### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #31 : Algebraic Concepts

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(b) is greater

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

**Correct answer:**

(a) and (b) are equal

Multiply both sides of the top equation by 3, and subtract both sides of the second equation.

Now substitute to find :

The two are equal.

### Example Question #32 : Algebraic Concepts

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

**Correct answer:**

(b) is greater

Triple both sides of the top equation, and subtract both sides of the bottom equation:

Now substitute to find :

This makes

### Example Question #33 : Algebraic Concepts

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

(b) is greater

**Correct answer:**

It is impossible to tell from the information given

We show that it cannot be determined which of and , if either, is greater, by showing one case in which and one case in which .

Case 1: . Then

and .

Case 2: . Then

and .

### Example Question #31 : How To Find The Solution To An Equation

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) is greater

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

**Correct answer:**

(a) is greater

If , then, since is nonnegative, .

If , then , so the equation becomes . Therefore, .

Either way, (a) is greater.

### Example Question #35 : Algebraic Concepts

Which is the greater quantity?

(a)

(b) 50

**Possible Answers:**

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

(a) is greater

**Correct answer:**

(a) and (b) are equal

By the commutative property,

By the multiplication property of equality,

By distribution,

### Example Question #36 : Algebraic Concepts

is a positive number. Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) is greater

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

**Correct answer:**

(b) is greater

If , then

If , then

If , then

Therefore, regardless of the value of , as long as is positive, , and (b) is greater.

### Example Question #37 : Algebraic Concepts

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

**Correct answer:**

(b) is greater.

From the multiplication property of equality,

.

By distribution,

.

### Example Question #38 : Algebraic Concepts

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(b) is greater.

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

**Correct answer:**

(a) is greater.

Take the absolute value of both sides:

This makes (a) greater.

### Example Question #39 : Algebraic Concepts

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

**Correct answer:**

(a) and (b) are equal.

(a)

(b)

Rewrite as a fraction:

### Example Question #40 : Algebraic Concepts

Define as follows:

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

**Correct answer:**

(b) is greater.

(a) can be evaluated by using the definition of for positive :

can be evaluated by using the definition of for nonpositive :

Add:

(b) can be evaluated by using the definition of for nonpositive :

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