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

## Example Questions

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### Example Question #161 : Equations

, and  all stand for negative quantities.

Which is the greater quantity?

(a)

(b)

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

(a) is the greater quantity

Explanation:

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

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

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

Explanation:

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)

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

(b) is the greater quantity

Explanation:

, so, setting ,

By definition,

so, by substitution,

Therefore, .

### Example Question #164 : Equations

Define  and .

Which is the greater quantity?

(a) 0

(b)

(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

(b) is the greater quantity

Explanation:

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)

(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

(a) is the greater quantity

Explanation:

, 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 :

Explanation:

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 .

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