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ISEE Upper Level Quantitative : Algebraic Concepts

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All ISEE Upper Level Quantitative Resources

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Example Questions

Example Question #131 : How To Find The Solution To An Equation

Function h

Let be the function whose graph is shown in the above figure. is defined by the equation

g(x) = f(-x) .

Give the -intercept of the graph of .

Possible Answers:

(0, -5)

(0, 5)

(0, 2)

(0, -2)

The graph of has no -intercept.

Correct answer:

(0, 5)

Explanation:

The -intercept of a function is the point at which x = 0 , so we can find this by evaluating g(0) .

g(x) = f(-x)

g(0) = f(-0) = f(0)

As seen in the diagram below, f(0) = 5 .

Function h1

Therefore, g(0) = f(0) = 5 , and the -intercept of the graph of is (0, 5) .

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Example Question #132 : How To Find The Solution To An Equation

Function h

Let be the function whose graph is shown in the above figure. is defined by the equation

g(x) = -f(x) .

Give the -intercept of the graph of .

Possible Answers:

The graph of has no -intercept.

(0, 2)

(0, 5)

(0, -2)

(0, -5)

Correct answer:

(0, -5)

Explanation:

The -intercept of a function is the point at which x = 0 , so we can find this by evaluating g(0) .

g(x) = -f(x)

g(0) = -f(0)

As seen in the diagram below, f(0) = 5 .

Function h1

Therefore, g(0) = -f(0) = -5 , and the -intercept of the graph of is (0, -5)

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Example Question #131 : How To Find The Solution To An Equation

Function h

Let be the function whose graph is shown in the above figure. is defined by the equation

$g(x) =- f^{-1}(x)$ .

Give the -intercept of the graph of .

Possible Answers:

The graph of has no -intercept.

(0, 5)

(0, -5)

(0, -2)

(0, 2)

Correct answer:

(0, 2)

Explanation:

The -intercept of a function is the point at which x = 0 , so we can find this by evaluating g(0) .

$g(x) =- f^{-1}(x)$

$g(0) = - f^{-1}(0)$

The graph of includes the point (-2,0) , as can be seen in the diagram below:

Function h1

Therefore, f(-2) = 0 and $g(0) =- f^{-1}(0)=- \left (-2 \right )= 2$ . The -intercept of the graph of is (0, 2) .

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Example Question #134 : How To Find The Solution To An Equation

Give the solution set of the equation

|70-t| - 70 = 70 .

Possible Answers:

$\left \{ -210 , - 70 \right \}$

$\left \{ -210 , 70 \right \}$

$\left \{ 70, 210\right \}$

The equation has no solution.

$\left \{ -70, 210\right \}$

Correct answer:

$\left \{ -70, 210\right \}$

Explanation:

|70-t| - 70 = 70

|70-t| - 70 + 70 = 70 + 70

|70-t| = 140

Either

70- t = 140 or 70- t = -140

so we solve each separately.

70- t = 140

70- t + t - 140 = 140 + t - 140

-70 = t

70- t+ t + 140 = -140+ t + 140

210 = t

The solution set is $\left \{ -70, 210\right \}$ .

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Example Question #131 : How To Find The Solution To An Equation

|5t - 16| = |23 - 5t|

How many solutions does this equation have?

Possible Answers:

Two solutions

Four solutions

Infinitely many solutions

One solution

No solutions

Correct answer:

One solution

Explanation:

Two numbers have the same absolute value if and only if the numbers are either equal or each other's opposite. We examine both possibilities.

Case 1:

5t - 16 = 23 - 5t

5t - 16 + 5t + 16 = 23 - 5t + 5t + 16

10t =39

$\frac{10t}{10} =\frac{39}{10}$

t = 3.9

Case 2:

$5t - 16 = -\left (23 - 5t \right )$

5t -16 = 5t-23

5t -16 -5t = 5t-23 -5t

-16 = -23 ,

a false statement. No additional solution is yielded.

The statement has one solution.

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Example Question #136 : How To Find The Solution To An Equation

Function 4

Let be the function whose graph is shown in the above figure. Give the -intercept of the graph of , which is defined by the equation

g(x) = f(x-7) .

Possible Answers:

(0, 2)

The graph of has no -intercept.

(0, -3)

(0, -11)

$\left ( 0, 2\frac{1}{2}\right )$

Correct answer:

(0, 2)

Explanation:

The -intercept of a function is the point at which x = 0 , so we can find this by evaluating g(0) .

g(x) = f(x-7)

g(0) = f(0-7)

g(0) = f(-7)

As can be seen in the diagram below, f(-7)= 2 .

Function 4a

The -intercept of the graph of is (0, 2) .

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Example Question #132 : How To Find The Solution To An Equation

|2t - 7| = |t + 13|

Which of the following is a true statement?

Possible Answers:

The equation has one solution, which is positive.

The equation has two solutions, one positive and one negative.

The equation has one solution, which is negative.

The equation has two solutions, both positive.

The equation has two solutions, both negative.

Correct answer:

The equation has two solutions, one positive and one negative.

Explanation:

Two numbers have the same absolute value if and only if the numbers are either equal or each other's opposite. We examine both possibilities.

Case 1:

2t - 7= t + 13

2t - 7- t + 7 = t + 13 - t + 7

t = 20

Case 2:

2t - 7=- (t + 13)

2t - 7= -t - 13

2t - 7+t + 7= -t - 13 +t + 7

3t = -6

$\frac{3t}{3} =\frac{ -6}{3}$

t = -2

The equation therefore has two solutions, one positive, one negative.

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Example Question #138 : How To Find The Solution To An Equation

|t - 5| = 3

$u^{2} -10u + 16 = 0$

Which is the greater quantity?

(a)

(b)

Possible Answers:

(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

Correct answer:

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

Explanation:

|t - 5| = 3 ,

so either

t-5 = 3 ,

in which case

t-5 + 5 = 3 + 5

t = 8

t-5 = -3 ,

in which case

t-5 + 5 = -3 + 5

t = 2

$u^{2} -10u + 16 = 0$

can be solved by factoring the trinomial. Since and have product 16 and sum , the statement can be rewritten as

(u-2)(u-8) = 0

Either u - 2 = 0 , in which case u = 2 ,

u- 8 = 0 , in which case u= 8 .

Each of and is equal to 2 or 8, but it cannot be determined which is which, or even if they are both the same. Therefore, it cannot be determined which is greater.

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Example Question #139 : How To Find The Solution To An Equation

| a + 5 | = 8

|b+ 8| = 5

and are both negative.

Which is the greater quantity?

(a)

(b)

Possible Answers:

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

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

Correct answer:

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

Explanation:

| a + 5 | = 8 , so either

a+5 = 8

a+5 -5 = 8 - 5

a = 3

a+5 = -8

a+5 -5 = - 8 - 5

a = -13

Since is negative, a = -13 is the only possibility.

|b+ 8| = 5 , so either

b+ 8 = 5

b+ 8 - 8 = 5 - 8

b = -3

b+ 8 =- 5

b+ 8 - 8 = -5 - 8

b = -13

is negative, so neither value can be eliminated.

a = -13 . If b = -13 , then a = b ; if b = -3 , then is the greater quantity. Therefore, it cannot be determined which is the greater.

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Example Question #140 : How To Find The Solution To An Equation

$x^{2}+ 12 = 7x$

$y^{2} + 4y+ 3 = 0$

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is the greater quantity

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

(b) is the greater quantity

(a) and (b) are equal

Correct answer:

(a) is the greater quantity

Explanation:

Both equations are quadratic, so solve as follows:

$x^{2}+ 12 = 7x$

$x^{2}+ 12- 7x = 7x - 7x$

$x^{2}- 7x+ 12 =0$

Since and have product 12 and sum , the trinomial factors to produce the equation:

(x-3)(x-4) = 0

Either x - 3 = 0 , in which case x= 3 ,

or x - 4= 0 , in which case x= 4 .

Similarly,

$y^{2} + 4y+ 3 = 0$

1 and 3 have product 3 and sum 4, so the above becomes

(y+ 1)(y+3) = 0

Either y+ 1= 0 , in which case y = -1 ,

or y+ 3= 0 , in which case y = -3 .

While the values of the variables cannot be determined with certainty, it can be determined that x >0 > y .

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Algebraic Concepts

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #131 : How To Find The Solution To An Equation

Example Question #132 : How To Find The Solution To An Equation

Example Question #131 : How To Find The Solution To An Equation

Example Question #134 : How To Find The Solution To An Equation

Example Question #131 : How To Find The Solution To An Equation

Example Question #136 : How To Find The Solution To An Equation

Example Question #132 : How To Find The Solution To An Equation

Example Question #138 : How To Find The Solution To An Equation

Example Question #139 : How To Find The Solution To An Equation

Example Question #140 : How To Find The Solution To An Equation

All ISEE Upper Level Quantitative Resources

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