### All ISEE Upper Level Quantitative Resources

## Example Questions

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

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(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

**Correct answer:**

(a) is the greater quantity

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)

**Possible Answers:**

(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

**Correct answer:**

(a) is the greater quantity

, 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)

**Possible Answers:**

(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

**Correct answer:**

(b) is the greater quantity

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:

**Possible Answers:**

The expression cannot be simplified further

**Correct answer:**

Group and combine like terms :

### Example Question #2 : Variables And Exponents

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

**Correct answer:**

(b) is greater.

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)

**Possible Answers:**

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

**Correct answer:**

It is impossible to tell from the information given.

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)

**Possible Answers:**

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

(b) is greater

**Correct answer:**

It is impossible to tell from the information given

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:

**Possible Answers:**

None of the answer choices are correct.

**Correct answer:**

None of the answer choices are correct.

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

.

None of the given expressions are correct.

### Example Question #6 : Variables And Exponents

Simplify:

**Possible Answers:**

**Correct answer:**

### Example Question #7 : Variables And Exponents

Which is greater?

(a)

(b)

**Possible Answers:**

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

**Correct answer:**

(b) is greater

If , then and

, so by transitivity, , and (b) is greater

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