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ISEE Upper Level Quantitative : Equations

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

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All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #141 : How To Find The Solution To An Equation

x = y + 1

Which is the greater quantity?

(a) $\frac{1}{x}$

(b) $\frac{1}{y}$

Possible Answers:

(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

Correct answer:

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

Explanation:

From these two scenarios, it can be seen that either (a) or (b) can be the greater.

Case 1: y = 2 . Then x = y+ 1 = 2 + 1 = 3 .

$\frac{1}{x} =\frac{1}{3} = \frac{2}{6}$

$\frac{1}{y} =\frac{1}{2} = \frac{3}{6}$

$\frac{1}{y} > \frac{1}{x}$

Case 2: $y = - \frac{1}{2}$ . $x = y+1 = - \frac{1}{2} + 1= \frac{1}{2}$

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

$\frac{1}{y} =\frac{1}{- \frac{1}{2}} =-2$

$\frac{1}{x} > \frac{1}{y}$

Therefore, it cannot be determined which is greater.

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Example Question #142 : How To Find The Solution To An Equation

Define $f(x) = \frac{1}{x}$ .

f(A) = 10

f(10 ) = B

Which is the greater quantity?

(a)

(b)

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:

(a) and (b) are equal

Explanation:

$f(x) = \frac{1}{x}$

$f(A) = \frac{1}{A}=10$

$\frac{1}{A} \cdot \frac{A}{10} =10 \cdot \frac{A}{10}$

$\frac{1} {10} =A$

Also,

$f(10) = \frac{1}{10} = B$

Therefore, A = B .

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Example Question #143 : How To Find The Solution To An Equation

Define $f(x) = x^{2}$ .

$f(A) = \frac{1}{16}$

$f \left ( \frac{1}{16} \right )= B$

Which is the greater quantity?

(a)

(b)

Possible Answers:

(b) is the greater quantity

(a) and (b) are equal

(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:

$f(A) = \frac{1}{16}$

$A^{2}= \frac{1}{16}$

Either

$A = \sqrt{\frac{1}{16}} = \frac{\sqrt{1}}{\sqrt{16}} = \frac{1}{4}$

$A =- \sqrt{\frac{1}{16}} =- \frac{\sqrt{1}}{\sqrt{16}} =- \frac{1}{4}$

$B = f \left ( \frac{1}{16} \right )= \left ( \frac{1}{16} \right )^{2} = \frac{1}{256}$

Since

$- \frac{1}{4}< \frac{1}{256} < \frac{1}{4}$ ,

it is not clear whether or is the greater.

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Example Question #144 : How To Find The Solution To An Equation

Function 4

Above is the graph of a function . f(4) = A and f(B) = 4 . Which is the greater quantity?

(a)

(b)

Possible Answers:

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

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

Correct answer:

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

Explanation:

Examine this diagram:

Function x

As indicated by the blue circle in the diagram, the only point on the graph of the function with domain element 4 is (4, -2) , so f(4) = -2 , and A = -2 .

As indicated by the red circles in the diagram, there are three points on the graph of the function with range element 4 - namely, (-6, 4) , (-4, 4) , and (8, 4) . Therefore, $B \in \left \{ -6, -4, 8 \right \}$ . It is unclear whether A < B or A > B without further information.

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Example Question #141 : Equations

$x^{3} + 7x^{2} = 49x + 343$

$y^{3} + 343 = 7y^{2}+ 49 y$

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is the greater quantity

(b) is the greater quantity

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

(a) and (b) are equal

Correct answer:

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

Explanation:

$x^{3} + 7x^{2} = 49x + 343$

$x^{3} + 7x^{2} - 49x - 343 = 49x + 343 - 49x - 343$

$x^{3} + 7x^{2} - 49x - 343 = 0$

$(x^{3} + 7x^{2}) -( 49x + 343 )= 0$

$x ^{2}(x + 7 ) -49 (x + 7 )= 0$

$(x ^{2} -49 )(x + 7 ) = 0$

$(x ^{2} -7 ^{2})(x + 7 ) = 0$

(x + 7 ) (x - 7 ) (x + 7 ) = 0

By the Zero Product Principle, either x - 7 = 0 , in which case x = 7 , or x+ 7 = 0 , in which case x =- 7 .

$y^{3} + 343 = 7y^{2}+ 49 y$

$y^{3} + 343- 7y^{2}- 49 y = 7y^{2}+ 49 y - 7y^{2}- 49 y$

$y^{3} - 7y^{2}- 49 y + 343 =0$

$(y^{3} - 7y^{2} )- (49 y - 343 )=0$

$y^{2} (y - 7 )- 49 (y - 7)=0$

$(y ^{2} -49 )(y-7 ) = 0$

(y+7 ) (y-7 ) (y-7 ) = 0

By the Zero Product Principle, either y- 7 = 0 , in which case y= 7 , or y+ 7 = 0 , in which case y =- 7 .

Both and are equal to either or 7, but it is unclear which is which, or whether both are even the same. Therefore, it cannot be determined which, if either, is the greater.

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Example Question #141 : Equations

| 5x + 12 | < 30

| 5y+ 12 | >30

Which is the greater quantity?

(a)

(b)

Possible Answers:

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

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

Correct answer:

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

Explanation:

| 5x + 12 | < 30

-30 < 5x+12 < 30

-30- 12 < 5x+12 - 12 < 30 - 12

-42 < 5x < 18

$-42\div 5 < 5x \div 5 < 18 \div 5$

-8.4 < x < 3.6

| 5y+ 12 | >30

Either

5y + 12 > 30

5y + 12- 12 > 30 - 12

5y > 18

$5y \div 5 > 18 \div 5$

y > 3.6 ,

in which case y > x ,

5y + 12< - 30

5y + 12 - 12 < - 30 - 12

5y < - 42

$5y \div 5 < - 42 \div 5$

y < -8.4 ,

in which case x > y .

It is therefore unclear which is the greater, or

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Example Question #142 : Equations

|x+ 13| = | x + 7|

|y + 21 | = | y + 1|

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is the greater quantity

(a) and (b) are equal

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

(b) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

|x+ 13| = | x + 7| , so either:

x+13 = x + 7

x+13 - x= x + 7 - x

13 = 7 , which is false, or

x+13 = - (x + 7)

x+13 = -x - 7

x+13+ x - 13 = -x - 7 + x - 13

2x = -20

$2x \div 2 = -20 \div 2$

x = -10

|y + 21 | = | y + 1| , so either

y + 21 = y + 1

y + 21 - y = y + 1 - y

21 = 1 , which is false, or

y + 21 = - ( y + 1)

y + 21 = - y - 1

y + 21 + y -21 = - y - 1 + y -21

2 y = -22

$2 y\div 2 = -22 \div 2$

y = -11

x = -10 and y = -11 , so x > y

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Example Question #143 : Equations

$\sqrt{x} + \frac{1}{2} = \frac{4}{7}$

Which is the greater quantity?

(a)

(b) $\frac{9}{25}$

Possible Answers:

(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

Correct answer:

(b) is the greater quantity

Explanation:

$\sqrt{x} + \frac{1}{2} = \frac{4}{7}$

$\sqrt{x} + \frac{1}{2} - \frac{1}{2} = \frac{4}{7} - \frac{1}{2}$

$\sqrt{x} = \frac{4 \cdot 2}{7 \cdot 2} - \frac{1 \cdot 7 }{2 \cdot 7 }$

$\sqrt{x} = \frac{8}{14} - \frac{7 }{14 } = \frac{1}{14 }$

$(\sqrt{x} )^{2}= \left ( \frac{1}{14 } \right )^{2}$

$x= \frac{1^{2} }{14^{2} } = \frac{1}{196} < \frac{9}{25}$

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Example Question #144 : Equations

$x^{2} + 100 = 20 x$

$y^{2} +20y+ 100 = 0$

Which is the greater quantity?

(a)

(b)

Possible Answers:

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

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

$x^{2} + 100 = 20 x$

$x^{2} - 20 x + 100 = 20 x - 20 x$

$x^{2} - 20 x + 100 = 0$

The quadratic trinomial fits the perfect square pattern:

$x^{2} - 2 \cdot 10 \cdot x + 10 ^{2} = 0$

$(x-10) ^{2} = 0$

x- 10 = 0

x = 10

$y^{2} +20y+ 100 = 0$

The quadratic trinomial also fits the perfect square pattern:

$y^{2} + 2 \cdot 10 \cdot y + 10 ^{2} = 0$

$(y+10) ^{2} = 0$

y+ 10 = 0

y = -10

Therefore, x >y .

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Example Question #145 : Equations

| 4x + 7 | = 0

|4y-7|=0

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is the greater quantity

(b) is the greater quantity

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

(a) and (b) are equal

Correct answer:

(b) is the greater quantity

Explanation:

| 4x + 7 | = 0 , so

4x+7 = 0

4x = -7

$x = - \frac{7}{4}$

|4y-7|=0 , so

4y - 7 = 0

4y = 7

$y= \frac{7}{4}$

$x = - \frac{7}{4}$ and $y= \frac{7}{4}$ , so y > x .

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Equations

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #141 : How To Find The Solution To An Equation

Example Question #142 : How To Find The Solution To An Equation

Example Question #143 : How To Find The Solution To An Equation

Example Question #144 : How To Find The Solution To An Equation

Example Question #141 : Equations

Example Question #141 : Equations

Example Question #142 : Equations

Example Question #143 : Equations

Example Question #144 : Equations

Example Question #145 : Equations

All ISEE Upper Level Quantitative Resources

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