How to add variables

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ISEE Upper Level Quantitative Reasoning › How to add variables

Questions 1 - 10
1

Define . The graph of is a line with slope .

.

Which is the greater quantity?

(a)

(b)

(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

Explanation

, so .

, so, by definition, , or .

The graph of is a line through the point with coordinates and with slope . The equation of the line can be determined by setting in the slope-intercept form:

.

The equation of the line is , which makes this the definition of . By setting ,

.

Therefore,

2

; and are both positive.

Which is the greater quantity?

(a)

(b)

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

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

Explanation

If ,

then

.

The absolute value of a negative number is its (positive) opposite, so

Also, if and are both positive, then is positive; the absolute value of a positive number is the number itself, so . Since , it follows that . Therefore,

Since is given to be positive,

.

and

3

Add:

Explanation

In order to simplify this expression, we will need to add like terms.

There is a lone positive three.

Combine all the terms.

The answer is:

4

Which of the following is the greater quantity?

(A)

(B)

(B) is greater

(A) is greater

(A) and (B) are equal

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

Explanation

(A) Substitute and evaluate, remembering that by order of operations, squaring precedes addition:

(B) Substitute and evaluate

(B) is greater

5

Which is the greater quantity?

(A)

(B)

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

(A) is greater

(B) is greater

(A) and (B) are equal

Explanation

Which quantity is the greater depends on the value of .

Case 1:

Then

and

Subsequently, (A) is greater

Case 2:

Then

and

Subsequently, (B) is greater

6

is a negative integer. Which is the greater quantity?

(A)

(B)

(a) is the greater quantity

(b) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(a) and (b) are equal

Explanation

,

so

and

regardless of the value of .

7

is a negative number.

Which is the greater quantity?

(a)

(b)

(a) and (b) are equal

It cannot be determined which of (a) and (b) is greater

(a) is the greater quantity

(b) is the greater quantity

Explanation

The expressions and are each other's opposite, so they have the same absolute value. Therefore, regardless of the value of ,

.

8

is a negative integer. Which is the greater quantity?

(a)

(b)

(b) is the greater quantity

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(a) and (b) are equal

Explanation

Since

,

it follows that

and

regardless of the value of .

9

is a positive number.

Which is the greater quantity?

(a)

(b)

It cannot be determined which of (a) and (b) is greater

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

Explanation

By examining two scenarios, we see that we cannot determine which is the greater quantity.

Case 1:

Then

and

This makes (a) the greater quantity.

Case 2:

Then

and

This makes (b) the greater quantity.

10

is a negative integer. Which is the greater quantity?

(A)

(B)

(b) is the greater quantity

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(a) and (b) are equal

Explanation

, so

and

is the greater quantity, regardless of .

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