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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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