -2, y = -3. x + y = -5

Satisfies statement 2.

Set 2: x = -3, y = -2. x + y = -5

Yes

x > y

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Both x and y are negative

Set 1: x = -2, y = -3. x + y = -5

Satisfies statement 2.

Set 2: x = -3, y = -2. x + y = -5

This set also satisfies statement 2.

Yes

x > y

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Both x and y are negative

Set 1: x = -2, y = -3. x + y = -5

Satisfies statement 2.

Set 2: x = -3, y = -2. x + y = -5

This set also satisfies statement 2.

Yes

x > y

No

x < y

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Both x and y are negative

Statement 2 alone is also NOT sufficient

Set 1: x = -2, y = -3. x + y = -5

Satisfies statement 2.

Set 2: x = -3, y = -2. x + y = -5

This set also satisfies statement 2.

Yes

x > y

No

x < y

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Both x and y are negative

Eliminate choice B Statement 2 alone is also NOT sufficient

Set 1: x = -2, y = -3. x + y = -5

Satisfies statement 2.

Set 2: x = -3, y = -2. x + y = -5

This set also satisfies statement 2.

Yes

x > y

No

x < y

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Both x and y are negative

Choices narrow down to C or E.

Eliminate choice B Statement 2 alone is also NOT sufficient

Set 1: x = -2, y = -3. x + y = -5

Satisfies statement 2.

Set 2: x = -3, y = -2. x + y = -5

This set also satisfies statement 2.

Yes

x > y

No

x < y

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Both x and y are negative One positive and one negative.

Choices narrow down to C or E.

Eliminate choice B Statement 2 alone is also NOT sufficient

Set 1: x = -2, y = -3. x + y = -5

Satisfies statement 2.

Set 2: x = -3, y = -2. x + y = -5

This set also satisfies statement 2.

Yes

x > y

No

x < y

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Both x and y are negative One positive and one negative.

Choices narrow down to C or E.

Eliminate choice B Statement 2 alone is also NOT sufficient

Set 1: x = -2, y = -3. x + y = -5

Satisfies statement 2.

Set 2: x = -3, y = -2. x + y = -5

This set also satisfies statement 2.

Yes

x > y

No

x < y

Set 1: x = 2, y = -3. x + y = -1

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Both x and y are negative One positive and one negative.

Choices narrow down to C or E.

Eliminate choice B Statement 2 alone is also NOT sufficient

Set 1: x = -2, y = -3. x + y = -5

Satisfies statement 2.

Set 2: x = -3, y = -2. x + y = -5

This set also satisfies statement 2.

Yes

x > y

No

x < y

Set 1: x = 2, y = -3. x + y = -1

Satisfies statement 2.

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Both x and y are negative One positive and one negative.

Choices narrow down to C or E.

Eliminate choice B Statement 2 alone is also NOT sufficient

Set 1: x = -2, y = -3. x + y = -5

Satisfies statement 2.

Set 2: x = -3, y = -2. x + y = -5

This set also satisfies statement 2.

Yes

x > y

No

x < y

Set 1: x = 2, y = -3. x + y = -1

Satisfies statement 2.

Yes

x > y

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Both x and y are negative One positive and one negative.

Choices narrow down to C or E.

Eliminate choice B Statement 2 alone is also NOT sufficient

Set 1: x = -2, y = -3. x + y = -5

Satisfies statement 2.

Set 2: x = -3, y = -2. x + y = -5

This set also satisfies statement 2.

Yes

x > y

No

x < y

Set 1: x = 2, y = -3. x + y = -1

Satisfies statement 2.

Yes

x > y

Set 2: x = -3, y = 2. x + y = -1

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Both x and y are negative One positive and one negative.

Choices narrow down to C or E.

Eliminate choice B Statement 2 alone is also NOT sufficient

Set 1: x = -2, y = -3. x + y = -5

Satisfies statement 2.

Set 2: x = -3, y = -2. x + y = -5

This set also satisfies statement 2.

Yes

x > y

No

x < y

Set 1: x = 2, y = -3. x + y = -1

Satisfies statement 2.

Yes

x > y

Set 2: x = -3, y = 2. x + y = -1

This set also satisfies statement 2.

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Both x and y are negative One positive and one negative.

Choices narrow down to C or E.

Eliminate choice B Statement 2 alone is also NOT sufficient

Set 1: x = -2, y = -3. x + y = -5

Satisfies statement 2.

Set 2: x = -3, y = -2. x + y = -5

This set also satisfies statement 2.

Yes

x > y

No

x < y

Set 1: x = 2, y = -3. x + y = -1

Satisfies statement 2.

Yes

x > y

Set 2: x = -3, y = 2. x + y = -1

This set also satisfies statement 2.

No

x < y

Is x > y?Statement 2 : x + y < -(x + y). Simplifies as x + y < 0

Both x and y are negative One positive and one negative.

Choices narrow down to C or E.

Eliminate choice B Statement 2 alone is also NOT sufficient

Set 1: x = -2, y = -3. x + y = -5

Satisfies statement 2.

Set 2: x = -3, y = -2. x + y = -5

This set also satisfies statement 2.

Yes

x > y

No

x < y

Set 1: x = 2, y = -3. x + y = -1

Satisfies statement 2.

Yes

x > y

Set 2: x = -3, y = 2. x + y = -1

This set also satisfies statement 2.

No

x < y

Evaluating one positive and one negative is NOT needed as we have already proved insufficiency when both x and y

are negative. We have done it only to illustrate how to evaluate such a case.

Step 4

Let’s combine data from both the

statements.

Is x > y?Statements Together : x + y > x – y and x + y < -(x + y)

Is x > y?Statements Together : x + y > x – y and x + y < -(x + y)

Gist of statement 1

Is x > y?Statements Together : x + y > x – y and x + y < -(x + y)

Gist of statement 1

y > 0

Is x > y?Statements Together : x + y > x – y and x + y < -(x + y)

Gist of statement 1 Gist of statement 2

y > 0

Is x > y?Statements Together : x + y > x – y and x + y < -(x + y)

Gist of statement 1 Gist of statement 2

y > 0 x + y < 0

Is x > y?Statements Together : x + y > x – y and x + y < -(x + y)

Gist of statement 1 Gist of statement 2 Taken together

y > 0 x + y < 0

Is x > y?Statements Together : x + y > x – y and x + y < -(x + y)

Gist of statement 1 Gist of statement 2 Taken together

y > 0 x + y < 0 ‘y’ is positive

Is x > y?Statements Together : x + y > x – y and x + y < -(x + y)

Gist of statement 1 Gist of statement 2 Taken together

y > 0 x + y < 0 ‘y’ is positive

x + y is negative

Is x > y?Statements Together : x + y > x – y and x + y < -(x + y)

Gist of statement 1 Gist of statement 2 Taken together

y > 0 x + y < 0 ‘y’ is positive

x + y is negative

So, x has to be negative

Is x > y?Statements Together : x + y > x – y and x + y < -(x + y)

Gist of statement 1 Gist of statement 2 Taken together

y > 0 x + y < 0 ‘y’ is positive

x + y is negative

So, x has to be negative

If y is positive and x is negative, x < y. Answer definite NO.

Is x > y?Statements Together : x + y > x – y and x + y < -(x + y)

Gist of statement 1 Gist of statement 2 Taken together

Statements together are SUFFICIENT

y > 0 x + y < 0 ‘y’ is positive

x + y is negative

So, x has to be negative

If y is positive and x is negative, x < y. Answer definite NO.

Is x > y?Statements Together : x + y > x – y and x + y < -(x + y)

Gist of statement 1 Gist of statement 2 Taken together

Eliminate choice EStatements together are SUFFICIENT

y > 0 x + y < 0 ‘y’ is positive

x + y is negative

So, x has to be negative

If y is positive and x is negative, x < y. Answer definite NO.

Is x > y?Statements Together : x + y > x – y and x + y < -(x + y)

Gist of statement 1 Gist of statement 2 Taken together

Answer is choice C

Eliminate choice EStatements together are SUFFICIENT

y > 0 x + y < 0 ‘y’ is positive

x + y is negative

So, x has to be negative

If y is positive and x is negative, x < y. Answer definite NO.

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