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

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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Example Questions

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Example Question #161 : Equations

, and  all stand for negative quantities.

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(a) is the greater quantity

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

(b) is the greater quantity

(a) and (b) are equal

Correct answer:

(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

Possible Answers:

(b) is the greater quantity

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

(a) is the greater quantity

(a) and (b) are equal

Correct answer:

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) 

Possible Answers:

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

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

Correct answer:

(b) is the greater quantity

Explanation:

, so, setting ,

By definition, 

so, by substitution,

Therefore, .

Example Question #161 : How To Find The Solution To An Equation

Define  and .

Which is the greater quantity?

(a) 0

(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:

(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) 

Possible Answers:

(a) and (b) are equal

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

(b) is the greater quantity

(a) is the greater quantity

Correct answer:

(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 :

Possible Answers:

Correct answer:

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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