# ISEE Upper Level Quantitative : How to add variables

## Example Questions

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### Example Question #41 : Operations

is a negative integer. Which is the greater quantity?

(A)

(B)

(a) and (b) are equal

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(b) is the greater quantity

(b) is the greater quantity

Explanation:

, so

and

is the greater quantity, regardless of .

### Example Question #42 : Operations

is a positive number.

Which is the greater quantity?

(a)

(b)

It cannot be determined which of (a) and (b) is greater

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

Explanation:

By examining two scenarios, we see that we cannot determine which is the greater quantity.

Case 1:

Then

and

This makes (a) the greater quantity.

Case 2:

Then

and

This makes (b) the greater quantity.

### Example Question #43 : Operations

is a negative number.

Which is the greater quantity?

(a)

(b)

It cannot be determined which of (a) and (b) is greater

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

Explanation:

The expressions  and  are each other's opposite, so they have the same absolute value. Therefore, regardless of the value of

.

### Example Question #891 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

is a negative integer. Which is the greater quantity?

(A)

(B)

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

It cannot be determined which of (a) and (b) is greater

(a) is the greater quantity

Explanation:

,

so

and

regardless of the value of .

### Example Question #892 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

is a negative integer. Which is the greater quantity?

(a)

(b)

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

It cannot be determined which of (a) and (b) is greater

(b) is the greater quantity

Explanation:

Since

,

it follows that

and

regardless of the value of .

### Example Question #893 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Which is the greater quantity?

(a)

(b)

(b) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(a) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

Explanation:

Suppose  is nonnegative.

Then

Consequently,

,

which must be positive,

and

,

which is the opposite of  and consequently must be negative. Therefore, (a) is greater.

Suppose  is negative.

Then .

Consequently,

,

and

.

, so

,

and (a) is greater.

(a) is the greater quantity either way.

### Example Question #11 : How To Add Variables

Define . The graph of  is a line with slope .

.

Which is the greater quantity?

(a)

(b)

(a) and (b) are equal

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

(a) is the greater quantity

(b) is the greater quantity

(a) is the greater quantity

Explanation:

, so .

, so, by definition, , or .

The graph of  is a line through the point with coordinates  and with slope . The equation of the line can be determined by setting  in the slope-intercept form:

.

The equation of the line is , which makes this the definition of . By setting ,

.

Therefore,

### Example Question #891 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

and  are both positive.

Which is the greater quantity?

(a)

(b)

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

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

(b) is the greater quantity

Explanation:

If ,

then

.

The absolute value of a negative number is its (positive) opposite, so

Also, if  and  are both positive, then  is positive; the absolute value of a positive number is the number itself, so . Since , it follows that . Therefore,

Since  is given to be positive,

and

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