# ISEE Upper Level Quantitative : How to find the solution to an equation

## Example Questions

### Example Question #31 : Algebraic Concepts

Which is the greater quantity?

(a)

(b)

(b) is greater

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

Explanation:

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)

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

(b) is greater

Explanation:

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)

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

(b) is greater

It is impossible to tell from the information given

Explanation:

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)

(a) is greater

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

(a) is greater

Explanation:

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

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

(a) is greater

(a) and (b) are equal

Explanation:

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)

(a) is greater

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

(b) is greater

Explanation:

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)

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

(b) is greater.

Explanation:

From the multiplication property of equality,

.

By distribution,

.

### Example Question #38 : Algebraic Concepts

Which is the greater quantity?

(a)

(b)

(b) is greater.

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

Explanation:

Take the absolute value of both sides:

This makes (a) greater.

### Example Question #39 : Algebraic Concepts

Which is the greater quantity?

(a)

(b)

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Explanation:

(a)

(b)

Rewrite as a fraction:

### Example Question #40 : Algebraic Concepts

Define  as follows:

Which is the greater quantity?

(a)

(b)

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

(b) is greater.

Explanation:

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

can be evaluated by using the definition of  for nonpositive :