# ISEE Middle Level Quantitative : Algebraic Concepts

## Example Questions

### Example Question #1 : How To Add Variables

Which quantity is greater if ?

and are equal

It is not possible to tell from the information given

is greater

is greater

is greater

Explanation:

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.

### Example Question #1 : How To Add Variables

Which quantity is greater if ?

It is not possible to tell from the information given

is greater

is greater

and are equal

It is not possible to tell from the information given

Explanation:

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.

### Example Question #1 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Which quantity is greater if ?

is greater

and are equal

It is not possible to tell from the information given

is greater

is greater

Explanation:

When we can write:

We know that and . Based on this, we can compare the two given quantities.

is the greater quantity.

### Example Question #1 : How To Add Variables

Which quantity is greater if ?

and are equal

is greater

It is not possible to tell from the information given

is greater

is greater

Explanation:

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 .

### Example Question #1 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Which of the following is equivalent to  ?

Explanation:

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.

### Example Question #1 : How To Add Variables

is a positive integer.

Which is the greater quantity?

(A)

(B)

(A) and (B) are equal

It is impossible to determine which is greater from the information given

(B) is greater

(A) is greater

It is impossible to determine which is greater from the information given

Explanation:

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.

### Example Question #1 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

is a positive integer.

Which is the greater quantity?

(A)

(B)

(A) is greater

(B) is greater

It is impossible to determine which is greater from the information given

(A) and (B) are equal

(A) is greater

Explanation:

Since , and  is positive,

then by the multiplication property of inequality,

making (A) greater regardless of the value of .

### Example Question #1 : Algebraic Concepts

is a positive integer.

Which is the greater quantity?

(A)

(B)

It is impossible to determine which is greater from the information given

(B) is greater

(A) and (B) are equal

(A) is greater

(A) and (B) are equal

Explanation:

Regardless of the value of , the expressions are equal.

### Example Question #1 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Which of the following is equivalent to  ?

None of the other responses is correct.

None of the other responses is correct.

Explanation:

The expression is the sum of two unlike terms, and therefore cannot be further simplified. None of these responses is correct.

### Example Question #2 : How To Add Variables

is a positive integer.

Which is the greater quantity?

(A)

(B)

(A) and (B) are equal

(A) is greater

It is impossible to determine which is greater from the information given

(B) is greater