Algebra Alıştırma Soruları — GRE General Test — Sayfa 9

Soru 161Soru →

An electronics manufacturer produces three types of circuit boards: Alpha, Beta, and Gamma. Production requires processing across three specialized workstations: Solder, Component Placement, and Inspection.

- Each Alpha board requires 2 hours of Solder, 3 hours of Component Placement, and 1 hour of Inspection.
- Each Beta board requires 1 hour of Solder, 4 hours of Component Placement, and 2 hours of Inspection.
- Each Gamma board requires 3 hours of Solder, 2 hours of Component Placement, and 4 hours of Inspection.

During a given production cycle, the Solder station was operated for 55 hours, the Component Placement station for 85 hours, and the Inspection station for 65 hours. If all three workstations were operated at full capacity with no downtime, what was the total number of circuit boards produced?

Cevabı ve açıklamayı göster

Cevap: 28

Cevap

The total number of circuit boards produced is 28.

Setting up equations for total machine hours yields

2x + y + 3z = 55

,

3x + 4y + 2z = 85

, and

x + 2y + 4z = 65

, where

x

,

y

, and

z

represent the quantities of Alpha, Beta, and Gamma boards produced, respectively. Subtracting the third equation from the first equation gives

(2x + y + 3z) - (x + 2y + 4z) = 55 - 65

, which simplifies to

x - y - z = -10

, or

x = y + z - 10

. Substituting

x = y + z - 10

into the second and third equations produces the 2x2 system

7y + 5z = 115

and

3y + 5z = 75

. Subtracting these two equations eliminates

z

, giving

4y = 40

, so

y = 10

. Substituting

y = 10

into

3y + 5z = 75

gives

30 + 5z = 75

, so

z = 9

. Substituting

y = 10

and

z = 9

into

x = y + z - 10

gives

x = 9

. The total number of circuit boards produced is

x + y + z = 9 + 10 + 9 = 28

.

Adım Adım Çözüm

1

Set up the 3x3 system of linear equations based on workstation hours.

2x + y + 3z = 55

,

3x + 4y + 2z = 85

, and

x + 2y + 4z = 65

Each equation models the total operational hours used across the three product types.

2

Subtract the third equation from the first equation to isolate

x

in terms of

y

and

z

.

x = y + z - 10

Eliminating terms directly reduces coefficient complexity.

3

Substitute

x = y + z - 10

into the second and third equations to construct a 2x2 system.

3y + 5z = 75

and

7y + 5z = 115

Reducing to a two-variable system allows direct elimination.

4

Subtract the two reduced equations to solve for

y

and

z

.

y = 10

and

z = 9

The

5z

terms cancel out upon subtraction.

5

Determine

x

and compute the total sum

x + y + z

.

x = 9

, total

= 9 + 10 + 9 = 28

The question asks for the total quantity of circuit boards produced.

Anahtar Kavram

Systems of Linear Equations

Soru 162Soru →

If $x$ is an integer that satisfies both $|2x - 3| \le 9$ and $\frac{5 - 3x}{-2} > 4$ , how many possible values of $x$ exist?

2

3

5

8

10

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

There are 2 possible integer values of

x

(specifically

x = 5

and

x = 6

).

Solving

|2x - 3| \le 9

yields

-3 \le x \le 6

. Solving

\frac{5 - 3x}{-2} > 4

requires reversing the inequality sign twice (first when multiplying by

-2

, giving

5 - 3x < -8

, and second when dividing by

-3

, giving

x > \frac{13}{3} \approx 4.33

). The integers satisfying both

-3 \le x \le 6

and

x > 4.33

are

5

and

6

, which gives exactly 2 possible values.

Adım Adım Çözüm

1

Solve the absolute value inequality

|2x - 3| \le 9

.

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

.

An absolute value inequality of the form

|u| \le k

unfolds into the compound inequality

-k \le u \le k

.

2

Solve the linear inequality

\frac{5 - 3x}{-2} > 4

.

5 - 3x < -8 \implies -3x < -13 \implies x > \frac{13}{3} \approx 4.33

.

Multiplying or dividing an inequality by a negative number requires reversing the direction of the inequality sign at each step.

3

Find the intersection of the two solution sets for integer values of

x

.

The range of valid values is

4.33 < x \le 6

. The integers in this interval are

x = 5

and

x = 6

.

Combining

-3 \le x \le 6

and

x > 4.33

restricts

x

to integer values strictly greater than

4.33

and less than or equal to

6

.

4

Count the valid integer values.

There are 2 integers (

5

and

6

).

Direct counting of the valid integer set

\{5, 6\}

.

Anahtar Kavram

Solving absolute value inequalities and compound linear inequalities with negative multipliers

Soru 163Soru →

A catering service charges a fixed base setup fee of $\$150$ plus $\$25$ per guest for a private dinner event. If a client was billed a total of $\$900$ for an event, how many guests attended?

24

30

34

36

42

Cevabı ve açıklamayı göster

Cevap: $30$

Cevap

30 guests attended the event.

The total billing model is represented by the equation

150 + 25g = 900

, where

g

is the number of guests. Subtracting the fixed setup fee of

\$150

gives

25g = 750

. Dividing

750

by

25

yields

g = 30

, showing that exactly

30

guests attended.

Adım Adım Çözüm

1

Formulate a linear modeling equation for total cost

\text{Total Cost} = \text{Base Fee} + (\text{Rate per Guest} \times g)

, where

g

represents the number of guests. Thus,

150 + 25g = 900

.

The total charge consists of a fixed constant component plus a variable component that depends on the number of guests.

2

Isolate the variable term by subtracting the fixed setup fee from both sides

25g = 900 - 150 \implies 25g = 750

.

Subtracting the fixed fee isolates the portion of the total cost spent strictly on guests.

3

Solve for

g

by dividing by the per-guest rate

g = \frac{750}{25} = 30

.

Dividing the variable total cost by the rate per guest yields the exact number of attending guests.

Anahtar Kavram

Linear Modeling and Single-Variable Word Problems

Tahmini Süre:1m 0s

Soru 164Soru →

For all real numbers $a$ and $b$ , the custom operation $\star$ is defined by $a \star b = a(b + 2) - b$ . What is the value of $4 \star (-3)$ ?

-7

-4

-1

1

7

Cevabı ve açıklamayı göster

Cevap: $-1$

Cevap

-1

Substituting

a = 4

and

b = -3

into the given operation gives

4((-3) + 2) - (-3) = 4(-1) + 3 = -4 + 3 = -1

. The value of the expression is

-1

.

Adım Adım Çözüm

1

Substitute

a = 4

and

b = -3

into the custom operation formula

a \star b = a(b + 2) - b

.

4 \star (-3) = 4((-3) + 2) - (-3)

The definition specifies replacing variable

a

with

4

and variable

b

with

-3

.

2

Evaluate the expression inside the parentheses.

-3 + 2 = -1

, simplifying the expression to

4(-1) - (-3)

Perform operations inside grouping symbols first according to the standard order of operations.

3

Perform the multiplication and simplify the double negative.

4(-1) = -4

and

-(-3) = +3

, yielding

-4 + 3 = -1

Multiplying positive and negative yields negative, and subtracting a negative number is equivalent to addition.

Anahtar Kavram

Evaluating Custom Binary Operations

Tahmini Süre:45s

Soru 165Soru →

For all real numbers

x

such that

x \neq 0

and

x \neq 1

, the function

f

satisfies the equation

f(x) + f\left(\frac{x - 1}{x}\right) = 1 + x

What is the value of

f(2)

?

Cevabı ve açıklamayı göster

Cevap: 0.75

Cevap

The value of

f(2)

is

0.75

(or

\frac{3}{4}

).

Evaluating the given equation at

x = 2

,

x = 1/2

, and

x = -1

generates a system of three linear equations:

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

,

f(1/2) + f(-1) = 1.5

, and

f(-1) + f(2) = 0

. Adding all three equations yields

2(f(2) + f(1/2) + f(-1)) = 4.5

, which simplifies to

f(2) + f(1/2) + f(-1) = 2.25

. Subtracting the equation

f(1/2) + f(-1) = 1.5

from

2.25

gives

f(2) = 0.75

.

Adım Adım Çözüm

1

Determine the cyclic sequence of inputs starting at

x = 2

Evaluating the inner transformation

g(x) = \frac{x-1}{x}

yields

g(2) = \frac{1}{2}

,

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

, and

g(-1) = 2

. The input sequence cycles every 3 steps.

Identifying that

g(g(g(x))) = x

allows constructing a closed system of linear equations.

2

Set up a system of three linear equations using the cycle values

For

x = 2

:

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

.
For

x = \frac{1}{2}

:

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

.
For

x = -1

:

f(-1) + f(2) = 0

.

Substituting each element of the cycle into the given relation produces three linear equations in three unknown function values.

3

Sum the equations and isolate

f(2)

Summing all three equations gives

2\left[f(2) + f\left(\frac{1}{2}\right) + f(-1)\right] = \frac{9}{2}

, so

f(2) + f\left(\frac{1}{2}\right) + f(-1) = \frac{9}{4}

. Subtracting

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

yields

f(2) = \frac{9}{4} - \frac{6}{4} = \frac{3}{4} = 0.75

.

Symmetric summation provides the total sum of the three function values, allowing direct elimination of the other two variables.

Anahtar Kavram

Solving cyclic functional equations by forming a system of linear equations

Soru 166Soru →

Consider the following system of three linear equations in variables

x

,

y

, and

z

, where

k

is a real constant:

\begin{aligned} 2x + 3y - z &= 11 \\ x - 2y + 4z &= -3 \\ 4x - y + 7z &= k \end{aligned}

For what value of

k

does the system have at least one solution

(x, y, z)

?

-5

5

8

17

23

Cevabı ve açıklamayı göster

Cevap: $5$

Cevap

5

The left-hand side of the third equation is a linear combination of the first two equations:

1 \cdot (2x + 3y - z) + 2 \cdot (x - 2y + 4z) = 4x - y + 7z

. For the linear system to be consistent and possess at least one solution, the same linear combination must hold for the constant terms on the right-hand side:

1(11) + 2(-3) = 11 - 6 = 5

. Therefore, the value of

k

must be

5

.

Adım Adım Çözüm

1

Analyze the variable coefficients across the three equations for linear dependence.

Notice that the coefficients of the third equation can be expressed as a linear combination of the first two equations.

If the left-hand side of the third equation is a linear combination of the first two equations, the system will only be consistent if the right-hand side constants satisfy the exact same linear combination.

2

Determine the multiplier needed to produce the third equation's left-hand side.

Multiply the second equation by

2

and add it to the first equation:

(2x + 3y - z) + 2(x - 2y + 4z) = 4x - y + 7z

.

This yields the exact expression

4x - y + 7z

present on the left-hand side of the third equation.

3

Apply the identical combination to the right-hand side constant terms.

The combined constant value is

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

.

For the system to have at least one solution (i.e., to avoid contradiction and be consistent),

k

must equal this computed value of

5

.

Anahtar Kavram

Linear Dependence and Consistency in 3x3 Linear Systems

Tahmini Süre:2m 0s

Soru 167Soru →

A retailer purchases items for $x$ dollars each and sets the regular selling price at a $40\%$ markup above the purchase price. During a clearance event, the retailer discounts the regular selling price by $15\%$ . Which of the following statements regarding the pricing of an item must be true? Select all such statements.

Geçerli olan tümünü seçin

The regular selling price of an item is

1.40x

dollars.

The clearance sale price of an item is

1.19x

dollars.

The clearance sale price represents a

25\%

markup above the original purchase price

x

.

The dollar profit earned on each item sold during the clearance event is

0.25x

dollars.

Cevabı ve açıklamayı göster

Cevap: The regular selling price of an item is $1.40x$ dollars.; The clearance sale price of an item is $1.19x$ dollars.

Cevap

The statements confirming that the regular selling price is

1.40x

dollars and that the clearance sale price is

1.19x

dollars are correct.

The regular selling price is computed by increasing the cost

x

by 40%, yielding

1.40x

. The clearance price is found by reducing

1.40x

by 15%, which equals

1.40x \times 0.85 = 1.19x

. Therefore, both the statement that the regular price is

1.40x

dollars and the statement that the clearance price is

1.19x

dollars are correct.

Adım Adım Çözüm

1

Model the regular selling price using the original purchase price

x

and the

40\%

markup.

Regular price

= x \times (1 + 0.40) = 1.40x

.

A 40% markup increases the base cost

x

by 0.40 times its original value.

2

Model the clearance sale price by applying a

15\%

discount to the regular selling price.

Clearance price

= 1.40x \times (1 - 0.15) = 1.40x \times 0.85 = 1.19x

.

A 15% discount reduces the regular price to 85% of its value.

3

Calculate the effective markup percentage and profit per item to evaluate all statements.

Profit

= 1.19x - x = 0.19x

, which is a

19\%

net markup over the purchase price

x

.

Comparing

1.19x

to the base cost

x

shows a net gain of

0.19x

per unit.

Anahtar Kavram

Applying successive percent changes in algebraic modeling by using correct base values.

Tahmini Süre:1m 0s

Soru 168Soru →

Two water pumps, Pump $A$ and Pump $B$ , operate at constant rates to fill a $910$ -gallon storage tank. The rate at which Pump $A$ fills the tank is $15$ gallons per hour more than twice the rate of Pump $B$ . Both pumps begin working together to fill the empty tank. After $4$ hours, Pump $A$ shuts down, and Pump $B$ continues working alone for another $5$ hours to finish filling the tank completely. What is the pumping rate of Pump $A$ , in gallons per hour?

50

85

100

115

130

Cevabı ve açıklamayı göster

Cevap: 115

Cevap

115 gallons per hour

Letting

r

represent the rate of Pump B in gallons per hour, the rate of Pump A is

2r + 15

. During the first 4 hours, both pumps operate together at a combined rate of

(3r + 15)

gallons per hour, contributing

4(3r + 15) = 12r + 60

gallons. In the remaining 5 hours, Pump B alone contributes

5r

gallons. Setting the total amount equal to the tank capacity gives

12r + 60 + 5r = 910

, which simplifies to

17r = 850

, so

r = 50

. Substituting

r = 50

into

2r + 15

yields

115

gallons per hour.

Adım Adım Çözüm

1

Define the variable for the unknown rate.

Let

r

be the pumping rate of Pump

B

in gallons per hour. Then the rate of Pump

A

is

2r + 15

gallons per hour.

Expressing both rates in terms of a single variable

r

allows us to formulate a linear equation in one variable.

2

Determine the combined pumping rate and the work done during the first phase.

Combined rate

= r + (2r + 15) = 3r + 15

gallons per hour. Work completed in 4 hours

= 4(3r + 15) = 12r + 60

gallons.

When both pumps work together, their individual rates add together.

3

Formulate the total work equation for filling the 910-gallon tank.

(12r + 60) + 5r = 910 \implies 17r + 60 = 910

The sum of the work done in the first 4 hours by both pumps and the work done in the next 5 hours by Pump B alone equals the total capacity.

4

Solve the linear equation for

r

.

17r = 850 \implies r = 50

gallons per hour.

Subtracting 60 from both sides and dividing by 17 gives the rate of Pump B.

5

Calculate the pumping rate of Pump A.

Rate of Pump

A = 2(50) + 15 = 115

gallons per hour.

Substitute

r = 50

into the expression for Pump A's rate (

2r + 15

).

Anahtar Kavram

Linear equation formulation from work and rate relationships

Soru 169Soru →

Container X contains a liquid solution that is $80\%$ alcohol by volume, and Container Y contains a liquid solution that is $25\%$ alcohol by volume. A chemist removes a specific volume of solution from Container X and mixes it with a solution from Container Y to produce $100$ liters of a new mixture that is $58\%$ alcohol by volume. Following this removal, $20$ liters of pure alcohol are added to the liquid remaining in Container X. If Container X initially held $120$ liters of solution, what is the concentration of alcohol, by volume, in Container X after the pure alcohol is added?

56.67\%

68\%

70\%

80\%

85\%

Cevabı ve açıklamayı göster

Cevap: $85\%$

Cevap

85%

To find the final concentration, we first determine the volume of solution removed from Container X. Using the weighted average for the 100-liter mixture:

0.80 V_x + 0.25 (100 - V_x) = 58

, which simplifies to

0.55 V_x = 33

, so

V_x = 60

liters. Container X originally held 120 liters, so removing 60 liters leaves 60 liters of solution containing

80\% \times 60 = 48

liters of alcohol. Adding 20 liters of pure alcohol increases the total alcohol to

48 + 20 = 68

liters and the total volume to

60 + 20 = 80

liters. The final concentration is

\frac{68}{80} = 85\%

.

Adım Adım Çözüm

1

Find the volume of solution

V_x

removed from Container X to make the 100-liter mixture.

V_x = 60

liters.

Let

V_x

be the volume from X and

100 - V_x

be the volume from Y. Setting up the alcohol concentration equation:

0.80 V_x + 0.25 (100 - V_x) = 0.58(100) \implies 0.55 V_x + 25 = 58 \implies 0.55 V_x = 33 \implies V_x = 60

liters.

2

Determine the remaining solution volume and alcohol volume in Container X after removing 60 liters.

Remaining solution =

60

liters; Remaining alcohol =

48

liters.

Container X initially had

120

liters. Removing

60

liters leaves

120 - 60 = 60

liters. Since the mixture is homogeneous, the remaining liquid retains an

80\%

alcohol concentration, yielding

0.80 \times 60 = 48

liters of alcohol.

3

Calculate the total alcohol volume and total solution volume in Container X after adding 20 liters of pure alcohol.

New alcohol volume =

68

liters; New total solution volume =

80

liters.

Adding

20

liters of pure alcohol increases both the alcohol amount (

48 + 20 = 68

liters) and the total liquid volume (

60 + 20 = 80

liters).

4

Calculate the final concentration of alcohol in Container X.

Concentration =

\frac{68}{80} = 0.85 = 85\%

.

The final alcohol concentration is the ratio of final alcohol volume to final total liquid volume.

Anahtar Kavram

Algebraic Modeling of Multi-Step Mixture Solutions

Tahmini Süre:2m 30s

Soru 170Soru →

If $x$ is a real number that satisfies both $|4 - x| < 3$ and $\frac{2x - 15}{-3} \le 1$ , which of the following inequalities expresses all possible values of $x$ ?

6 \le x < 7

1 < x \le 6

1 < x < 7

x \ge 6

6 < x < 7

Cevabı ve açıklamayı göster

Cevap: $6 \le x < 7$

Cevap

The inequality expressing all possible values of

x

is

6 \le x < 7

.

The correct inequality is determined by finding the intersection of both given inequalities. The absolute value inequality

|4 - x| < 3

simplifies to

1 < x < 7

. The linear inequality

\frac{2x - 15}{-3} \le 1

simplifies to

x \ge 6

after reversing the inequality sign when multiplying by

-3

. Combining

1 < x < 7

and

x \ge 6

yields the compound inequality stating

x

is greater than or equal to

6

and strictly less than

7

.

Adım Adım Çözüm

1

Solve the absolute value inequality

|4 - x| < 3

.

-3 < 4 - x < 3 \implies -7 < -x < -1 \implies 1 < x < 7

.

An absolute value inequality

|u| < k

translates to

-k < u < k

. Multiplying or dividing by

-1

flips the inequality signs.

2

Solve the linear inequality

\frac{2x - 15}{-3} \le 1

.

2x - 15 \ge -3 \implies 2x \ge 12 \implies x \ge 6

.

Multiplying both sides of an inequality by a negative number reverses the direction of the inequality sign.

3

Find the intersection of the two solution sets.

1 < x < 7

and

x \ge 6

combine to give

6 \le x < 7

.

Values of

x

must satisfy both inequalities simultaneously.

Anahtar Kavram

Solving systems of absolute value inequalities and linear inequalities, including sign reversal rules for negative multipliers.

Soru 171Soru →

If $x$ is an integer that satisfies both $|3x - 12| \le 15$ and $|2x + 4| \ge 6$ , what is the sum of the maximum possible value of $x$ and the minimum possible value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The correct answer is 10.

Solving the first inequality

|3x - 12| \le 15

yields

-1 \le x \le 9

. Solving the second inequality

|2x + 4| \ge 6

yields

x \ge 1

or

x \le -5

. The values of

x

that satisfy both constraints are those in the interval

[1, 9]

. Within this interval, the minimum integer value is

1

and the maximum integer value is

9

. Adding these values together gives

1 + 9 = 10

.

Adım Adım Çözüm

1

Solve the bounded absolute value inequality

|3x - 12| \le 15

.

-15 \le 3x - 12 \le 15 \implies -3 \le 3x \le 27 \implies -1 \le x \le 9.

An inequality of the form

|A| \le B

(where

B \ge 0

) unwraps to the compound inequality

-B \le A \le B

.

2

Solve the boundary absolute value inequality

|2x + 4| \ge 6

.

2x + 4 \ge 6 \implies 2x \ge 2 \implies x \ge 1

, or

2x + 4 \le -6 \implies 2x \le -10 \implies x \le -5$.

An inequality of the form

|A| \ge B

(where

B > 0

) splits into two separate cases:

A \ge B

or

A \le -B

.

3

Find the intersection of the solution sets from Step 1 and Step 2.

The intersection of

[-1, 9]

and

(-\infty, -5] \cup [1, \infty)

is

[1, 9]

.

To satisfy both inequalities simultaneously,

x

must belong to both set

[-1, 9]

and set

(-\infty, -5] \cup [1, \infty)

.

4

Identify the extreme integer values of

x

and compute their sum.

\text{Minimum integer } x = 1, \quad \text{Maximum integer } x = 9; \quad 1 + 9 = 10.

The range of valid integer values for

x

is all integers from 1 to 9 inclusive.

Anahtar Kavram

Linear Inequalities and Absolute Value

Tahmini Süre:2m 0s

Soru 172Soru →

If

x

and

y

are real numbers such that

x > y > 0

and they satisfy the following system of equations:

\frac{3}{x+y} + \frac{4}{x-y} = \frac{11}{4}

\frac{5}{x+y} - \frac{2}{x-y} = \frac{1}{4}

what is the value of

x

?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The value of

x

is 3.

Substituting

u = \frac{1}{x+y}

and

v = \frac{1}{x-y}

transforms the non-linear looking equations into the linear system

3u + 4v = \frac{11}{4}

and

5u - 2v = \frac{1}{4}

. Solving this system yields

u = \frac{1}{4}

and

v = \frac{1}{2}

. Consequently,

x + y = 4

and

x - y = 2

. Adding these two equations gives

2x = 6

, so

x = 3

.

Adım Adım Çözüm

1

Introduce auxiliary variables to linearize the system.

Let

u = \frac{1}{x+y}

and

v = \frac{1}{x-y}

. The system becomes

3u + 4v = \frac{11}{4}

and

5u - 2v = \frac{1}{4}

.

Replacing non-linear reciprocal terms with simple variables allows elimination or substitution methods for linear systems.

2

Solve the system of linear equations for

u

and

v

using elimination.

Multiply

5u - 2v = \frac{1}{4}

by 2 to get

10u - 4v = \frac{1}{2}

. Add this to

3u + 4v = \frac{11}{4}

:

13u = \frac{11}{4} + \frac{2}{4} = \frac{13}{4} \implies u = \frac{1}{4}

. Then

4v = \frac{11}{4} - 3\left(\frac{1}{4}\right) = 2 \implies v = \frac{1}{2}

.

Eliminating

v

yields a single equation in

u

, which provides the values of both auxiliary variables.

3

Convert auxiliary values back to equations in

x

and

y

.

Since

u = \frac{1}{x+y} = \frac{1}{4}

, we get

x + y = 4

. Since

v = \frac{1}{x-y} = \frac{1}{2}

, we get

x - y = 2

.

Inverting the fractions restores the original variables in a standard 2x2 linear system.

4

Solve for

x

by adding the two linear equations.

(x + y) + (x - y) = 4 + 2 \implies 2x = 6 \implies x = 3

.

Adding the equations eliminates

y

directly, isolating

x

.

Anahtar Kavram

Solving systems of linear equations using substitution variables for algebraic simplification

Soru 173Soru →

For all real numbers $x$ , what is the numerical value of the expression $\frac{3^{x+2} - 3^x}{3^{x-1} + 3^{x-2}}$ ?

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

The numerical value of the expression is 18.

Factoring

3^x

from both terms in the numerator gives

3^x(3^2 - 1) = 8 \cdot 3^x

. Factoring

3^x

from both terms in the denominator gives

3^x(3^{-1} + 3^{-2}) = 3^x\left(\frac{1}{3} + \frac{1}{9}\right) = \frac{4}{9} \cdot 3^x

. Dividing the two expressions cancels

3^x

entirely, resulting in

\frac{8}{\frac{4}{9}} = 8 \times \frac{9}{4} = 18

.

Adım Adım Çözüm

1

Factor out

3^x

from the numerator.

3^{x+2} - 3^x = 3^x(3^2 - 1) = 3^x(9 - 1) = 8 \cdot 3^x

Applying the exponent rule

a^{m+n} = a^m \cdot a^n

allows factoring out the common exponential factor

3^x

.

2

Factor out

3^x

from the denominator.

3^{x-1} + 3^{x-2} = 3^x(3^{-1} + 3^{-2}) = 3^x\left(\frac{1}{3} + \frac{1}{9}\right) = 3^x\left(\frac{4}{9}\right)

Applying negative exponent rules

a^{-n} = \frac{1}{a^n}

allows evaluating the remaining numerical terms inside the parentheses.

3

Simplify the overall fraction by dividing the factored numerator by the factored denominator.

\frac{8 \cdot 3^x}{\frac{4}{9} \cdot 3^x} = \frac{8}{\frac{4}{9}} = 8 \times \frac{9}{4} = 18

The non-zero common term

3^x

cancels from both numerator and denominator, leaving a constant integer.

Anahtar Kavram

Factoring and simplifying exponential expressions with variable exponents.

Soru 174Soru →

For all real numbers $a$ and $b$ , the custom operation $\diamondsuit$ is defined by $a \diamondsuit b = a^2 - b^2 + 2ab$ . The functions $f$ and $g$ are defined by $f(x) = x \diamondsuit 2$ and $g(x) = 2 \diamondsuit x$ . If $k$ is a positive real number such that $f(k) = g(k)$ , what is the value of $f(g(-1))$ ?

-15

-7

-1

1

9

Cevabı ve açıklamayı göster

Cevap: $-7$

Cevap

-7

Applying the custom symbol rule gives

f(x) = x^2 + 4x - 4

and

g(x) = -x^2 + 4x + 4

. Equating

f(k) = g(k)

yields

2k^2 = 8

, so the positive solution is

k = 2

. Evaluating the inner function gives

g(-1) = -(-1)^2 + 4(-1) + 4 = -1

. Substituting this value into

f

gives

f(-1) = (-1)^2 + 4(-1) - 4 = -7

.

Adım Adım Çözüm

1

Express

f(x)

and

g(x)

using the definition of the custom operation

\diamondsuit

.

f(x) = x \diamondsuit 2 = x^2 - 2^2 + 2(x)(2) = x^2 + 4x - 4

and

g(x) = 2 \diamondsuit x = 2^2 - x^2 + 2(2)(x) = -x^2 + 4x + 4

.

Applying

a \diamondsuit b = a^2 - b^2 + 2ab

with

(a, b) = (x, 2)

and

(a, b) = (2, x)

separately.

2

Set

f(k) = g(k)

to solve for the positive constant

k

.

k^2 + 4k - 4 = -k^2 + 4k + 4 \implies 2k^2 = 8 \implies k^2 = 4 \implies k = 2

(since

k > 0

).

Equating the two algebraic function expressions and solving the resulting quadratic equation.

3

Evaluate the inner function expression

g(-1)

.

g(-1) = -(-1)^2 + 4(-1) + 4 = -1 - 4 + 4 = -1

.

Substituting

x = -1

into the formula for

g(x)

.

4

Evaluate the outer function

f(g(-1)) = f(-1)

.

f(-1) = (-1)^2 + 4(-1) - 4 = 1 - 4 - 4 = -7

.

Substituting the result from Step 3 into the formula for

f(x)

.

Anahtar Kavram

Evaluating algebraic custom operations, solving functional equalities, and applying nested function compositions.

Soru 175Soru →

Consider the following system of linear equations in variables

x

,

y

, and

z

, where

a

is a real constant:

\begin{aligned} x + y + z &= 6 \\ x + 2y + 3z &= 10 \\ x + 2y + (a^2 - 1)z &= a + 8 \end{aligned}

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

Geçerli olan tümünü seçin

If

a = 2

, the system has infinitely many solutions.

If

a = -2

, the system has no solution.

If

a = 2

, every solution to the system satisfies

2x + y = 8

.

If

a = 3

, the unique solution has

y = 3

.

The system has a unique solution for all real values of

a

except

a = 2

.

Cevabı ve açıklamayı göster

Cevap: If $a = 2$ , the system has infinitely many solutions.; If $a = -2$ , the system has no solution.; If $a = 2$ , every solution to the system satisfies $2x + y = 8$ .

Cevap

The correct statements are that

a = 2

yields infinitely many solutions,

a = -2

results in no solution, and for

a = 2

every solution satisfies

2x + y = 8

.

The reduced equation

(a^2 - 4)z = a - 2

determines the behavior of the system. Setting

a = 2

gives

0 = 0

, leading to infinitely many solutions where

x = z + 2

and

y = 4 - 2z

, which identically satisfies

2x + y = 8

. Setting

a = -2

gives

0 = -4

, an inconsistency yielding no solutions.

Adım Adım Çözüm

1

Eliminate

x

and

y

using elimination between the second and third equations.

(x + 2y + (a^2 - 1)z) - (x + 2y + 3z) = (a + 8) - 10 \implies (a^2 - 4)z = a - 2

Isolating the parameter dependence onto a single variable

z

reveals existence and uniqueness conditions.

2

Analyze the equation

(a - 2)(a + 2)z = a - 2

for key parameter values.

If

a = 2

,

0 \cdot z = 0

(infinitely many solutions). If

a = -2

,

0 \cdot z = -4

(no solution). If

a \neq \pm 2

,

z = \frac{1}{a + 2}

(unique solution).

Determining system consistency depends on whether the leading coefficient and right-hand side evaluate to zero.

3

Express

x

and

y

in terms of

z

for the consistent case

a = 2

.

Subtracting the first equation from the second gives

y + 2z = 4 \implies y = 4 - 2z

. Substituting into the first gives

x = z + 2

.

Parameterizing the solution set allows verification of linear combinations.

4

Evaluate the linear combination

2x + y

when

a = 2

.

2x + y = 2(z + 2) + (4 - 2z) = 2z + 4 + 4 - 2z = 8

.

Verifies that

2x + y = 8

is an invariant across all parametric solutions.

Anahtar Kavram

Parametric Analysis of 3x3 Systems of Linear Equations

Soru 176Soru →

In the $xy$ -plane, line $m$ passes through the point $(4, -2)$ and has a $y$ -intercept of $6$ . What is the slope of line $m$ ?

Cevabı ve açıklamayı göster

Cevap: -2

Cevap

The slope of line

m

is

-2

.

The

y

-intercept of

6

indicates that the line passes through

(0, 6)

. Substituting

(0, 6)

and

(4, -2)

into the slope formula yields

m = \frac{6 - (-2)}{0 - 4} = \frac{8}{-4} = -2

.

Adım Adım Çözüm

1

Identify the coordinates of two points on line

m

.

The line passes through

(4, -2)

and the

y

-intercept point

(0, 6)

.

The

y

-intercept is the point where

x = 0

.

2

Apply the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

.

Substitute

(x_1, y_1) = (4, -2)

and

(x_2, y_2) = (0, 6)

into the formula:

m = \frac{6 - (-2)}{0 - 4}

.

Slope represents the ratio of vertical change to horizontal change between two points on a line.

3

Simplify the fractional expression.

m = \frac{8}{-4} = -2

.

Dividing positive

8

by negative

4

gives

-2

.

Anahtar Kavram

Slope of a line given two points or a point and intercept

Soru 177Soru →

Three water pumps, A, B, and C, are used to drain a large industrial reservoir. Operating alone at its constant rate, Pump A can drain the reservoir in $x$ hours, where $x > 0$ . Pump B operating alone takes $x + 4$ hours to drain the reservoir. When operating together for 2 hours, Pump A and Pump B complete the exact same fraction of the total job that Pump C completes operating alone in 3 hours. If all three pumps working simultaneously at their respective constant rates can drain the entire reservoir in $\frac{9}{4}$ hours, what is the value of $x$ ?

4

6

8

10

12

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

x

is 6.

The correct answer 6 is derived by properly formulating the work rates

R_A = \frac{1}{x}

,

R_B = \frac{1}{x+4}

, and

R_C = \frac{4(x+2)}{3x(x+4)}

. Setting their sum

\frac{10(x+2)}{3x(x+4)}

equal to the combined rate

\frac{4}{9}

forms the quadratic

2x^2 - 7x - 30 = 0

, which yields the unique positive solution

x = 6

.

Adım Adım Çözüm

1

Express the individual work rates of Pump A and Pump B in terms of

x

.

R_A = \frac{1}{x}

and

R_B = \frac{1}{x+4}

. Combined rate

R_{A+B} = \frac{1}{x} + \frac{1}{x+4} = \frac{2x+4}{x(x+4)}

.

Work rate is defined as the fraction of the job completed per hour.

2

Determine the work rate of Pump C using the given relationship.

Work done by A and B in 2 hours is

2 \cdot \frac{2x+4}{x(x+4)} = \frac{4x+8}{x(x+4)}

. Since Pump C does this in 3 hours,

R_C = \frac{1}{3} \cdot \frac{4x+8}{x(x+4)} = \frac{4(x+2)}{3x(x+4)}

.

Pump C's hourly rate is one-third of the total work completed by A and B in 2 hours.

3

Sum all three rates to find the total combined rate and set it equal to the given combined rate.

R_{total} = \frac{2x+4}{x(x+4)} + \frac{4x+8}{3x(x+4)} = \frac{3(2x+4) + 4x+8}{3x(x+4)} = \frac{10(x+2)}{3x(x+4)}

. Given total time is

\frac{9}{4}

hours, total rate is

\frac{4}{9}

. Thus,

\frac{10(x+2)}{3x(x+4)} = \frac{4}{9}

.

The sum of individual rates equals the inverse of total time required when working simultaneously.

4

Solve the algebraic equation for

x

.

Multiply both sides by 9 to get

\frac{90(x+2)}{3x(x+4)} = 4 \implies \frac{30(x+2)}{x(x+4)} = 4 \implies 15(x+2) = 2x(x+4) \implies 15x + 30 = 2x^2 + 8x \implies 2x^2 - 7x - 30 = 0

. Factoring gives

(2x + 5)(x - 6) = 0

. Since

x > 0

,

x = 6

.

Solving the quadratic yields the valid positive real root for time

x

.

Anahtar Kavram

Work Rate Modeling and Rational Equation Systems

Soru 178Soru →

In the $xy$ -plane, line $k$ passes through the points $(2, 5)$ and $(6, 13)$ . What is the slope of line $k$ ?

2

\frac{1}{2}

-2

-\frac{1}{2}

\frac{9}{4}

Cevabı ve açıklamayı göster

Cevap: $2$

Cevap

The slope of line

k

is

2

.

The slope of a straight line passing through points

(x_1, y_1)

and

(x_2, y_2)

is calculated as

m = \frac{y_2 - y_1}{x_2 - x_1}

. Substituting

(2, 5)

and

(6, 13)

gives

m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2

.

Adım Adım Çözüm

1

Identify the given points on the line.

(x_1, y_1) = (2, 5)

and

(x_2, y_2) = (6, 13)

.

These coordinates provide the required values for the slope formula.

2

Apply the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

.

m = \frac{13 - 5}{6 - 2} = \frac{8}{4}

.

The slope is defined as the change in vertical position (

y

) divided by the change in horizontal position (

x

).

3

Simplify the fraction.

m = 2

.

Dividing

8

by

4

yields the simplified slope of the line.

Anahtar Kavram

Slope of a Line in Coordinate Geometry

Tahmini Süre:45s

Soru 179Soru →

If $x$ is a real number that satisfies both $|3 - x| \le 5$ and $\frac{2x - 8}{-3} \le -2$ , which of the following could be the value of $x$ ? Indicate all such values.

Geçerli olan tümünü seçin

-1

4

7

7.5

8

Cevabı ve açıklamayı göster

Cevap: $7$ ; $7.5$ ; $8$

Cevap

The values

7

,

7.5

, and

8

satisfy both inequalities.

Solving

|3 - x| \le 5

gives

-2 \le x \le 8

. Solving

\frac{2x - 8}{-3} \le -2

by multiplying by

-3

and reversing the inequality sign gives

2x - 8 \ge 6

, or

x \ge 7

. Taking the intersection of both conditions yields

7 \le x \le 8

. Among the choices, the values

7

,

7.5

, and

8

fall within this range.

Adım Adım Çözüm

1

Solve the absolute value inequality

|3 - x| \le 5

.

-2 \le x \le 8

Rewrite as a compound inequality

-5 \le 3 - x \le 5

. Subtracting

3

gives

-8 \le -x \le 2

. Multiplying by

-1

and reversing inequality signs yields

-2 \le x \le 8

.

2

Solve the linear inequality

\frac{2x - 8}{-3} \le -2

.

x \ge 7

Multiply both sides by

-3

, making sure to flip the inequality sign:

2x - 8 \ge 6

. Adding

8

yields

2x \ge 14

, so

x \ge 7

.

3

Find the intersection of the two solution sets.

7 \le x \le 8

Combining

-2 \le x \le 8

and

x \ge 7

gives the range

7 \le x \le 8

.

4

Test the given options against the combined range

7 \le x \le 8

.

The values

7

,

7.5

, and

8

fall within

[7, 8]

, while

-1

and

4

do not.

Only numbers greater than or equal to

7

and less than or equal to

8

satisfy both conditions.

Anahtar Kavram

Solving compound linear and absolute value inequalities, ensuring inequality signs are reversed when multiplying or dividing by negative numbers.

Soru 180Soru →

If $k$ is a real constant such that the inequality $|2x - 3| + |x + 4| \le k$ has no real solutions for $x$ , which of the following inequality statements expresses all possible values of $k$ ?

k < \frac{11}{2}

k \le \frac{11}{2}

k < 7

k < 11

k > \frac{11}{2}

Cevabı ve açıklamayı göster

Cevap: $k < \frac{11}{2}$

Cevap

The statement expressing all possible values of

k

is

k < \frac{11}{2}

.

The function

f(x) = |2x - 3| + |x + 4|

represents a continuous piecewise linear curve. Evaluating

f(x)

at its critical points

x = -4

and

x = \frac{3}{2}

yields

f(-4) = 11

and

f\left(\frac{3}{2}\right) = 5.5 = \frac{11}{2}

. Since the slope is

-3

for

x < -4

,

-1

for

-4 < x < \frac{3}{2}

, and

+3

for

x > \frac{3}{2}

, the global minimum value of

f(x)

across all real numbers is

\frac{11}{2}

. Consequently, the inequality

f(x) \le k

has no real solutions if and only if

k

is strictly less than this minimum value, leading to

k < \frac{11}{2}

.

Adım Adım Çözüm

1

Identify the critical points of the absolute value terms.

The terms

|2x - 3|

and

|x + 4|

change behavior at

x = \frac{3}{2}

and

x = -4

, respectively.

Absolute value functions

f(x) = |ax + b|

reach zero and change slope at their roots.

2

Evaluate

f(x) = |2x - 3| + |x + 4|

at the critical points and analyze its piecewise behavior.

At

x = -4

,

f(-4) = |-11| + |0| = 11

. At

x = \frac{3}{2}

,

f\left(\frac{3}{2}\right) = |0| + |\frac{11}{2}| = \frac{11}{2}

. For

x < -4

,

f(x) = (3 - 2x) - (x + 4) = -3x - 1

. For

-4 \le x \le \frac{3}{2}

,

f(x) = (3 - 2x) + (x + 4) = -x + 7

. For

x > \frac{3}{2}

,

f(x) = (2x - 3) + (x + 4) = 3x + 1

.

Because

f(x)

is a convex piecewise linear function that grows to

\infty

as

x \to \pm\infty

, its global minimum must occur at one of its critical points.

3

Determine the global minimum value of

f(x)

.

Comparing values,

f\left(\frac{3}{2}\right) = \frac{11}{2}

is smaller than

f(-4) = 11

, so the minimum value of

|2x - 3| + |x + 4|

for all real

x

is

\frac{11}{2}

.

The function output is always greater than or equal to

\frac{11}{2}

for any real number

x

.

4

Apply the condition for no real solutions.

For

|2x - 3| + |x + 4| \le k

to have no solutions,

k

must be strictly less than the absolute minimum value of the expression, so

k < \frac{11}{2}

.

If

k \ge \frac{11}{2}

, there is at least one

x

value (such as

x = \frac{3}{2}

) satisfying the inequality.

Anahtar Kavram

Minimizing Sums of Absolute Values and Boundary Conditions of Inequalities

Algebra

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