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

Example Question #11 : How To Find The Solution To An Equation

Possible Answers:

Correct answer:

Explanation:

First, rewrite the quadratic equation in standard form by distributing the  through the product on the left and 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  and whose product is . These integers are , so:

Set each expression equal to 0 and solve:

or 

The solution set is .

Example Question #11 : Algebraic Concepts

Consider the line of the equation 

Which is the greater quantity?

(a) The -coordinate of the -intercept

(b) The -coordinate of the -intercept

Possible Answers:

(b) is greater

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

Correct answer:

(b) is greater

Explanation:

(a) To find the -coordinate of the -intercept, substitute :

(b) To find the -coordinate of the -intercept, substitute :

 

(b) is the greater quantity.

Example Question #13 : Equations

Which is the greater quantity?

(a) 

(b) 0 

Possible Answers:

(a) is greater

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

Correct answer:

(a) is greater

Explanation:

 can be rewritten as a compound statement:

 or 

Solve both:

or 

Either way, , so (a) is the greater quantity

Example Question #14 : Equations

Consider the line of the equation 

Which is the greater quantity?

(a) The -coordinate of the -intercept

(b) The -coordinate of the -intercept

Possible Answers:

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

Correct answer:

(b) is greater

Explanation:

(a) To find the -coordinate of the -intercept, substitute :

(b) To find the -coordinate of the -intercept, substitute :

This makes (b) the greater quantity

 

Example Question #15 : Equations

 refers to the greatest integer less than or equal to .

 and  are integers. Which is greater?

(a) 

(b) 

Possible Answers:

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Correct answer:

(a) and (b) are equal

Explanation:

If  is an integer, then  by definition.

Since , and, by closure,  are all integers, 

 and , making (a) and (b) equal.

Example Question #11 : Algebraic Concepts

Consider the line of the equation .

Which is the greater quantity?

(a) The -coordinate of the -intercept.

(b) The -coordinate of the -intercept.

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) is greater.

Explanation:

(a) To find the -coordinate of the -intercept, substitute :

(b) To find the -coordinate of the -intercept, substitute :

Therefore (a) is the greater quantity.

Example Question #12 : 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:

It is impossible to tell from the information given.

Explanation:

Each can be rewritten as a compound statement. Solve separately:

or 

 

Similarly:

 

 

Therefore, it cannot be determined with certainty which of  and  is the greater.

Example Question #13 : 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:

It is impossible to tell from the information given.

Explanation:

If , then either  or  . Solve for  in both equations:

or 

Therefore, either (a) and (b) are equal or (b) is the greater quantity, but it cannot be determined with certainty.

Example Question #19 : Equations

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:

(a) and (b) are equal

Explanation:

Example Question #14 : Algebraic Concepts

Consider the line of the equation .

Which is the greater quantity?

(a) The -coordinate of the -intercept

(b) The -coordinate of the -intercept

Possible Answers:

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

(a) To find the -coordinate of the -intercept, substitute :

(b) To find the -coordinate of the -intercept, substitute :

(a) is the greater quantity.

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