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ISEE Upper Level Quantitative : How to add variables

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

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All ISEE Upper Level Quantitative Resources

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Example Questions

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Example Question #41 : Operations

is a negative integer. Which is the greater quantity?

(A) 100 - (X + 25 )

(B) (100 - X )+ 25

Possible Answers:

(a) is the greater quantity

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

(a) and (b) are equal

(b) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

100 - (X + 25 )

=100 - X - 25

=75 - X

(100 - X )+ 25

= 100 - X + 25

= 125 - X

125 > 75 , so

125 -X > 75 - X

and

(100 - X )+ 25 is the greater quantity, regardless of .

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Example Question #41 : Variables

is a positive number.

Which is the greater quantity?

(a) |7x - 9|

(b) |9x - 7|

Possible Answers:

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

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

Correct answer:

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

Explanation:

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

Case 1: $x = \frac{1}{2}$

Then $|7x - 9| =\left | 7 \cdot \frac{1}{2}- 9 \right |= \left | 3 \frac{1}{2}- 9 \right |= \left | -5 \frac{1}{2} \right |=5 \frac{1}{2}$

and $|9x - 7|=\left | 9 \cdot \frac{1}{2}- 7 \right |= \left | 4 \frac{1}{2}- 7 \right |= \left | -2 \frac{1}{2} \right |=2 \frac{1}{2}$

This makes (a) the greater quantity.

Case 2: x= 2

This makes (b) the greater quantity.

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Example Question #41 : Variables

is a negative number.

Which is the greater quantity?

(a) |7-11x|

(b) |11x - 7|

Possible Answers:

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

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

Correct answer:

(a) and (b) are equal

Explanation:

- (7-11x ) = - 7 + 11x = 11x -7

The expressions 7 - 11x and 11x-7 are each other's opposite, so they have the same absolute value. Therefore, regardless of the value of ,

|7-11x| = |11x - 7| .

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Example Question #41 : Variables

is a negative integer. Which is the greater quantity?

(A) $10 + X \div 5$

(B) $(10 + X )\div 5$

Possible Answers:

(a) and (b) are equal

(b) is the greater quantity

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

(a) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

$10 + X \div 5 = 10 + \frac{X}{5}$

$(10 + X )\div 5 = \frac{(10 + X )}{5} = 2 + \frac{X}{5}$

10 > 2 ,

$10 + \frac{X}{5}> 2 + \frac{X}{5}$

and

$10 + X \div 5> (10 + X )\div 5$ regardless of the value of .

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Example Question #42 : Variables

is a negative integer. Which is the greater quantity?

(a) 3X + 10

(b) 3(X+10)

Possible Answers:

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

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

Correct answer:

(b) is the greater quantity

Explanation:

$3(X+10) = 3 \cdot X + 3 \cdot 10 = 3X + 30$

Since

30 > 10 ,

it follows that

3X+ 30 > 3X+ 10

and

3(X+10) > 3 X+10

regardless of the value of .

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Example Question #51 : Variables

Which is the greater quantity?

(b) $(-7) \cdot x+ (-4)$

Possible Answers:

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

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

Correct answer:

(a) is the greater quantity

Explanation:

Suppose is nonnegative.

Then |x| = x .

Consequently,

which must be positive,

and

$(-7) \cdot x+ (-4) = -7x-4$ ,

which is the opposite of 7 x +4 and consequently must be negative. Therefore, (a) is greater.

Suppose is negative.

Then |x| = -x .

Consequently,

and

$(-7) \cdot x+ (-4) = -7x-4$ .

4 > -4 , so

-7x+4> -7x-4 ,

and (a) is greater.

(a) is the greater quantity either way.

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Example Question #51 : Variables

Define f (x) = 3x . The graph of g(x) is a line with slope $\frac{1}{2}$ .

$(g \circ f) (2) = 6$ .

Which is the greater quantity?

(a) f(2)

(b) g(2)

Possible Answers:

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

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

f (x) = 3x , so $f (2) = 3 \cdot 2 = 6$ .

$(g \circ f) (2) = 6$ , so, by definition, g[ f (2) ] = 6 , or g(6) = 6 .

The graph of g(x) is a line through the point with coordinates (6,6) and with slope $\frac{1}{2}$ . The equation of the line can be determined by setting $g(x) = x = 6 , m = \frac{1}{2}$ in the slope-intercept form:

g(x) = mx+ b

$6 = \frac{1}{2} \cdot 6+ b$

6= 3+ b

b = 3 .

The equation of the line is $g(x) = \frac{1}{2} x+ 3$ , which makes this the definition of g(x) . By setting x= 2 ,

$g(2) = \frac{1}{2} \cdot 2+ 3 = 1 + 3 = 4$ .

Therefore, f(2)> g(2)

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Example Question #53 : Variables

x- y = 125 ; and are both positive.

Which is the greater quantity?

(a) | y - x|

(b) | y + x|

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:

If x- y = 125 ,

then

y - x = - (x-y) = -125 .

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

|y - x |= | -125| = 125

Also, if and are both positive, then y + x is positive; the absolute value of a positive number is the number itself, so | y + x| = y+ x . Since x- y = 125 , it follows that x = 125 + y . Therefore,

| y + x| = y+ x = y + (125+y ) = 125 + 2y

Since is given to be positive,

125 + 2y > 125 .

and

| y + x| > |y - x|

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : How to add variables

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #41 : Operations

Example Question #41 : Variables

Example Question #41 : Variables

Example Question #41 : Variables

Example Question #42 : Variables

Example Question #51 : Variables

Example Question #51 : Variables

Example Question #53 : Variables

All ISEE Upper Level Quantitative Resources

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