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

Example Questions

Example Question #11 : Algebraic Concepts

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 #12 : 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

It is impossible to tell from the information given

(a) is greater

(b) is greater

(a) and (b) are equal

(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 : Algebraic Concepts

Which is the greater quantity?

(a)

(b) 0

(b) is greater

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

(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 #11 : Equations

Consider the line of the equation

Which is the greater quantity?

(a) The -coordinate of the -intercept

(b) The -coordinate of the -intercept

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

(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 : Algebraic Concepts

refers to the greatest integer less than or equal to .

and  are integers. Which is greater?

(a)

(b)

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

(b) is greater

(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 #16 : 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.

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

(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 #17 : 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.

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 #18 : Algebraic Concepts

Which is the greater quantity?

(a)

(b)

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

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 : Algebraic Concepts

Which is the greater quantity?

(a)

(b)

(b) is greater

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

(a) and (b) are equal

Explanation:

Example Question #20 : 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

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.