Algebra - GMAT Quantitative
Card 1 of 1488
Define an operation
on two real numbers as follows:

Is
positive, negative, or zero?
Statement 1:
is negative.
Statement 2:
is positive.
Define an operation on two real numbers as follows:
Is positive, negative, or zero?
Statement 1: is negative.
Statement 2: is positive.
Tap to reveal answer
If we know Statement 1 only - that
is negative - then, since
must be nonnegative,
must be a negative number minus a nonnegative number. This makes
negative.
If we know Statement 2 only - that
is positive - then we do not have a definite answer. For example,

but

If we know Statement 1 only - that is negative - then, since
must be nonnegative,
must be a negative number minus a nonnegative number. This makes
negative.
If we know Statement 2 only - that is positive - then we do not have a definite answer. For example,
but
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Given two functions
and
defined on the set of real numbers, which, if either, is the greater quantity,
or
?
Statement 1: 
Statement 2:
for all real values of
.
Given two functions and
defined on the set of real numbers, which, if either, is the greater quantity,
or
?
Statement 1:
Statement 2: for all real values of
.
Tap to reveal answer
Statement 1 is a consequence of Statement 2, so we need only show that Statement 2 provides insufficient information.
Let
. Then

and
.
However, without knowing the sign of
, we cannot determine whether
or
is the greater quantity.
Statement 1 is a consequence of Statement 2, so we need only show that Statement 2 provides insufficient information.
Let . Then
and
.
However, without knowing the sign of , we cannot determine whether
or
is the greater quantity.
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Define an operation
on two real numbers as follows:

Is
positive, negative, or zero?
Statement 1: 
Statement 2: 
Define an operation on two real numbers as follows:
Is positive, negative, or zero?
Statement 1:
Statement 2:
Tap to reveal answer
is positive or zero, so if Statement 1 is assumed, and
is negative, then
, the product of a negative number and a nonnegative number, must be nonpositive. It can be negative or zero, however.
If Statement 2 is assumed, the expression can be of any sign, since, although
must be positive, no information is given about the sign of
.
If both statements are assumed, then since
is nonzero,
is positive; also,
is negative.
is the product of two numbers of unlike sign, and therefore, it can be determined that it is a negative number.
is positive or zero, so if Statement 1 is assumed, and
is negative, then
, the product of a negative number and a nonnegative number, must be nonpositive. It can be negative or zero, however.
If Statement 2 is assumed, the expression can be of any sign, since, although must be positive, no information is given about the sign of
.
If both statements are assumed, then since is nonzero,
is positive; also,
is negative.
is the product of two numbers of unlike sign, and therefore, it can be determined that it is a negative number.
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Define an operation
on two real numbers as follows:
For all real numbers
,
.
True or false: 
Statement 1: 
Statement 2: 
Define an operation on two real numbers as follows:
For all real numbers ,
.
True or false:
Statement 1:
Statement 2:
Tap to reveal answer

Similarly,
.
Therefore, for
, it must hold that
.




Statement 1 contradicts this; Statement 2 neither confirms nor contradict this.
Similarly, .
Therefore, for , it must hold that
.
Statement 1 contradicts this; Statement 2 neither confirms nor contradict this.
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John is writing out an arithmetic sequence. How many terms does he need to write out before he writes a term greater than or equal to 1,000?
Statement 1: The fourth term is 50.
Statement 2: The twentieth term is 418.
John is writing out an arithmetic sequence. How many terms does he need to write out before he writes a term greater than or equal to 1,000?
Statement 1: The fourth term is 50.
Statement 2: The twentieth term is 418.
Tap to reveal answer
Knowing one term of a sequence will not help you find any other terms, so neither statement alone will answer the question. But knowing two terms and knowing that the sequence is arithmetic will allow you to find the common difference
.
The twentieth term is
greater than the fourth term, so take the difference and divide by 16:

Now solve this inequality for
to find the minimum number of terms needed to exceed 1,000:

Knowing one term of a sequence will not help you find any other terms, so neither statement alone will answer the question. But knowing two terms and knowing that the sequence is arithmetic will allow you to find the common difference .
The twentieth term is greater than the fourth term, so take the difference and divide by 16:
Now solve this inequality for to find the minimum number of terms needed to exceed 1,000:
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Define an operation
on the real numbers as follows:

Is
positive, negative, or zero?
Statement 1: 
Statement 2: 
Define an operation on the real numbers as follows:
Is positive, negative, or zero?
Statement 1:
Statement 2:
Tap to reveal answer
, and
. Therefore, since
is a positive number, the sign of
is the sign of
. This makes Statement 1 neither necessary nor helpful. We need to know whether
is greater than, equal to, or less than 0, or, equivalently, whether
is greater than, equal to, or less than 10. Statement 2 does not tell us this either.
, and
. Therefore, since
is a positive number, the sign of
is the sign of
. This makes Statement 1 neither necessary nor helpful. We need to know whether
is greater than, equal to, or less than 0, or, equivalently, whether
is greater than, equal to, or less than 10. Statement 2 does not tell us this either.
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What is
?
(1) 
(2)

What is ?
(1)
(2)
Tap to reveal answer
Statement (1) does not give us any information about g(x), so it is not sufficient.
Statement (2) alone gives us the relationship between the functions f and g but does not give us any information about f(x), so it is not sufficient.
Both statement together allow us to get an expression for both f and g:


With an expression for both functions we can estimate f(g(1)):
and 
So the correct answer is C.
Statement (1) does not give us any information about g(x), so it is not sufficient.
Statement (2) alone gives us the relationship between the functions f and g but does not give us any information about f(x), so it is not sufficient.
Both statement together allow us to get an expression for both f and g:
With an expression for both functions we can estimate f(g(1)):
and
So the correct answer is C.
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is a real number. True or false:
is positive.
Statement 1: 
Statement 2: 
is a real number. True or false:
is positive.
Statement 1:
Statement 2:
Tap to reveal answer
Assume Statement 1 alone. If
is positive, then
and
; since
is the sum of three positive numbers, then
, and
is a false statement. Therefore,
cannot be positive.
Assume Statement 2. If
is positive, then so is
, and the inequality can be rewritten as

Consequently,

,
a contradiction since
is positive. Therefore,
is not positive.
Assume Statement 1 alone. If is positive, then
and
; since
is the sum of three positive numbers, then
, and
is a false statement. Therefore,
cannot be positive.
Assume Statement 2. If is positive, then so is
, and the inequality can be rewritten as
Consequently,
,
a contradiction since is positive. Therefore,
is not positive.
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Of distinct integers
, which is the greatest of the three?
Statement 1: 
Statement 2:
and
are negative.
Of distinct integers , which is the greatest of the three?
Statement 1:
Statement 2: and
are negative.
Tap to reveal answer
Statement 1 alone gives insufficient information.
Case 1: 


, which is true.
Case 2: 


, which is true.
But in the first case,
is the greatest of the three. In the second,
is the greatest.
Statement 2 gives insuffcient information, since no information is given about the sign of
.
Assume both statements to be true.
, and from Statement 1,
; by transitivity,
. From Statement 2,
. This makes
the greatest of the three.
Statement 1 alone gives insufficient information.
Case 1:
, which is true.
Case 2:
, which is true.
But in the first case, is the greatest of the three. In the second,
is the greatest.
Statement 2 gives insuffcient information, since no information is given about the sign of .
Assume both statements to be true. , and from Statement 1,
; by transitivity,
. From Statement 2,
. This makes
the greatest of the three.
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What are the solutions of
in the most simplified form?
What are the solutions of in the most simplified form?
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This is a quadratic formula problem. Use equation
. For our problem,
Plug these values into the equation, and simplify:
. Here, we simplified the radical by 
This is a quadratic formula problem. Use equation . For our problem,
Plug these values into the equation, and simplify:
. Here, we simplified the radical by
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Solve for 
Statement 1: 
Statement 2: 
Solve for
Statement 1:
Statement 2:
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To solve for three unknowns, we need three equations. Here we have three equations if we use both statements 1 and 2. We don't need to solve any further. Because this is a data sufficiency question, it doesn't matter what the actual values of x, y, and z are. The important fact is the we could find them if we wanted to.
To solve for three unknowns, we need three equations. Here we have three equations if we use both statements 1 and 2. We don't need to solve any further. Because this is a data sufficiency question, it doesn't matter what the actual values of x, y, and z are. The important fact is the we could find them if we wanted to.
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What is the value of z?
Statement 1: 
Statement 2: 
What is the value of z?
Statement 1:
Statement 2:
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To solve for three variables, you must have three equations. Statements 1 and 2 together only give two equations, so the statements together are not sufficient.
To solve for three variables, you must have three equations. Statements 1 and 2 together only give two equations, so the statements together are not sufficient.
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Is the equation linear?
Statement 1: 
Statement 2:
is a constant
Is the equation linear?
Statement 1:
Statement 2: is a constant
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If we only look at statement 1, we might think the equation is not linear because of the
term. But statement 2 tells us the
is a constant. Then the equation is linear. We need both statements to answer this question.
If we only look at statement 1, we might think the equation is not linear because of the term. But statement 2 tells us the
is a constant. Then the equation is linear. We need both statements to answer this question.
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Data sufficiency question- do not actually solve the question
Solve for
:

1. 
2. 
Data sufficiency question- do not actually solve the question
Solve for :
1.
2.
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When solving an equation with 2 variables, a second equation or the solution of 1 variable is necessary to solve.
When solving an equation with 2 variables, a second equation or the solution of 1 variable is necessary to solve.
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Data Sufficiency Question
Solve for
and
.

1. 
2. Both
and
are positive integers
Data Sufficiency Question
Solve for and
.
1.
2. Both and
are positive integers
Tap to reveal answer
Using statement 1 we can set up a series of equations and solve for both
and
. 
Additionally, the information in statement 2 indicates that there is only one possible solution that satisfies the requirement that both are positive integers.
Using statement 1 we can set up a series of equations and solve for both and
.
Additionally, the information in statement 2 indicates that there is only one possible solution that satisfies the requirement that both are positive integers.
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Solve the following for x:
4x+7y = 169
1. x > y
2. x - y = 12
Solve the following for x:
4x+7y = 169
1. x > y
2. x - y = 12
Tap to reveal answer
To solve with 2 unknowns, we must create a system of equations with at least 2 equations. Using statement 2 as a second equation we can easily get our answer. Solve statement 2 for x or y, and plug in for the corresponding variable in the equation given by the problem.
So, solving statement 2 for x, we get x=12+y. Replacing x in the equation from the problem, we get 4(12+y) + 7y=169. We can distribute the 4, and combine terms to find 48+11y=169. Subtract 48 from both sides, we get 11y=121. So y=11. Reusing either equation and plugging in our y value gives our x value. So x - 11=12, or x=23. This shows that x > y, and statement 1 is true. But even though it's true, it is completely unneccessary information. Therefore the answer is that we only need the information from statement 2, and statement 1 is not needed.
To solve with 2 unknowns, we must create a system of equations with at least 2 equations. Using statement 2 as a second equation we can easily get our answer. Solve statement 2 for x or y, and plug in for the corresponding variable in the equation given by the problem.
So, solving statement 2 for x, we get x=12+y. Replacing x in the equation from the problem, we get 4(12+y) + 7y=169. We can distribute the 4, and combine terms to find 48+11y=169. Subtract 48 from both sides, we get 11y=121. So y=11. Reusing either equation and plugging in our y value gives our x value. So x - 11=12, or x=23. This shows that x > y, and statement 1 is true. But even though it's true, it is completely unneccessary information. Therefore the answer is that we only need the information from statement 2, and statement 1 is not needed.
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How many solutions does this system of equations have: one, none, or infinitely many?


Statement 1: 
Statement 2: 
How many solutions does this system of equations have: one, none, or infinitely many?
Statement 1:
Statement 2:
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If the slopes of the lines are not equal, then the lines intersect at one solution; if they are equal, then they do not intersect, or the lines are the same line. Write each equation in slope-intercept form,
:









The slopes of the lines are
.
We need to know both
and
in order to determine their equality or inequality, and only if they are unequal can we answer the question.
Set
and
.


The slopes are unequal, so the lines intersect at one point; the system has exactly one solution.
If the slopes of the lines are not equal, then the lines intersect at one solution; if they are equal, then they do not intersect, or the lines are the same line. Write each equation in slope-intercept form, :
The slopes of the lines are .
We need to know both and
in order to determine their equality or inequality, and only if they are unequal can we answer the question.
Set and
.
The slopes are unequal, so the lines intersect at one point; the system has exactly one solution.
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Given that both
, how many solutions does this system of equations have: one, none, or infinitely many?


Statement 1: 
Statement 2: 
Given that both , how many solutions does this system of equations have: one, none, or infinitely many?
Statement 1:
Statement 2:
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If the slopes of the lines are not equal, then the lines intersect at one solution. If the slopes are equal, then there are two possibilties: either they do not intersect or they are the same line. Write each equation in slope-intercept form:










The slopes of these lines are
.
If Statement 1 is true, then we can rewrite the first slope as
, meaning that the lines have unequal slopes, and that there is only one solution. Statement 2 tells us the value of
, which is irrelevant.
If the slopes of the lines are not equal, then the lines intersect at one solution. If the slopes are equal, then there are two possibilties: either they do not intersect or they are the same line. Write each equation in slope-intercept form:
The slopes of these lines are .
If Statement 1 is true, then we can rewrite the first slope as , meaning that the lines have unequal slopes, and that there is only one solution. Statement 2 tells us the value of
, which is irrelevant.
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is an integer. Is there a real number
such that
?
Statement 1:
is negative
Statement 2:
is even
is an integer. Is there a real number
such that
?
Statement 1: is negative
Statement 2: is even
Tap to reveal answer
The equivalent question is "does
have a real
root?"
If you know only that
is negative, you need to know whether
is even or odd; negative numbers have real odd-numbered roots, but not real even-numbered roots.
If you know only that
is even, you need to know whether
is negative or nonnegative; negative numbers do not have real even-numbered roots, but nonnegative numbers do.
If you know both, however, then you know that the answer is no, since as stated before, negative numbers do not have real even-numbered roots.
Therefore, the answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.
The equivalent question is "does have a real
root?"
If you know only that is negative, you need to know whether
is even or odd; negative numbers have real odd-numbered roots, but not real even-numbered roots.
If you know only that is even, you need to know whether
is negative or nonnegative; negative numbers do not have real even-numbered roots, but nonnegative numbers do.
If you know both, however, then you know that the answer is no, since as stated before, negative numbers do not have real even-numbered roots.
Therefore, the answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.
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Data Sufficiency Question
Solve for
and
:

1. 
2. 
Data Sufficiency Question
Solve for and
:
1.
2.
Tap to reveal answer
In order to solve an equation set, one requires a number of equations equal to the number of variables. Therefore, either of the statements allow the problem to be solved.
In order to solve an equation set, one requires a number of equations equal to the number of variables. Therefore, either of the statements allow the problem to be solved.
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