### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #11 : Algebraic Concepts

**Possible Answers:**

**Correct answer:**

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

**Possible Answers:**

It is impossible to tell from the information given

(a) is greater

(b) is greater

(a) and (b) are equal

**Correct answer:**

(b) is greater

(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

**Possible Answers:**

(b) is greater

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

**Correct answer:**

(a) is greater

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

**Possible Answers:**

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

**Correct answer:**

(b) is greater

(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)

**Possible Answers:**

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

(b) is greater

**Correct answer:**

(a) and (b) are equal

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.

**Possible Answers:**

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

**Correct answer:**

(a) is greater.

(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)

**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.

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)

**Possible Answers:**

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

**Correct answer:**

It is impossible to tell from the information given.

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)

**Possible Answers:**

(b) is greater

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

**Correct answer:**

(a) and (b) are equal

### 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

**Possible Answers:**

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

**Correct answer:**

(a) is greater.

(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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