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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 #101 : How To Find The Solution To An Equation

Define $f (x) = \frac{1}{x+ 6}$ and $g(x) = \frac{1}{x + 8 }$ .

Which is the greater quantity?

(A) $\left (f+g \right ) (1)$

(B) $\left (f+g \right ) (-1)$

Possible Answers:

(B) is greater

(A) is greater

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

(A) and (B) are equal

Correct answer:

(B) is greater

Explanation:

$\left (f+g \right ) (x) = f(x) + g(x) = \frac{1}{x+6}+ \frac{1}{x+8}$

$\left (f+g \right ) (1) = f(1) + g(1) = \frac{1}{1+6}+ \frac{1}{1+8} = \frac{1}{7}+ \frac{1}{9}$

$\left (f+g \right ) (-1) = f(-1) + g(-1) = \frac{1}{-1+6}+ \frac{1}{-1+8} = \frac{1}{5}+ \frac{1}{7}$

Since $\frac{1}{9} < \frac{1}{5}$ ,

$\frac{1}{7}+ \frac{1}{9} < \frac{1}{5} + \frac{1}{7}$

and

$\left (f+g \right ) (1) < \left (f+g \right ) (-1)$

making (B) greater.

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Example Question #102 : How To Find The Solution To An Equation

Define f (x) = 4x - 7 and g (x) = 3x - 8

Let be a positive number. Which is the greater quantity?

(A) $\left ( f \circ g\right ) (t)$

(B) $\left ( g \circ f\right ) (t)$

Possible Answers:

(A) is greater

(A) and (B) are equal

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

(B) is greater

Correct answer:

(B) is greater

Explanation:

$\left ( f \circ g\right ) (t) = f (g(t)) = f (3t-8) = 4(3t-8) - 7 = 12t -32 - 7 = 12t -39$

$\left ( g \circ f\right ) (t) = g (f(t)) =g (4t-7) = 3 (4t-7) -8 = 12 t - 21 - 8 = 12t - 29$

Regardless of the value of ,

$\left ( g \circ f\right ) (t) = \left ( f \circ g\right ) (t) + 10$ ,

which means that

$\left ( g \circ f\right ) (t) > \left ( f \circ g\right ) (t)$ .

That is, (B) is greater.

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Example Question #103 : How To Find The Solution To An Equation

Two lines have -intercept (0, 9) . Line A has -intercept (-5,0) ; Line B has -intercept (-7,0) . Which is the greater quantity?

(A) The slope of Line A

(B) The slope of Line B

Possible Answers:

(A) and (B) are equal

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

(A) is greater

(B) is greater

Correct answer:

(A) is greater

Explanation:

To get the slope of each line, use the slope formula

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1 }}$

For Line A, $x_{1} =-5, y_{1} =0, x_{2} = 0, y_{2} = 9$ . Substitute in the slope formula.

The slope is

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{9-0}{0-(-5)} = \frac{9}{5}$

For Line B, $x_{1} =-7, y_{1} =0, x_{2} = 0, y_{2} = 9$ . Substitute in the slope formula.

The slope is

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{9-0}{0-(-7)} = \frac{9}{7}$

Since

$\frac{9}{5} = \frac{63}{35}> \frac{45}{35} = \frac{9}{7}$ ,

Line A has the greater slope, and (A) is greater.

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Example Question #104 : How To Find The Solution To An Equation

Line A has -intercept (5, 0) and -intercept (0, 7) . Line C has -intercept (14, 0) and -intercept (0, -10) ; Line B is perpendicular to Line C. Which is the greater quantity?

(A) The slope of Line A

(B) The slope of Line B

Possible Answers:

(A) is greater

(A) and (B) are equal

(B) is greater

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

Correct answer:

(A) and (B) are equal

Explanation:

To get the slope of Line A and Line C, use the slope formula

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1 }}$

For Line A, $x_{1} =5, y_{1} =0, x_{2} = 0, y_{2} = 7$ . Substitute in the slope formula.

The slope is

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1 }}= \frac{7-0}{0-5} = \frac{7}{-5} = - \frac{7}{5}$

For Line C, $x_{1} =14, y_{1} =0, x_{2} = 0, y_{2} = -10$ . Substitute in the slope formula.

The slope is

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1 }}= \frac{-10-0}{0-14} = \frac{-10}{-14} = \frac{5}{7}$

Since Line B is perpendicular to Line C, its slope is the opposite of the reciprocal of that of Line C; this is $- \frac{5}{7}$ , which is equal to the slope of Line A.

The two quantities are equal.

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Example Question #105 : How To Find The Solution To An Equation

One-third of the sum of a number and sixty is ninety-three. What is the number?

Possible Answers:

339

219

459

Correct answer:

219

Explanation:

If we let be the number, "the sum of a number and sixty" can be written as

x+60

"One-third of the sum of a number and sixty" can be written as

$\frac{1}{3} (x+60)$

Set this equal to ninety-three and solve:

$\frac{1}{3} (x+60) = 93$

$3 \cdot \frac{1}{3} (x+60) =3 \cdot 93$

x+60 = 279

x+60 - 60= 279 - 60

x = 219

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Example Question #106 : How To Find The Solution To An Equation

Twelve added to two-fifths of a number is equal to sixty. What is that number?

Possible Answers:

120

162

180

102

Correct answer:

120

Explanation:

If we let be the number, "two-fifths of a number" can be written as

$\frac{2}{5} x$ .

"Twelve added to two-fifths of a number" can be written as

$\frac{2}{5} x + 12$ .

Then "Twelve added to two-fifths of a number is equal to sixty" can be written and solved for as follows:

$\frac{2}{5} x + 12 = 60$

$\frac{2}{5} x + 12 - 12 = 60 - 12$

$\frac{2}{5} x =48$

$\frac{5} {2}\cdot \frac{2}{5}x = \frac{5} {2}\cdot 48$

x = 120

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Example Question #101 : Equations

Ninety-seven is five less than two-fifths of a number. What is the number?

Possible Answers:

$237\frac{1}{2}$

255

$247\frac{1}{2}$

The correct answer is not among the other choices.

230

Correct answer:

255

Explanation:

If we let be the number, "two-fifths of a number" can be written as

$\frac{2}{5}x$ .

"Five less than two-fifths of a number can be written as

$\frac{2}{5}x -5$

"Ninety-seven is five less than two-fifths of a number" can be written and solved as follows:

$97 = \frac{2}{5}x -5$

$97 + 5 = \frac{2}{5}x -5 + 5$

$102 = \frac{2}{5}x$

$\frac{5}{2} \cdot 102 = \frac{5}{2} \cdot \frac{2}{5}x$

255 = x

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Example Question #108 : How To Find The Solution To An Equation

Define $f (x) = \sqrt{3x+8 }$ and $g(x) = \sqrt{2x+7}$ .

What is the domain of the function f + g ?

Possible Answers:

$\left [-2 \frac{2}{3}, \infty \right )$

$\left [-3 \frac{1} {2},-2 \frac{2}{3} \right ]$

$\left [-3 \frac{1} {2}, \infty \right )$

$\varnothing$

$(- \infty,\infty )$

Correct answer:

$\left [-2 \frac{2}{3}, \infty \right )$

Explanation:

$f (x) = \sqrt{3x+8 }$ has as its domain the set of values of for which its radicand is nonnegative; that is,

$3x+8 \geq 0$

$3x+8-8 \geq 0 -8$

$3x \geq -8$

$3x\div 3 \geq -8 \div 3$

$x \geq -2 \frac{2}{3}$ or $\left [-2 \frac{2}{3}, \infty \right )$

Similarly, $g(x) = \sqrt{2x+7}$ has as its domain the set of values of for which its radicand is nonnegative; that is,

$2x+7 \geq 0$

$2x+7-7 \geq 0 -7$

$2x \geq -7$

$2x \div 2 \geq -7 \div 2$

$x \geq -3\frac{1}{2}$ or $\left [-3 \frac{1} {2}, \infty \right )$

The domain of the sum of two functions is the intersection of the domains of the two individual functions. This intersection is

$\left [-2 \frac{2}{3}, \infty \right ) \cap \left [-3 \frac{1} {2}, \infty \right ) = \left [-2 \frac{2}{3}, \infty \right )$

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Example Question #109 : How To Find The Solution To An Equation

For all real numbers and , define an operation $\ast$ as follows:

$a \ast b = \frac{a b + 1}{a+ b + 1 }$

For which value of is the expression $N \ast \left (-7 \right )$ undefined?

Possible Answers:

$N = \frac{1}{7}$

N = 7

$N = - \frac{1}{7}$

N = 8

N = 6

Correct answer:

N = 6

Explanation:

$a \ast b = \frac{a b + 1}{a+ b + 1 }$ , so

$N \ast \left (-7 \right ) = \frac{N (-7) + 1}{N+ (-7) + 1 } = \frac{-7N + 1}{N-6 }$

This is is undefined if and only if the denominator is equal to zero, which happens when

N - 6 = 0 , or N = 6

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Example Question #110 : How To Find The Solution To An Equation

For all real numbers and , define an operation $\Lambda$ as follows:

$a \Lambda b = \frac{a^{2} b - 16}{16a^{2} + b}$

For which value of is the expression $N \Lambda 4$ undefined?

Possible Answers:

N = -2

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

None of the other responses gives a correct answer.

N = 2

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

Correct answer:

None of the other responses gives a correct answer.

Explanation:

$a \Lambda b = \frac{a^{2} b - 16}{16a^{2} + b}$ , so

$N \Lambda 4 = \frac{N^{2} \cdot 4 - 16}{16 \cdot N^{2} +4} = \frac{4N^{2} - 16}{16 N^{2} +4}$

This expression is undefined if and only if the denominator is equal to 0. However, for all values of ,

$16 N^{2} \geq 0$ , so

$16 N^{2} +4 > 0$

It is impossible for $N \Lambda 4$ to be undefined, so none of the four values of given gives a correct response.

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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 #101 : How To Find The Solution To An Equation

Example Question #102 : How To Find The Solution To An Equation

Example Question #103 : How To Find The Solution To An Equation

Example Question #104 : How To Find The Solution To An Equation

Example Question #105 : How To Find The Solution To An Equation

Example Question #106 : How To Find The Solution To An Equation

Example Question #101 : Equations

Example Question #108 : How To Find The Solution To An Equation

Example Question #109 : How To Find The Solution To An Equation

Example Question #110 : How To Find The Solution To An Equation

All ISEE Upper Level Quantitative Resources

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