Algebra Alıştırma Soruları — GRE General Test — Sayfa 8

Soru 141Soru →

For all non-zero real numbers $x$ and $y$ , the custom operation $\diamond$ is defined by $x \diamond y = \frac{x^2 + y^2}{xy}$ . Which of the following statements must be true for all non-zero real numbers $a$ and $b$ ? Select all such statements.

Geçerli olan tümünü seçin

a \diamond b = b \diamond a

a \diamond (-a) = -2

(2a) \diamond (2b) = 2(a \diamond b)

a \diamond a = 1

Cevabı ve açıklamayı göster

Cevap: $a \diamond b = b \diamond a$ ; $a \diamond (-a) = -2$

Cevap

The correct statements are that

a \diamond b = b \diamond a

and that

a \diamond (-a) = -2

.

The operation is symmetric with respect to

a

and

b

, making

a \diamond b = b \diamond a

true. Substituting

-a

into the operation yields

\frac{2a^2}{-a^2} = -2

, which makes

a \diamond (-a) = -2

true as well.

Adım Adım Çözüm

1

Evaluate the commutative statement

a \diamond b

a \diamond b = \frac{a^2 + b^2}{ab}

and

b \diamond a = \frac{b^2 + a^2}{ba} = \frac{a^2 + b^2}{ab}

.

Addition and multiplication of real numbers are commutative, so the expressions are identical.

2

Evaluate the statement

a \diamond (-a)

a \diamond (-a) = \frac{a^2 + (-a)^2}{a(-a)} = \frac{a^2 + a^2}{-a^2} = \frac{2a^2}{-a^2} = -2

.

Squaring a negative quantity

(-a)^2

yields positive

a^2

, while the denominator evaluates to

-a^2

.

3

Evaluate the scaling statement

(2a) \diamond (2b)

(2a) \diamond (2b) = \frac{4a^2 + 4b^2}{4ab} = \frac{a^2 + b^2}{ab} = a \diamond b

.

The factor of 4 in the numerator and denominator cancels out, showing

(2a) \diamond (2b) = a \diamond b

, which is not equal to

2(a \diamond b)

.

4

Evaluate the self-operation statement

a \diamond a

a \diamond a = \frac{a^2 + a^2}{a^2} = \frac{2a^2}{a^2} = 2

.

Summing identical squared terms in the numerator produces

2a^2

, which divides by

a^2

to give 2.

Anahtar Kavram

Evaluating custom binary operations by algebraic substitution and simplifying terms.

Soru 142Soru →

If $x$ is a real number that satisfies the inequality $|3 - 2x| \le 7$ , what is the maximum possible value of $5 - 3x$ ?

-10

-1

5

11

20

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

The maximum possible value of

5 - 3x

is

11

.

Solving

|3 - 2x| \le 7

gives

-7 \le 3 - 2x \le 7

. Subtracting

3

yields

-10 \le -2x \le 4

. Dividing by

-2

and flipping the inequality signs gives

-2 \le x \le 5

. Because

5 - 3x

has a negative coefficient for

x

, it decreases as

x

increases. Therefore, the maximum value of

5 - 3x

occurs at the smallest value in the domain,

x = -2

. Substituting

x = -2

into

5 - 3x

gives

5 - 3(-2) = 11

.

Adım Adım Çözüm

1

Rewrite the absolute value inequality

|3 - 2x| \le 7

as a compound inequality.

-7 \le 3 - 2x \le 7

An absolute value inequality of the form

|u| \le k

(where

k \ge 0

) is equivalent to

-k \le u \le k

.

2

Isolate

x

by subtracting

3

from all parts of the inequality and then dividing by

-2

.

-10 \le -2x \le 4 \implies 5 \ge x \ge -2

, which is equivalent to

-2 \le x \le 5

Dividing an inequality by a negative number reverses the direction of the inequality signs.

3

Determine which value of

x

in the interval

-2 \le x \le 5

maximizes the linear expression

5 - 3x

.

The expression reaches its maximum at the minimum bound

x = -2

.

Since the coefficient of

x

is negative (

-3

), smaller values of

x

result in larger values for

5 - 3x

.

4

Substitute

x = -2

into

5 - 3x

.

5 - 3(-2) = 5 + 6 = 11

Evaluating the linear expression at the lower bound yields its maximum value.

Anahtar Kavram

Solving linear absolute value inequalities and optimizing linear expressions over a closed interval.

Tahmini Süre:1m 30s

Soru 143Soru →

A company allocates a total budget of $\$4,000$ between its marketing and research departments. The amount allocated to marketing is $\$400$ more than three times the amount allocated to research. How many dollars are allocated to research?

Cevabı ve açıklamayı göster

Cevap: 900

Cevap

The amount allocated to research is

900

dollars.

If

x

represents the research budget, the marketing budget is

3x + 400

. Summing both department allocations gives

x + (3x + 400) = 4,000

. Simplifying this linear equation gives

4x + 400 = 4,000

, leading to

4x = 3,600

and

x = 900

.

Adım Adım Çözüm

1

Define the unknown variable and express both allocations algebraically.

Let

x

be the research budget. The marketing budget is

3x + 400

.

The marketing allocation is defined relative to the research allocation.

2

Formulate a linear equation representing the combined budget.

x + (3x + 400) = 4000

The total budget allocated across both departments is

\$4,000

.

3

Solve the linear equation for

x

.

4x + 400 = 4000 \implies 4x = 3600 \implies x = 900

Combine like terms, isolate the variable term by subtracting

400

, and divide by

4

.

Anahtar Kavram

Setting up and solving a linear equation in one variable from a real-life word problem context.

Soru 144Soru →

If

x

is a positive real number satisfying the equation

\sqrt[3]{\frac{x^3 \sqrt{x\sqrt{x}}}{x^{-1/4}}} = 16

what is the value of

x

?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

By converting all radicals into fractional exponents and systematically applying exponent rules, the expression under the cube root simplifies to

x^4

. Taking the cube root gives

x^{4/3} = 16

. Solving for

x

by raising both sides to

3/4

yields

x = 16^{3/4} = (2^4)^{3/4} = 2^3 = 8

.

Adım Adım Çözüm

1

Express the inner nested radical using fractional exponents

\sqrt{x\sqrt{x}} = \sqrt{x \cdot x^{1/2}} = \sqrt{x^{3/2}} = x^{3/4}

Applying the product and power rules of exponents:

x^a \cdot x^b = x^{a+b}

and

(x^a)^b = x^{ab}

.

2

Simplify the numerator inside the outer radical

x^3 \cdot x^{3/4} = x^{3 + 3/4} = x^{15/4}

Multiplying exponential terms with the same base requires adding their exponents.

3

Divide by the negative exponent in the denominator

\frac{x^{15/4}}{x^{-1/4}} = x^{15/4 - (-1/4)} = x^{16/4} = x^4

Dividing exponential terms with the same base requires subtracting the denominator exponent from the numerator exponent.

4

Apply the outer cube root to the simplified expression

\sqrt[3]{x^4} = (x^4)^{1/3} = x^{4/3}

The

n

-th root of an expression is equivalent to raising that expression to the power of

1/n

.

5

Solve the resulting exponential equation for

x

x^{4/3} = 16 \implies x = 16^{3/4} = (2^4)^{3/4} = 2^3 = 8

Raise both sides of

x^{4/3} = 16

to the power of

3/4

to isolate

x

.

Anahtar Kavram

Simplifying nested algebraic radicals and solving equations with fractional exponents using exponent rules.

Soru 145Soru →

A bookstore sells hardcover books for $\$15$ each and paperback books for $\$10$ each. A customer purchased a total of $12$ books and spent $\$150$ in total. Let $h$ represent the number of hardcover books purchased and $p$ represent the number of paperback books purchased. Which of the following equations correctly model this situation? Select all such equations.

Geçerli olan tümünü seçin

h + p = 12

15h + 10p = 150

3h + 2p = 30

10h + 15p = 150

15h + 10p = 12

Cevabı ve açıklamayı göster

Cevap: $h + p = 12$ ; $15h + 10p = 150$ ; $3h + 2p = 30$

Cevap

The correct equations are

h + p = 12

,

15h + 10p = 150

, and

3h + 2p = 30

.

The system of equations that models this situation requires one equation for the total count of books (

h + p = 12

) and one for the total monetary expenditure (

15h + 10p = 150

). Dividing the cost equation

15h + 10p = 150

by 5 gives another mathematically equivalent relation

3h + 2p = 30

. Thus, all three of these statements accurately represent the given conditions.

Adım Adım Çözüm

1

Formulate the total quantity equation

h + p = 12

The total number of books bought is 12, which is the sum of hardcover books

h

and paperback books

p

.

2

Formulate the total cost equation

15h + 10p = 150

Hardcover books cost

\$15

each and paperbacks cost

\$10

each, yielding a total cost of

\$150

.

3

Simplify the total cost equation by dividing by the greatest common divisor

3h + 2p = 30

Dividing all coefficients in

15h + 10p = 150

by

5

produces an equivalent simplified linear equation.

Anahtar Kavram

Linear Modeling and Equivalent Equations

Tahmini Süre:1m 0s

Soru 146Soru →

If $x$ is an integer that satisfies both $|2x - 9| \le 15$ and $\frac{5 - 2x}{-3} \le 3$ , what is the product of the smallest and largest possible values of $x$ ?

-21

84

35

-36

21

Cevabı ve açıklamayı göster

Cevap: $-21$

Cevap

The product of the smallest and largest possible integer values of

x

is

-21

.

Solving

|2x - 9| \le 15

gives

-3 \le x \le 12

. Solving

\frac{5 - 2x}{-3} \le 3

requires reversing the inequality sign twice (first when multiplying by

-3

, then when dividing by

-2

), which yields

x \le 7

. Taking the intersection of both intervals gives

-3 \le x \le 7

. The smallest integer in this range is

-3

and the largest is

7

, giving a product of

(-3) \times 7 = -21

.

Adım Adım Çözüm

1

Solve the absolute value inequality

|2x - 9| \le 15

.

-15 \le 2x - 9 \le 15 \implies -6 \le 2x \le 24 \implies -3 \le x \le 12

.

An absolute value inequality

|u| \le a

(for

a \ge 0

) is equivalent to the compound inequality

-a \le u \le a

.

2

Solve the linear inequality

\frac{5 - 2x}{-3} \le 3

.

Multiply by

-3

and flip the inequality direction:

5 - 2x \ge -9

. Subtract

5

:

-2x \ge -14

. Divide by

-2

and flip the inequality direction again:

x \le 7

.

Multiplying or dividing an inequality by a negative number reverses the direction of the inequality sign.

3

Determine the intersection of the two solution sets.

Combining

-3 \le x \le 12

and

x \le 7

gives

-3 \le x \le 7

.

The value of

x

must satisfy both conditions simultaneously.

4

Identify the smallest and largest integer values of

x

and calculate their product.

Smallest integer

= -3

, largest integer

= 7

. Product

= (-3) \times 7 = -21

.

Both endpoints

-3

and

7

are included in the closed interval

[-3, 7]

.

Anahtar Kavram

Solving compound linear inequalities involving absolute values and applying the rule for reversing inequality signs when multiplying or dividing by negative numbers.

Tahmini Süre:2m 0s

Soru 147Soru →

If $x$ is a real number that satisfies the inequality $3|x - 4| + 2|x + 5| \le 26$ , what is the maximum possible value of $|x - 7|$ ?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

11

Solving the piecewise linear inequality

3|x - 4| + 2|x + 5| \le 26

yields the interval

[-4, 5.6]

. The distance function

|x - 7|

reaches its maximum at the endpoint farthest from

7

, which is

x = -4

. Evaluating

|-4 - 7|

yields 11.

Adım Adım Çözüm

1

Identify the critical points of the absolute value terms.

The critical points are

x = -5

(where

x + 5 = 0

) and

x = 4

(where

x - 4 = 0

).

Critical points mark where the linear expressions inside the absolute values change sign.

2

Analyze the inequality piecewise across the three regions defined by the critical points.

For

x < -5

:

3(4 - x) + 2(-5 - x) \le 26 \implies 2 - 5x \le 26 \implies x \ge -4.8

. This produces no solution since

x

cannot be simultaneously

< -5

and

\ge -4.8

.
For

-5 \le x < 4

:

3(4 - x) + 2(x + 5) \le 26 \implies 22 - x \le 26 \implies x \ge -4

, yielding

-4 \le x < 4

.
For

x \ge 4

:

3(x - 4) + 2(x + 5) \le 26 \implies 5x - 2 \le 26 \implies x \le 5.6

, yielding

4 \le x \le 5.6

.

Expanding absolute value terms according to their regional sign definitions removes the absolute values.

3

Combine the valid regional solutions to establish the complete solution interval for

x

.

The set of all satisfying real numbers is

x \in [-4, 5.6]

.

Taking the union of the non-empty piecewise solution intervals yields the total solution set.

4

Find the maximum value of

|x - 7|

over

x \in [-4, 5.6]

.

At

x = -4

,

|-4 - 7| = |-11| = 11

. At

x = 5.6

,

|5.6 - 7| = |-1.4| = 1.4

. The maximum possible value is 11.

The expression

|x - 7|

measures distance from

7

. The maximum distance on a closed interval occurs at the endpoint farthest from

7

, which is

x = -4

.

Anahtar Kavram

Piecewise analysis of linear absolute value inequalities and optimization of absolute value distance functions.

Tahmini Süre:2m 30s

Soru 148Soru →

For all real numbers $p$ and $q$ , the custom operation $\odot$ is defined by $p \odot q = 2p^2 - 3q$ . What is the value of $3 \odot (-4)$ ?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

The value of

3 \odot (-4)

is

30

.

By applying the definition of the custom operation

p \odot q = 2p^2 - 3q

with

p = 3

and

q = -4

, we obtain

2(3)^2 - 3(-4) = 2(9) + 12 = 18 + 12 = 30

.

Adım Adım Çözüm

1

Substitute

p = 3

and

q = -4

into the expression

2p^2 - 3q

.

2(3)^2 - 3(-4)

The custom operation defines how to process the two inputs

p

and

q

.

2

Simplify the powers and products according to order of operations.

2(9) - (-12) = 18 + 12

Exponents must be calculated before multiplication, and multiplying two negative numbers yields a positive value.

3

Add the terms together to get the final numerical result.

30

18 + 12 = 30

.

Anahtar Kavram

Custom Symbol Operations

Soru 149Soru →

Consider the linear equation in

x

shown below, where

k

is a real constant:

\frac{x - k}{2} - \frac{2x + 1}{3} = \frac{k(x + 1)}{6} - 1

Which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

If

k = 1

, then

x = 0

is the unique solution to the equation.

If

k = -1

, the equation has infinitely many solutions.

If

k = 0

, the solution to the equation is

x = 4

.

If

k = 3

, the solution to the equation is

x = 2

.

If

k = 2

, the solution to the equation is

x = -\frac{4}{3}

.

Cevabı ve açıklamayı göster

Cevap: If $k = 1$ , then $x = 0$ is the unique solution to the equation.; If $k = 0$ , the solution to the equation is $x = 4$ .; If $k = 2$ , the solution to the equation is $x = -\frac{4}{3}$ .

Cevap

The correct statements are: if

k = 1

, then

x = 0

is the unique solution; if

k = 0

, the solution is

x = 4

; and if

k = 2

, the solution is

x = -\frac{4}{3}

.

Multiplying the equation by

6

yields

3(x - k) - 2(2x + 1) = k(x + 1) - 6

. Expanding and collecting terms in

x

leads to

(k + 1)x = 4(1 - k)

. Testing the proposed values of

k

shows that when

k = 1

,

2x = 0

gives

x = 0

; when

k = 0

,

x = 4

; and when

k = 2

,

3x = -4

gives

x = -\frac{4}{3}

. Therefore, the statements corresponding to

k = 1

,

k = 0

, and

k = 2

are all correct.

Adım Adım Çözüm

1

Clear fractions by multiplying the entire equation by the least common denominator,

6

.

3(x - k) - 2(2x + 1) = k(x + 1) - 6

Eliminating fractions simplifies algebraic expansion and grouping of like terms.

2

Expand all expressions and combine like terms on both sides.

3x - 3k - 4x - 2 = kx + k - 6 \implies -x - 3k - 2 = kx + k - 6

Distribute terms carefully to prevent sign errors.

3

Isolate terms containing

x

on one side and parameter/constant terms on the other side.

-x - kx = k + 3k - 6 + 2 \implies -x(k + 1) = 4k - 4 \implies (k + 1)x = 4(1 - k)

Factoring out

x

provides the general solution form

x = \frac{4(1 - k)}{k + 1}

for

k \neq -1

.

4

Evaluate each specified value of

k

against

(k + 1)x = 4(1 - k)

.

For

k = 1

:

2x = 0 \implies x = 0

.
For

k = -1

:

0x = 8 \implies

no solution.
For

k = 0

:

1x = 4 \implies x = 4

.
For

k = 3

:

4x = -8 \implies x = -2

.
For

k = 2

:

3x = -4 \implies x = -\frac{4}{3}

.

Direct substitution verifies which given statements are true.

Anahtar Kavram

Solving linear equations with parameters and analyzing existence and uniqueness of solutions.

Tahmini Süre:2m 30s

Soru 150Soru →

Consider the following system of linear equations in

x

,

y

, and

z

, where

c

is a real constant:

\frac{3}{2}x - y + 2z = 5

x + \frac{1}{3}y - z = 2

6x - y + z = c

If the system has at least one solution $(x, y, z)$ , what is the value of $7x - \frac{2}{3}y$ ?

18

16

24

7

2

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

18

To find the value of

7x - \frac{2}{3}y

without individual values for

x, y,

and

z

, we express

7x - \frac{2}{3}y

as a linear combination

m(\text{Eq. 1}) + n(\text{Eq. 2})

. Matching the

z

-coefficients requires

2m - n = 0 \implies n = 2m

. Matching the

y

-coefficients yields

-m + \frac{1}{3}(2m) = -\frac{2}{3} \implies m = 2

, which gives

n = 4

. Verifying the

x

-coefficient gives

2\left(\frac{3}{2}\right) + 4(1) = 7

. Applying these multipliers to the right-hand sides gives

2(5) + 4(2) = 10 + 8 = 18

.

Adım Adım Çözüm

1

Identify the target expression

7x - \frac{2}{3}y

as a linear combination of the first two equations

Express

m\left(\frac{3}{2}x - y + 2z\right) + n\left(x + \frac{1}{3}y - z\right) = 7x - \frac{2}{3}y + 0z

Because the system is dependent when consistent, individual variable values cannot be uniquely determined, but specific linear combinations independent of

z

can be evaluated.

2

Set up a system of equations for the scalar multipliers

m

and

n

Equating coefficients of

z

:

2m - n = 0 \implies n = 2m

. Equating coefficients of

y

:

-m + \frac{1}{3}n = -\frac{2}{3}

.

Eliminating

z

requires the net coefficient of

z

to equal 0.

3

Solve for

m

and

n

Substitute

n = 2m

into the

y

-coefficient equation:

-m + \frac{2}{3}m = -\frac{1}{3}m = -\frac{2}{3} \implies m = 2

, which gives

n = 4

.

Determining the exact linear multipliers needed to match the target expression.

4

Verify

x

-coefficient consistency and compute the target value

x

-coefficient:

2\left(\frac{3}{2}\right) + 4(1) = 3 + 4 = 7

. Value:

2(5) + 4(2) = 10 + 8 = 18

.

Applying the scalars

m = 2

and

n = 4

to the right-hand side constants gives the exact numerical value of

7x - \frac{2}{3}y

.

Anahtar Kavram

Linear combinations of dependent systems of equations

Alternatif Yöntem

Multiply the first equation by 2 to clear fractions:

3x - 2y + 4z = 10

. Multiply the second equation by 3 to clear fractions:

3x + y - 3z = 6

. Eliminate

z

by forming

3(3x - 2y + 4z) + 4(3x + y - 3z) = 3(10) + 4(6) \implies 21x - 2y = 54

. Dividing both sides of

21x - 2y = 54

by 3 directly gives

7x - \frac{2}{3}y = 18

.

Tahmini Süre:2m 0s

Soru 151Soru →

A manufacturing facility uses Machine A and Machine B to process standard orders of raw materials. Operating independently at its constant rate, Machine A requires $x$ hours to process one standard order, where $x > 0$ . Machine B operates at a constant rate and requires $x + 2$ hours to process one standard order. When both machines operate simultaneously at their respective constant rates for $6$ hours, the total number of standard orders processed is $2$ fewer than the number of standard orders Machine A would process operating alone for $18$ hours. What is the value of $x$ ?

2

3

4

6

8

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4

The rate of Machine A is

\frac{1}{x}

orders per hour, and the rate of Machine B is

\frac{1}{x+2}

orders per hour. In 18 hours, Machine A processes

\frac{18}{x}

orders. Working together for 6 hours, both machines process

6\left(\frac{1}{x} + \frac{1}{x+2}\right)

orders. Setting up the difference:

\frac{18}{x} - 6\left(\frac{1}{x} + \frac{1}{x+2}\right) = 2

, which simplifies to

\frac{12}{x} - \frac{6}{x+2} = 2

. Multiplying both sides by

x(x+2)

yields

12(x+2) - 6x = 2x^2 + 4x

, leading to

2x^2 - 2x - 24 = 0

or

x^2 - x - 12 = 0

. Factoring gives

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

. Because time must be positive,

x = 4

.

Adım Adım Çözüm

1

Express the individual work rates of Machine A and Machine B.

Machine A completes

\frac{1}{x}

orders per hour; Machine B completes

\frac{1}{x+2}

orders per hour.

Work rate is the reciprocal of the total time required to complete one unit of work.

2

Formulate the equation based on the total orders processed in the given time frames.

18 \cdot \frac{1}{x} - 6\left(\frac{1}{x} + \frac{1}{x+2}\right) = 2

Machine A alone in 18 hours processes

\frac{18}{x}

orders. Together in 6 hours, they process

6\left(\frac{1}{x} + \frac{1}{x+2}\right)

orders, which is 2 orders less.

3

Simplify the algebraic equation.

\frac{12}{x} - \frac{6}{x+2} = 2

Subtracting

6 \cdot \frac{1}{x}

from

18 \cdot \frac{1}{x}

yields

\frac{12}{x}

.

4

Clear the denominators by multiplying through by

x(x+2)

and solve the resulting quadratic equation.

12(x+2) - 6x = 2x(x+2) \implies 6x + 24 = 2x^2 + 4x \implies 2x^2 - 2x - 24 = 0 \implies x^2 - x - 12 = 0 \implies (x-4)(x+3) = 0

Clearing denominators transforms the rational equation into a standard quadratic equation.

5

Select the physically meaningful solution for time

x

.

x = 4

hours (rejecting

x = -3

since

x > 0

).

Time must be positive.

Anahtar Kavram

Algebraic Work-Rate Modeling and Quadratic Solution

Tahmini Süre:2m 0s

Soru 152Soru →

A water reservoir is filled by Pipe A and Pipe B operating simultaneously at their respective constant rates. Operating together at their original rates, the two pipes can fill the empty reservoir completely in $12$ hours. On a certain day, both pipes begin filling the empty reservoir together at their original rates. After $4$ hours, Pipe A's rate decreases by $25\%$ , while Pipe B's rate increases by $50\%$ . Operating at these new constant rates, the two pipes require an additional $7$ hours to fill the remainder of the reservoir. How many hours would it take Pipe A, operating alone at its original rate, to fill the entire reservoir?

Cevabı ve açıklamayı göster

Cevap: 25.2

Cevap

It would take Pipe A 25.2 hours operating alone at its original rate to fill the entire reservoir.

By defining the original work rates

a

and

b

in reservoirs per hour, the initial condition yields

a + b = \frac{1}{12}

. In the first 4 hours,

\frac{1}{3}

of the job is completed, leaving

\frac{2}{3}

. Setting up the equation for the remaining job with modified rates

0.75a

and

1.5b

over 7 hours produces

7(0.75a + 1.5b) = \frac{2}{3}

. Solving this system of two linear equations yields

a = \frac{5}{126}

reservoirs per hour. Taking the reciprocal gives the time required for Pipe A alone to fill the reservoir, which is

25.2

hours.

Adım Adım Çözüm

1

Set up equations for the original rates of Pipe A (

a

) and Pipe B (

b

).

The combined original rate is

a + b = \frac{1}{12}

reservoir per hour.

Together they complete

1

reservoir in

12

hours.

2

Determine the fraction of the reservoir filled in the first 4 hours and the remaining fraction.

Work completed =

4 \times \frac{1}{12} = \frac{1}{3}

; Remaining work =

\frac{2}{3}

.

The pipes worked at their original combined rate for 4 hours.

3

Set up an equation for the work done during the remaining 7 hours at the adjusted rates.

7 \left(0.75a + 1.5b\right) = \frac{2}{3} \implies 5.25a + 10.5b = \frac{2}{3} \implies 63a + 126b = 8

.

Pipe A's rate decreases by

25\%

to

0.75a

, and Pipe B's rate increases by

50\%

to

1.5b

.

4

Solve the system of linear equations for

a

.

a = \frac{5}{126}

reservoir per hour.

Multiplying

a + b = \frac{1}{12}

by

126

yields

126a + 126b = 10.5

. Subtracting

63a + 126b = 8

gives

63a = 2.5

, so

a = \frac{2.5}{63} = \frac{5}{126}

.

5

Calculate the time for Pipe A alone to fill the entire reservoir.

Time

= \frac{1}{a} = \frac{126}{5} = 25.2

hours.

Time equals total work divided by individual rate.

Anahtar Kavram

Algebraic modeling of combined work and rates with mid-process rate modifications

Soru 153Soru →

If $x$ is a real number satisfying the radical equation $\sqrt{x + 7} - x = 1$ , what is the value of $x$ ?

-3

-2

2

3

6

Cevabı ve açıklamayı göster

Cevap: $2$

Cevap

The value of

x

is

2

.

Isolating the square root yields

\sqrt{x + 7} = x + 1

. Squaring both sides produces

x + 7 = x^2 + 2x + 1

, which reduces to the quadratic equation

x^2 + x - 6 = 0

. Factoring yields

(x + 3)(x - 2) = 0

, giving potential roots of

x = -3

and

x = 2

. Testing

x = 2

in the original expression yields

\sqrt{2+7} - 2 = 3 - 2 = 1

, confirming it is correct.

Adım Adım Çözüm

1

Isolate the radical expression on one side of the equation.

\sqrt{x + 7} = x + 1

Isolating the radical allows squaring both sides cleanly to eliminate the radical sign.

2

Square both sides of the equation.

x + 7 = (x + 1)^2 = x^2 + 2x + 1

Squaring removes the square root on the left side.

3

Rearrange into standard quadratic form

ax^2 + bx + c = 0

.

x^2 + x - 6 = 0

Subtracting

x + 7

from both sides collects all terms on one side.

4

Factor the quadratic equation and solve for potential values of

x

.

(x + 3)(x - 2) = 0 \implies x = -3 \text{ or } x = 2

Setting each factor to zero identifies potential solutions.

5

Substitute each potential solution back into the original equation

\sqrt{x + 7} - x = 1

to check for extraneous roots.

For

x = 2

:

\sqrt{2 + 7} - 2 = 3 - 2 = 1

(Valid). For

x = -3

:

\sqrt{-3 + 7} - (-3) = 2 + 3 = 5 \neq 1

(Extraneous).

Squaring an equation can introduce false solutions that must be eliminated.

Anahtar Kavram

Solving radical equations and checking for extraneous solutions

Tahmini Süre:1m 30s

Soru 154Soru →

For all real numbers $x$ , the function $g$ is defined by $g(x) = 5 - 2x^2$ . What is the value of $g(g(2))$ ?

23

-13

11

3

9

Cevabı ve açıklamayı göster

Cevap: -13

Cevap

-13

Evaluating the inner expression

g(2)

gives

5 - 2(2)^2 = 5 - 8 = -3

. Substituting

-3

back into the function

g(x)

gives

g(-3) = 5 - 2(-3)^2 = 5 - 2(9) = -13

. Thus, the correct value is

-13

.

Adım Adım Çözüm

1

Evaluate the inner function

g(2)

g(2) = 5 - 2(2)^2 = 5 - 2(4) = 5 - 8 = -3

Substitute

x = 2

into the definition of

g(x)

and follow standard order of operations (exponents before multiplication).

2

Substitute the result into the outer function to evaluate

g(-3)

g(-3) = 5 - 2(-3)^2 = 5 - 2(9) = 5 - 18 = -13

Squaring a negative number yields a positive result, so

(-3)^2 = 9

, which is then multiplied by

2

.

Anahtar Kavram

Nested Function Evaluation and Order of Operations

Tahmini Süre:45s

Soru 155Soru →

If

x

satisfies the linear equation

\frac{2(3x - 1)}{5} - \frac{4x - 3}{3} = \frac{x + 18}{15} - 2

what is the value of

3x - 5

?

7

16

-14

-12

40

Cevabı ve açıklamayı göster

Cevap: $16$

Cevap

16

Multiplying the entire equation by the common denominator

15

clears all fractions, yielding

6(3x - 1) - 5(4x - 3) = (x + 18) - 30

. Expanding both sides produces

18x - 6 - 20x + 15 = x - 12

, which simplifies to

-2x + 9 = x - 12

. Rearranging terms yields

-3x = -21

, so

x = 7

. Substituting

x = 7

into

3x - 5

gives

3(7) - 5 = 16

.

Adım Adım Çözüm

1

Clear the denominators by multiplying both sides of the equation by the least common multiple (LCM) of 5, 3, and 15, which is 15.

15 \cdot \left(\frac{2(3x - 1)}{5}\right) - 15 \cdot \left(\frac{4x - 3}{3}\right) = 15 \cdot \left(\frac{x + 18}{15}\right) - 15 \cdot 2

Eliminating fractions simplifies the linear equation into standard integer polynomial terms.

2

Simplify the products and distribute coefficients across parentheses.

3 \cdot 2(3x - 1) - 5(4x - 3) = (x + 18) - 30 \implies 6(3x - 1) - 5(4x - 3) = x - 12

Perform fractional reduction and simplify constants on the right side.

3

Expand both groupings and combine like terms on the left-hand side.

18x - 6 - 20x + 15 = x - 12 \implies -2x + 9 = x - 12

Ensure the negative sign is properly distributed to both terms inside

-5(4x - 3)

.

4

Isolate the variable

x

by subtracting

x

and

9

from both sides.

-3x = -21 \implies x = 7

Solve for the single variable

x

.

5

Substitute

x = 7

into the target expression

3x - 5

.

3(7) - 5 = 21 - 5 = 16

The question asks for the value of the algebraic expression

3x - 5

, not

x

alone.

Anahtar Kavram

Linear Equations in One Variable with Fractional Coefficients

Tahmini Süre:2m 0s

Soru 156Soru →

If $x$ and $y$ are real numbers that satisfy the absolute value inequalities $|2x - 6| \le 4$ and $|y + 3| \le 5$ , which of the following values could be equal to the product $xy$ ? Indicate all such values.

Geçerli olan tümünü seçin

-45

-35

0

8

15

Cevabı ve açıklamayı göster

Cevap: $-35$ ; $0$ ; $8$

Cevap

The possible values for the product

xy

are

-35

,

0

, and

8

.

Solving

|2x - 6| \le 4

yields

1 \le x \le 5

, and solving

|y + 3| \le 5

yields

-8 \le y \le 2

. The product

xy

attains its minimum at

5 \times (-8) = -40

and its maximum at

5 \times 2 = 10

. Since

xy

can take any value in the continuous interval

[-40, 10]

, the values

-35

,

0

, and

8

are all valid choices.

Adım Adım Çözüm

1

Solve the inequality

|2x - 6| \le 4

for

x

.

-4 \le 2x - 6 \le 4 \implies 2 \le 2x \le 10 \implies 1 \le x \le 5

.

Unfold the absolute value into a compound inequality to determine the valid range for

x

.

2

Solve the inequality

|y + 3| \le 5

for

y

.

-5 \le y + 3 \le 5 \implies -8 \le y \le 2

.

Unfold the absolute value into a compound inequality to determine the valid range for

y

.

3

Determine the minimum and maximum possible values of the product

xy

.

Evaluating the extreme product combinations of endpoints:

1 \times (-8) = -8

,

1 \times 2 = 2

,

5 \times (-8) = -40

, and

5 \times 2 = 10

. Thus,

-40 \le xy \le 10

.

The continuous product of two real intervals

[a, b]

and

[c, d]

spans from the minimum endpoint product to the maximum endpoint product.

4

Select all options that fall within the interval

[-40, 10]

.

The values

-35

,

0

, and

8

lie within

[-40, 10]

, while

-45

and

15

fall outside.

Any real number within the closed interval

[-40, 10]

can be formed by valid choices of

x

and

y

.

Anahtar Kavram

Determining the range of a product from two independent absolute value inequalities.

Soru 157Soru →

For all non-zero real numbers $x$ , the function $f$ satisfies the relation $2f(x) + f\left(\frac{1}{x}\right) = 3x$ . What is the value of $f(2)$ ?

\frac{7}{2}

\frac{8}{3}

\frac{12}{5}

\frac{11}{4}

\frac{5}{2}

Cevabı ve açıklamayı göster

Cevap: $\frac{7}{2}$

Cevap

The value of

f(2)

is

\frac{7}{2}

.

To solve for

f(2)

, set up a system of equations by evaluating the given relation

2f(x) + f(1/x) = 3x

at

x = 2

and at

x = 1/2

. Substituting

x = 2

gives

2f(2) + f(1/2) = 6

, and substituting

x = 1/2

gives

2f(1/2) + f(2) = 3/2

. Expressing

f(1/2)

from the first equation as

6 - 2f(2)

and substituting it into the second yields

2(6 - 2f(2)) + f(2) = 3/2

, which simplifies to

-3f(2) = -21/2

, so

f(2) = 7/2

.

Adım Adım Çözüm

1

Substitute

x = 2

into the functional relation.

2f(2) + f\left(\frac{1}{2}\right) = 3(2) = 6

This yields a linear relationship involving

f(2)

and

f\left(\frac{1}{2}\right)

.

2

Substitute

x = \frac{1}{2}

into the original functional relation.

2f\left(\frac{1}{2}\right) + f(2) = 3\left(\frac{1}{2}\right) = \frac{3}{2}

This provides a second independent linear equation with the same two unknown quantities,

f(2)

and

f\left(\frac{1}{2}\right)

.

3

Multiply the second equation by

2

to eliminate

f\left(\frac{1}{2}\right)

.

4f\left(\frac{1}{2}\right) + 2f(2) = 3

Preparing to eliminate

f(2)

or

f\left(\frac{1}{2}\right)

using elimination.

4

Solve the system of equations for

f(2)

.

From

2f(2) + f\left(\frac{1}{2}\right) = 6

, we get

f\left(\frac{1}{2}\right) = 6 - 2f(2)

. Substituting this into the second equation

2(6 - 2f(2)) + f(2) = \frac{3}{2} \implies 12 - 4f(2) + f(2) = \frac{3}{2} \implies -3f(2) = \frac{3}{2} - 12 = -\frac{21}{2} \implies f(2) = \frac{7}{2}

.

Algebraic reduction isolates

f(2)

.

Anahtar Kavram

Functional Equations and System of Equations via Variable Substitution

Soru 158Soru →

For all positive real numbers $a$ and $b$ , which of the following expressions are equivalent to $\left( \frac{a^{-2} b^3}{\sqrt{a^4 b^{-2}}} \right)^{-1/2}$ ? Select all that apply.

Geçerli olan tümünü seçin

\frac{a^2}{b^2}

\left( \frac{b}{a} \right)^{-2}

\sqrt{\frac{a^4}{b^4}}

\frac{a^{-2}}{b^{-2}}

\frac{b^2}{a^2}

Cevabı ve açıklamayı göster

Cevap: $\frac{a^2}{b^2}$ ; $\left( \frac{b}{a} \right)^{-2}$ ; $\sqrt{\frac{a^4}{b^4}}$

Cevap

The expressions equivalent to the given quantity are

\frac{a^2}{b^2}

,

\left( \frac{b}{a} \right)^{-2}

, and

\sqrt{\frac{a^4}{b^4}}

.

Simplifying the original expression step-by-step yields

\frac{a^2}{b^2}

. The expressions

\frac{a^2}{b^2}

,

\left( \frac{b}{a} \right)^{-2} = \left( \frac{a}{b} \right)^2 = \frac{a^2}{b^2}

, and

\sqrt{\frac{a^4}{b^4}} = \frac{a^2}{b^2}

are all identical to the simplified form.

Adım Adım Çözüm

1

Simplify the radical in the denominator of the inner expression.

Since

a > 0

and

b > 0

,

\sqrt{a^4 b^{-2}} = \sqrt{a^4} \cdot \sqrt{b^{-2}} = a^2 b^{-1}

.

Applying the square root to each variable power individually.

2

Simplify the expression inside the outer parenthesis.

\frac{a^{-2} b^3}{a^2 b^{-1}} = a^{-2 - 2} b^{3 - (-1)} = a^{-4} b^4.

Subtracting exponents of like bases according to the quotient rule of exponents.

3

Apply the outer exponent of

-\frac{1}{2}

.

(a^{-4} b^4)^{-1/2} = (a^{-4})^{-1/2} (b^4)^{-1/2} = a^2 b^{-2} = \frac{a^2}{b^2}.

Multiplying internal exponents by

-\frac{1}{2}

according to the power rule of exponents.

4

Evaluate each choice against the simplified form

\frac{a^2}{b^2}

.

The expressions

\frac{a^2}{b^2}

,

\left(\frac{b}{a}\right)^{-2}

, and

\sqrt{\frac{a^4}{b^4}}

are all algebraically equivalent to

\frac{a^2}{b^2}

.

Testing algebraic equivalence using standard exponent and radical laws.

Anahtar Kavram

Properties of exponents and radicals, including power of a quotient, quotient rule, negative exponents, and square roots of powers.

Tahmini Süre:1m 30s

Soru 159Soru →

A total initial capital of $\$100,000$ is split between Fund X and Fund Y. Fund X yields an annual simple interest rate of $r\%$ , while Fund Y yields an annual simple interest rate of $(r + 2)\%$ . Under the original capital allocation, the ratio of the annual interest earned from Fund X to the annual interest earned from Fund Y is $15 : 14$ . If the initial allocation to Fund X had been increased by $25\%$ by transferring funds directly from Fund Y, the total annual interest earned from both funds combined would have been $\$5,500$ . What was the original amount allocated to Fund X?

\$40,000

\$50,000

\$60,000

\$64,000

\$75,000

Cevabı ve açıklamayı göster

Cevap: $\$60,000$

Cevap

The original amount allocated to Fund X was

\$60,000

.

The correct option is

\$60,000

. Substituting

X = 60,000

gives Fund Y allocation

Y = 40,000

. From the modified interest condition,

100,000r - 2.5(60,000) = 350,000

, giving

r = 5\%

. Fund X yields

60,000 \times 0.05 = \$3,000

and Fund Y yields

40,000 \times 0.07 = \$2,800

, which satisfies the ratio

3,000 : 2,800 = 15 : 14

. Under the modified allocation (

75,000

in X and

25,000

in Y), interest is

75,000(0.05) + 25,000(0.07) = 3,750 + 1,750 = \$5,500

.

Adım Adım Çözüm

1

Formulate variables and express initial conditions.

Let

X

be the initial capital in Fund X and

Y = 100,000 - X

be the initial capital in Fund Y. The interest rates are

\frac{r}{100}

and

\frac{r+2}{100}

respectively.

Establish a single-variable representation for the fund allocations.

2

Set up the interest ratio equation.

\frac{X \cdot r}{(100,000 - X)(r+2)} = \frac{15}{14} \implies 14 X r = 15(100,000 - X)(r+2).$

Relate the original interest outputs according to the

15:14

ratio.

3

Model the modified allocation scenario.

Fund X becomes

1.25X

and Fund Y becomes

100,000 - 1.25X

. Total interest equation:

(1.25X)\left(\frac{r}{100}\right) + (100,000 - 1.25X)\left(\frac{r+2}{100}\right) = 5,500.

Express total combined interest under the hypothetical

25\%

transfer.

4

Simplify the total interest equation to express

r

in terms of

X

.

1.25Xr + 100,000r + 200,000 - 1.25Xr - 2.5X = 550,000 \implies 100,000r - 2.5X = 350,000 \implies r = 3.5 + 0.000025X.$

Eliminate the

Xr

product term to solve for

r

linearly.

5

Substitute

r

into the ratio equation and solve for

X

.

Expanding

29Xr + 30X = 1,500,000r + 3,000,000

with

r = 3.5 + 0.000025X

yields

0.000725X^2 + 94X - 8,250,000 = 0

, which factors to give

X = 60,000

.

Determine the exact value for the initial allocation to Fund X.

Anahtar Kavram

Algebraic modeling of multi-variable financial rate and allocation systems.

Tahmini Süre:3m 0s

Soru 160Soru →

Two automated data processing algorithms, Algorithm X and Algorithm Y, operate at their respective constant rates. Algorithm X takes $x$ hours to process a full dataset when working alone. Algorithm Y operates at a constant rate that is $50\%$ faster than Algorithm X.

In Scenario 1, Algorithm X processes the dataset alone for $2$ hours, after which Algorithm Y joins, and both algorithms work together for an additional $t$ hours to complete the dataset.

In Scenario 2, Algorithm Y processes the dataset alone for $3$ hours, after which Algorithm X joins, and both algorithms complete the remaining work together in $\frac{4}{5}t$ hours.

Which of the following statements must be true? Indicate all such statements.

Geçerli olan tümünü seçin

Algorithm X operating alone requires 14 hours and 30 minutes to process one full dataset.

Algorithm Y operating alone requires 10 hours and 15 minutes to process one full dataset.

If both algorithms operate together from the beginning, they will complete one full dataset in 5 hours and 48 minutes.

The total time spent by Algorithm X working across Scenario 1 is 5 hours.

Algorithm Y processes exactly 60 percent of the dataset in Scenario 2.

Cevabı ve açıklamayı göster

Cevap: Algorithm X operating alone requires 14 hours and 30 minutes to process one full dataset.; If both algorithms operate together from the beginning, they will complete one full dataset in 5 hours and 48 minutes.

Cevap

The true statements are that Algorithm X operating alone requires 14 hours and 30 minutes to process one full dataset, and that operating together from the start, both algorithms complete one full dataset in 5 hours and 48 minutes.

The statement specifying that Algorithm X takes 14 hours and 30 minutes alone is correct because solving the system yields

x = 14.5

hours. The statement regarding the combined time of 5 hours and 48 minutes is correct because

\frac{1}{R_X + R_Y} = 5.8

hours, which equals 5 hours and 48 minutes.

Adım Adım Çözüm

1

Define individual and combined work rates in terms of

x

.

Rate of Algorithm X is

R_X = \frac{1}{x}

datasets/hr. Since Algorithm Y is

50\%

faster, its rate is

R_Y = 1.5 R_X = \frac{3}{2x}

datasets/hr. Their combined rate is

R_{X+Y} = \frac{1}{x} + \frac{3}{2x} = \frac{5}{2x}

datasets/hr.

Establishing accurate rate expressions is essential for modeling combined work scenarios.

2

Formulate equations for both scenarios and solve for

t

and

x

.

From Scenario 1:

2 \cdot \frac{1}{x} + t \cdot \frac{5}{2x} = 1 \implies 2 + 2.5t = x

.
From Scenario 2:

3 \cdot \frac{3}{2x} + \frac{4}{5}t \cdot \frac{5}{2x} = 1 \implies 4.5 + 2t = x

.
Equating the two expressions:

2 + 2.5t = 4.5 + 2t \implies 0.5t = 2.5 \implies t = 5

hours.
Substituting

t = 5

yields

x = 2 + 2.5(5) = 14.5

hours (14 hours 30 minutes).

Setting work completed equal to 1 dataset in both scenarios yields a solvable system of equations.

3

Evaluate the given statement choices using the derived parameters.

1. Algorithm X alone time:

x = 14.5

hours = 14 hours 30 minutes (True).
2. Algorithm Y alone time:

\frac{1}{R_Y} = \frac{2x}{3} = \frac{29}{3} = 9

hours 40 minutes (False).
3. Combined time from start:

\frac{1}{R_{X+Y}} = \frac{2x}{5} = \frac{29}{5} = 5.8

hours = 5 hours 48 minutes (True).
4. Total time Algorithm X works in Scenario 1:

2 + t = 2 + 5 = 7

hours (False).
5. Work fraction by Algorithm Y in Scenario 2:

(3 + \frac{4}{5}(5)) \cdot \frac{3}{2x} = 7 \cdot \frac{3}{29} = \frac{21}{29} \approx 72.4\%

(False).

Direct numerical verification reveals which statements hold true.

Anahtar Kavram

Combined work rate equations with variable initial delays and relative rate multipliers.

Algebra

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Sık Yapılan Hatalar

Alternatif Yöntem

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