# ISEE Upper Level Quantitative : Variables

## Example Questions

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

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

(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 #12 : How To Add Variables

Define . The graph of  is a line with slope .

.

Which is the greater quantity?

(a)

(b)

(b) is the greater quantity

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

(a) and (b) are equal

(a) 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 #13 : How To Add Variables

and  are both positive.

Which is the greater quantity?

(a)

(b)

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

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

(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

### Example Question #1 : Variables And Exponents

Simplify:

The expression cannot be simplified further

Explanation:

Group and combine like terms :

### Example Question #2 : Variables And Exponents

Which is the greater quantity?

(a)

(b)

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

(b) is greater.

Explanation:

Since  and  have different signs,

, and, subsequently,

Therefore,

This makes (b) the greater quantity.

### Example Question #3 : Variables And Exponents

Assume that  and  are not both zero. Which is the greater quantity?

(a)

(b)

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

It is impossible to tell from the information given.

Explanation:

Simplify the expression in (a):

Therefore, whether (a) or (b) is greater depends on the values of  and , neither of which are known.

### Example Question #4 : Variables And Exponents

Which is the greater quantity?

(a)

(b)

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

Explanation:

We give at least one positive value of  for which (a) is greater and at least one positive value of  for which (b) is greater.

Case 1:

(a)

(b)

Case 2:

(a)

(b)

Therefore, either (a) or (b) can be greater.

### Example Question #5 : Variables And Exponents

Assume all variables to be nonzero.

Simplify:

None of the answer choices are correct.

None of the answer choices are correct.

Explanation:

Any nonzero expression raised to the power of 0 is equal to 1. Therefore,

.

None of the given expressions are correct.

Simplify:

Explanation:

### Example Question #7 : Variables And Exponents

Which is greater?

(a)

(b)

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater