How to add variables - ISEE Middle Level Quantitative Reasoning
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Which quantity is greater if 
?
Which quantity is greater if
We know that 
is always positive for all values of 
. Therefore 
would be negative for all values of 
. From this conclusion, we know:
So we have:
is the greater quantity.
We know that
So we have:
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Which quantity is greater if 
?
Which quantity is greater if
A positive number raised to the third power will be positive, while a negative number raised to the third power will remain negative.
If 
, then 
and 
.
If 
, then 
and 
.
Since we do not know if 
is positive or negative, we cannot draw a conclusion about which option is greater.
If 
, then 
is greater.
If 
, then 
is greater.
A positive number raised to the third power will be positive, while a negative number raised to the third power will remain negative.
If
If
Since we do not know if
If
If
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Which quantity is greater if 
?
Which quantity is greater if
When 
we can write:
We know that 
and 
. Based on this, we can compare the two given quantities.
is the greater quantity.
When
We know that
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Which quantity is greater if 
?
Which quantity is greater if
We know that 
is greater than 
. We can easily test a few values for to determine if the values are increasing or decreasing.
If 
:
If 
:
If 
:
The value of 
is increasing, with the smallest possible value being 
. From this, we know that 
, so 
.
We know that
If
If
If
The value of
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Which of the following is equivalent to 
?
Which of the following is equivalent to
Using the distributive property:
and
Using the associative property of multiplication:
We can rewrite 
as 
; using the commutative and associative properties of multiplication:
is the sum of unlike terms and cannot be simplified.
is the correct choice.
Using the distributive property:
and
Using the associative property of multiplication:
We can rewrite
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Which of the following is equivalent to 
?
Which of the following is equivalent to
The expression is the sum of two unlike terms, and therefore cannot be further simplified. None of these responses is correct.
The expression is the sum of two unlike terms, and therefore cannot be further simplified. None of these responses is correct.
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is a positive integer.
Which is the greater quantity?
(A) 
(B) 
Which is the greater quantity?
(A)
(B)
Depending on the value of 
, it is possible for either expression to be greater or for both to be equal.
Case 1: 
and
So the two are equal.
Case 2: 
and
So (B) is greater.
The correct response is that it cannot be determined which is greater.
Depending on the value of
Case 1:
and
So the two are equal.
Case 2:
and
So (B) is greater.
The correct response is that it cannot be determined which is greater.
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is a positive integer.
Which is the greater quantity?
(A) 
(B) 
Which is the greater quantity?
(A)
(B)
Since 
, 
, so (A) is greater regardless of the value of 
.
Since
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is a positive integer.
Which is the greater quantity?
(A) 
(B) 
Which is the greater quantity?
(A)
(B)
Since 
, and 
is positive,
then by the multiplication property of inequality,
making (A) greater regardless of the value of 
.
Since
then by the multiplication property of inequality,
making (A) greater regardless of the value of
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is a positive integer.
Which is the greater quantity?
(A) 
(B) 
Which is the greater quantity?
(A)
(B)
Regardless of the value of 
, the expressions are equal.
Regardless of the value of
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Define an operation on the real numbers as follows:
For all real values of 
and 
,
is a positive number. Which is the greater quantity?
(a) 
(b) 
Define an operation on the real numbers as follows:
For all real values of
(a)
(b)
so
and
The two are equal regardless of the value of 
.
so
and
The two are equal regardless of the value of
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Which is the greater quantity?
(a) 
(b) 18
Which is the greater quantity?
(a)
(b) 18
The information is insufficient, as we see by exploring two cases:
Case 1: 
Case 2: 
Remember, the three variables need not stand for whole numbers.
The information is insufficient, as we see by exploring two cases:
Case 1:
Case 2:
Remember, the three variables need not stand for whole numbers.
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and 
are both negative numbers. Which is the greater quantity?
(a) 
(b) 
(a)
(b)
The two quantities are equal regardless of the values of 
and 
. To see this, we note that
and
Therefore, by the addition property of equality,
The two quantities are equal regardless of the values of
and
Therefore, by the addition property of equality,
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Simplify
Simplify
In order to add variables the terms must be like. In order for terms to be like, the variables must be exactly alike also being raised to the same power by the exponent.
In this case the like terms are 
and 
. Just because there is a 1 in the exponent for the first term doesnt mean it is different from the second term. With exponents if a variable does not show an exponent, that means it is still to the first power.
We add the coefficients of the like terms. The coefficient is the number in front of the first variable, in this case it is 1 for both terms because of the identity property of multiplication stating any variable, term, or number multiplied by 1 is itself.
Our last term is not like because the 
variable is raised to a different power than the other two. In this case we do not combine it to the like terms, we just add it to the end of the term.
In order to add variables the terms must be like. In order for terms to be like, the variables must be exactly alike also being raised to the same power by the exponent.
In this case the like terms are
We add the coefficients of the like terms. The coefficient is the number in front of the first variable, in this case it is 1 for both terms because of the identity property of multiplication stating any variable, term, or number multiplied by 1 is itself.
Our last term is not like because the
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Simplify the following:
Simplify the following:
When solving this problem we need to remember our order of operations, or PEMDAS.
PEMDAS stands for parentheses, exponents, multiplication/division, and addition/subtraction. When you have a problem with several different operations, you need to solve the problem in this order and you work from left to right for multiplication/division and addition/subtraction.
Parentheses: We are not able to add a variable to a number, so we move to the next step.
Multiplication: We can distribute (or multiply) the 
.
Addition/Subtraction: Remember, we can't add a variable to a number, so the 
is left alone.
Now we have 
When solving this problem we need to remember our order of operations, or PEMDAS.
PEMDAS stands for parentheses, exponents, multiplication/division, and addition/subtraction. When you have a problem with several different operations, you need to solve the problem in this order and you work from left to right for multiplication/division and addition/subtraction.
Parentheses: We are not able to add a variable to a number, so we move to the next step.
Multiplication: We can distribute (or multiply) the
Addition/Subtraction: Remember, we can't add a variable to a number, so the
Now we have
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Add the numbers and keep the variable:
Answer: 
Add the numbers and keep the variable:
Answer:
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Add the numbers and keep the variable:
Answer: 
Add the numbers and keep the variable:
Answer:
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Add the numbers and keep the variable:
Answer: 
Add the numbers and keep the variable:
Answer:
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Simplify:
Simplify:
First, group together your like variables:
The only like variables needing to be combined are the x-variables. You can do this in steps or all at once:
First, group together your like variables:
The only like variables needing to be combined are the x-variables. You can do this in steps or all at once:
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Simplify:
Simplify:
First, move the like terms to be next to each other:
Now, combine the x-variables and the y-variables:
First, move the like terms to be next to each other:
Now, combine the x-variables and the y-variables:
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