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

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All ISEE Upper Level Quantitative Resources

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Example Questions

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Example Question #21 : Algebraic Concepts

$\left \lceil N\right \rceil$ refers to the least integer greater than or equal to .

and are integers. C = A - 0.01; D = B + 0.01

Which is the greater quantity?

(a) $\left \lceil C \right \rceil + \left \lceil D\right \rceil$

(b) $\left \lceil C+ D \right \rceil$

Possible Answers:

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Correct answer:

(a) is greater.

Explanation:

(a) Since is an integer, $\left \lceil C\right \rceil = \left \lceil A - 0.01 \right \rceil = A$ .

Since is an integer, $\left \lceil D\right \rceil = \left \lceil B+ 0.01 \right \rceil = B + 1$ .

$\left \lceil C \right \rceil + \left \lceil D\right \rceil = A + (B + 1) = (A+B)+ 1$

(b) By closure, A + B is an integer, so

$\left \lceil C+ D \right \rceil = \left \lceil (A - 0.01) + (B + 0.01)\right \rceil = \left \lceil A + B\right \rceil = A + B$ .

(a) is the greater quantity.

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Example Question #22 : How To Find The Solution To An Equation

$\left \lfloor N\right \rfloor$ refers to the greatest integer less than or equal to .

and are integers.

a = x + 0.5, b= y + 0.5

Which is greater?

(a) $\left \lfloor a\right \rfloor + \left \lfloor b \right \rfloor$

(b) $\left \lfloor a+ b \right \rfloor$

Possible Answers:

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

Correct answer:

(b) is greater.

Explanation:

(a) Since is an integer, $\left \lfloor a \right \rfloor = \left \lfloor x + 0.5 \right \rfloor = x$ .

Since is an integer, $\left \lfloor b \right \rfloor = \left \lfloor y + 0.5 \right \rfloor = y$ .

$\left \lfloor a\right \rfloor + \left \lfloor b \right \rfloor = x + y$

(b) By closure, x+y is an integer, so

$\left \lfloor a+b \right \rfloor = \left \lfloor x + 0.5 +y+0.5 \right \rfloor = \left \lfloor x +y+1 \right \rfloor = x +y+1$ .

This makes (b) greater.

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Example Question #23 : How To Find The Solution To An Equation

$x + 7 \sqrt{x} - 30 = 0$

Which is the greater quantity?

(a)

(b)

Possible Answers:

(b) is greater.

(a) is greater.

(a) and (b) are equal.

It cannot be determined from the information given.

Correct answer:

(a) and (b) are equal.

Explanation:

Substitute $t = \sqrt {x}$ and, subsequently, $t^{2} = x$ :

$t^{2} + 7 t- 30 = 0$

Factor as (t + ?) (t + ?) , replacing the two question marks with integers whose product is and whose sum is . These integers are -3,10 .

(t-3)(t +10) = 0

Break this up into two equations, replacing $\sqrt{x}$ for :

t-3= 0

t = 3

$\sqrt{x} = 3$

$\left ( \sqrt{x} \right )^{2} = 3^{2}$

x=9

t +10 = 0

t = -10

$\sqrt{x} = -10$

This has no solution, since $\sqrt {x}$ must be nonnegative. x=9

is the only solution, so (a) and (b) must be equal.

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Example Question #24 : How To Find The Solution To An Equation

Consider the line through points (3,4) and (7,2) .

Which is the greater quantity?

(a) The -coordinate of the -intercept of this line

(b) The -coordinate of the -intercept of this line

Possible Answers:

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

Correct answer:

(a) is greater.

Explanation:

The slope of this line is

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}= \frac{4-2}{3-7}= \frac{2}{-4} = -\frac{1}{2}$ .

We will use the point-slope form of the line, with this slope and point (3,4) :

$y - 4 = -\frac{1}{2} (x - 3)$

The -coordinate of the -intercept of this line can be found by substituting y = 0 and solving for :

$y - 4 = -\frac{1}{2} (x - 3)$

$0- 4 = -\frac{1}{2} \cdot x + \frac{1}{2} \cdot 3$

$- 4 = -\frac{1}{2} x + \frac{3}{2}$

$- 4- \frac{3}{2} = -\frac{1}{2} x + \frac{3}{2} - \frac{3}{2}$

$- \frac{11}{2} = -\frac{1}{2} x$

$- \frac{11}{2}\cdot (-2) = -\frac{1}{2} x \cdot (-2)$

x = 11

The -coordinate of the -intercept of this line can be found by substituting x= 0 and solving for :

$y - 4 = -\frac{1}{2} (0 - 3)$

$y - 4 = -\frac{1}{2} (- 3)$

$y - 4 = \frac{3}{2}$

$y - 4 + 4 = \frac{3}{2} + 4$

$y = \frac{11}{2} = 5 \frac{1}{2}$

This makes (a) the greater quantity.

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Example Question #21 : How To Find The Solution To An Equation

The slope of a line is 2; the line does not pass through the origin.

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:

It is impossible to tell from the information given.

Explanation:

Let (b,0),(0,a) be the - and -intercepts, respectively. We know that the line does not pass through the origin - so $a,b \neq 0$ .

Then the slope is:

$m = \frac{a-0}{0-b} = \frac{a}{-b} = - \frac{a}{b}$

$- \frac{a}{b} = 2$

a = -2b

Either or can be the greater. For example, if b = 1 , then a = -2 , and if b =- 1 , then a = 2 .

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Example Question #26 : How To Find The Solution To An Equation

$6^{3x}= 36 ^{x+ 3}$ .

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) and (b) are equal.

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

Correct answer:

(a) and (b) are equal.

Explanation:

$36 = 6^{2}$ , so substitute and use the power of a power rule.

$6^{3x}= 36 ^{x+ 3}$

$6^{3x}=\left ( 6^{2} \right )^{x+ 3}$

$6^{3x}= 6^{2(x+ 3)}$

$6^{3x}= 6^{2x + 6}$

3x = 2x + 6

3x -2x= 2x-2x + 6

x = 6

This makes (a) and (b) equal.

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Example Question #21 : Algebraic Concepts

Which is the greater quantity?

(a) The slope of the line of the equation y = 5x + 4

(b) The slope of the line of the equation y = 4x + 5

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) is greater.

Explanation:

Both equations are in slope-intercept form, so compare the coefficients of . The coefficients in (a) and (b) are 5 and 4, respectively, so these are the slopes of the lines. The line in (a) has the greater slope.

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Example Question #28 : How To Find The Solution To An Equation

y = 2x + 1

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

Correct answer:

It is impossible to tell from the information given.

Explanation:

Using two different cases, we show that it is impossible to tell which is greater.

Case 1: x = 1 . Then $y = 2x + 1 = 2 \cdot 1+ 1 = 2 + 1 = 3$ , and y > x .

Case 2: x = -3 . Then $y = 2x + 1 = 2 \cdot (-3)+ 1 = -6+ 1 = -5$ , and y < x .

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Example Question #29 : How To Find The Solution To An Equation

x + y = 10

2x - y = 14

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:

(a) is greater.

Explanation:

To solve the system of equations, add the left and right sides of the equation separately:

x + y = 10

$\underline{2x - y = 14}$

$3x\; \; \; \; \; \; = 24$

Divide:

$3x \div 3 = 24\div 3$

x=8

Substitute to get :

x+y = 10

8+y = 10

8-8+y = 10-8

y = 2

is greater.

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Example Question #30 : How To Find The Solution To An Equation

t + 1 = |u|

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

Correct answer:

It is impossible to tell from the information given

Explanation:

We show that it is possible for either or to be the greater by giving one of each case.

Case 1: u = 1 . Then $t + 1 = \left | 1\right | = 1$ , so t = 0

Case 2: u = -1 . Then $t + 1 = \left | -1\right | = 1$ , so t = 0

In Case 1, t < u ; in Case 2, t > u

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All ISEE Upper Level Quantitative Resources

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

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #21 : Algebraic Concepts

Example Question #22 : How To Find The Solution To An Equation

Example Question #23 : How To Find The Solution To An Equation

Example Question #24 : How To Find The Solution To An Equation

Example Question #21 : How To Find The Solution To An Equation

Example Question #26 : How To Find The Solution To An Equation

Example Question #21 : Algebraic Concepts

Example Question #28 : How To Find The Solution To An Equation

Example Question #29 : How To Find The Solution To An Equation

Example Question #30 : How To Find The Solution To An Equation

All ISEE Upper Level Quantitative Resources

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