### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #161 : Equations

, , and all stand for negative quantities.

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

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

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

**Correct answer:**

(a) is the greater quantity

Solve the first equation for in terms of , using the properties of equality to isolate the :

Solve for in the second equation similarly:

, so by the properties of inequality,

### Example Question #162 : Equations

Define and .

Which is the greater quantity?

(a)

(b) 2

**Possible Answers:**

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

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

**Correct answer:**

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

The definition can be rewritten by noting that this is the product of a sum and a difference of the same two terms, and that the product is the difference of their squares:

By definition,

Since , it holds that

, or

We can factor the trinomial using two integers whose sum is 2 and whose product is ; by a little trial and error we find 4 and , so

.

By the Zero Product Principle,

, in which case ; or,

, in which case .

It is therefore unclear whether is less than or equal to 2.

### Example Question #163 : Equations

Define .

.

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

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

**Correct answer:**

(b) is the greater quantity

, so, setting ,

By definition,

so, by substitution,

Therefore, .

### Example Question #164 : Equations

Define and .

Which is the greater quantity?

(a) 0

(b)

**Possible Answers:**

(a) and (b) are equal

(a) is the greater quantity

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

(b) is the greater quantity

**Correct answer:**

(b) is the greater quantity

can be rewritten using the square of a binomial pattern:

By definition,

So

Since

, it holds that

Solving for :

, which is less than 0.

### Example Question #165 : Equations

Define .

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

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

**Correct answer:**

(a) is the greater quantity

, so, by substitution,

.

By way of the definition of a composition of functions,

.

Since , it follows that .

Also, by substitution,

.

Therefore, .

### Example Question #166 : Equations

Solve for :

**Possible Answers:**

**Correct answer:**

First, rewrite the quadratic equation in standard form by distributing the through the product on the left, then collecting all of the terms on the left side:

Use the method to factor the quadratic expression ; we are looking to split the linear term by finding two integers whose sum is 7 and whose product is . These integers are , so:

Set each expression equal to 0 and solve:

or

The solution set is .

Certified Tutor

Certified Tutor