All practice questions

Question 1341Question →

For all positive real numbers $x$ and $y$ , the expression $\frac{(x^{3} y^{2})^{2/3} \cdot (x^{-1} y^{4})^{1/6}}{(x^2 y)^{1/2}}$ can be written in the equivalent form $x^a y^b$ , where $a$ and $b$ are constants. What is the value of $6a + 2b$ ?

Show answer & explanation

Answer: 8

Answer

The correct answer is 8.

Applying the rules of exponents systematically yields the simplified expression

x^{5/6} y^{3/2}

. By setting

a = \frac{5}{6}

and

b = \frac{3}{2}

, the linear combination

6a + 2b

evaluates to

6\left(\frac{5}{6}\right) + 2\left(\frac{3}{2}\right) = 5 + 3 = 8

.

Step-by-Step Solution

1

Apply the power of a power rule to the first term in the numerator.

(x^3 y^2)^{2/3} = x^{3 \cdot \frac{2}{3}} y^{2 \cdot \frac{2}{3}} = x^2 y^{4/3}

To raise a product to a power, raise each factor to that power by multiplying their exponents.

2

Apply the power of a power rule to the second term in the numerator.

(x^{-1} y^4)^{1/6} = x^{-1 \cdot \frac{1}{6}} y^{4 \cdot \frac{1}{6}} = x^{-1/6} y^{2/3}

To raise a product to a power, raise each factor to that power by multiplying their exponents.

3

Multiply the two simplified terms in the numerator.

(x^2 y^{4/3})(x^{-1/6} y^{2/3}) = x^{2 - \frac{1}{6}} y^{\frac{4}{3} + \frac{2}{3}} = x^{11/6} y^2

When multiplying expressions with the same base, add their exponents.

4

Simplify the denominator.

(x^2 y)^{1/2} = x^{2 \cdot \frac{1}{2}} y^{1 \cdot \frac{1}{2}} = x y^{1/2}

To raise a product to a power, raise each factor to that power by multiplying their exponents.

5

Divide the numerator by the denominator.

\frac{x^{11/6} y^2}{x y^{1/2}} = x^{\frac{11}{6} - 1} y^{2 - \frac{1}{2}} = x^{5/6} y^{3/2}

When dividing expressions with the same base, subtract the exponent of the denominator from the exponent of the numerator.

6

Identify the values of

a

and

b

and calculate

6a + 2b

.

a = \frac{5}{6}

,

b = \frac{3}{2}

, so

6a + 2b = 6\left(\frac{5}{6}\right) + 2\left(\frac{3}{2}\right) = 5 + 3 = 8

Matching the simplified expression

x^{5/6} y^{3/2}

to

x^a y^b

yields the values of the constants

a

and

b

, which are then used to calculate the required expression.

Key Concept

Simplifying rational expressions with fractional exponents using exponent rules.

Question 1342Question →

In 2024, a local coalition of historical preservationists in Philadelphia began a comprehensive project to restore several neglected structures from the colonial era. Although the diverse coalition of architects, historians, and community organizers has encountered numerous logistical obstacles, _______ remains determined to complete the restoration before the city's upcoming semiquincentennial celebration.

Which choice completes the text so that it conforms to the conventions of Standard English?

it

they

them

these

Show answer & explanation

Answer: it

Answer

it

The singular pronoun 'it' correctly agrees in number with the singular collective noun 'coalition', which serves as the antecedent. Additionally, 'it' is in the subjective case, fitting as the subject of the verb 'remains'.

Step-by-Step Solution

1

Identify the antecedent of the pronoun needed in the blank.

The antecedent is 'the diverse coalition', which is a singular collective noun.

The pronoun must agree in number and gender with its antecedent.

2

Determine the grammatical function of the pronoun in the subordinate clause.

The pronoun serves as the subject of the singular verb 'remains'.

The pronoun must be in the subjective case rather than the objective case.

3

Select the option that is both singular and in the subjective case.

The singular subjective pronoun 'it' is the correct choice.

This establishes proper pronoun-antecedent agreement and correct pronoun case.

Key Concept

Pronoun-Antecedent Agreement and Case

Question 1343Question →

Desert ants of the genus *Cataglyphis* navigate harsh, featureless environments using a method called path integration, which relies on an internal odometer. While foraging, the ants continuously track their direction and distance relative to the nest ______ this mental log allows them to return home in a direct line rather than retracing their outward path.

Which choice completes the text so that it conforms to the conventions of Standard English?

, but this

; this

, this

this

Show answer & explanation

Answer: ; this

Answer

The choice containing a semicolon followed by the pronoun 'this' is correct because it properly separates two independent clauses without creating a comma splice or a run-on sentence.

The correct option properly uses a semicolon to connect two independent clauses, which is grammatically correct and maintains the logical connection between the ants tracking their location and the resulting benefit of that tracking.

Step-by-Step Solution

1

Analyze the clause structure before and after the blank.

The clause before the blank ('While foraging, the ants continuously track their direction and distance relative to the nest') is an independent clause. The clause after the blank ('this mental log allows them to return home...') is also an independent clause.

Identifying clause types is necessary to determine the required punctuation for linking them.

2

Determine valid grammatical linkers for independent clauses.

Two independent clauses must be joined using a period, a semicolon, a colon, or a comma paired with a coordinating conjunction.

This prevents grammatical errors such as comma splices and run-on sentences.

3

Evaluate the choices based on grammatical correctness and logical relationship.

A semicolon alone is a valid linker. A comma alone creates a comma splice, and no punctuation creates a run-on. Using 'but' is logically incorrect because there is no contrast.

The correct option must satisfy both punctuation rules and transition logic.

Key Concept

Clause Boundaries and Linking

Question 1344Question →

Based on the rules of Standard English conventions, which punctuation mark should be inserted in the blank to grammatically connect the two independent clauses in the passage?

Fill in the blanks below

In

1908

, excavations at the ancient palace of Phaistos on Crete yielded a mysterious clay disc covered in enigmatic stamped symbols. Despite over a century of intense scrutiny by linguists and cryptographers, the Phaistos Disc remains undeciphered its true purpose and the language it records continue to elude scholarly consensus.

Show answer & explanation

Answer

A semicolon should be used to connect the two independent clauses.

A semicolon is the correct punctuation mark to join two independent clauses that are closely related and not joined by a coordinating conjunction. Here, the clauses on both sides of the blank are independent, and a semicolon properly links them without causing a run-on or comma splice.

Step-by-Step Solution

1

Identify the grammatical structure of the clauses surrounding the blank.

The clause before the blank ('Despite over a century of intense scrutiny by linguists and cryptographers, the Phaistos Disc remains undeciphered') and the clause after the blank ('its true purpose and the language it records continue to elude scholarly consensus') are both independent clauses.

Determining clause type dictates which punctuation marks are grammatically permissible.

2

Determine the appropriate punctuation mark to link two independent clauses without a coordinating conjunction.

A semicolon is the standard punctuation mark used to link two closely related independent clauses when no coordinating conjunction (like 'and' or 'but') is present.

Using a comma alone would create a comma splice, whereas a colon is incorrect because the second clause does not explain, illustrate, or define the first clause.

Key Concept

Semicolons are used to link two independent clauses that are closely related in thought.

Question 1345Question →

If the expression $(x + 5)^2 - (x - 3)^2$ is equivalent to $ax + b$ for all values of $x$ , where $a$ and $b$ are constants, what is the value of $a$ ?

Show answer & explanation

Answer: 16

Answer

16

Expanding

(x + 5)^2

gives

x^2 + 10x + 25

, and expanding

(x - 3)^2

gives

x^2 - 6x + 9

. Subtracting the second expression from the first requires distributing the negative sign across all terms:

x^2 + 10x + 25 - (x^2 - 6x + 9) = x^2 + 10x + 25 - x^2 + 6x - 9

. Combining like terms yields

16x + 16

. Comparing this to

ax + b

shows that the coefficient of

x

,

a

, is 16.

Step-by-Step Solution

1

Expand the first squared binomial term

(x + 5)^2 = x^2 + 10x + 25

To express the binomial square as a trinomial using the perfect square identity

(u + v)^2 = u^2 + 2uv + v^2

.

2

Expand the second squared binomial term

(x - 3)^2 = x^2 - 6x + 9

To express the binomial square as a trinomial using the perfect square identity

(u - v)^2 = u^2 - 2uv + v^2

.

3

Subtract the expanded expressions and distribute the negative sign

x^2 + 10x + 25 - x^2 + 6x - 9

To combine the terms while correctly applying the distributive property to the subtracted expression.

4

Combine like terms to simplify the polynomial

16x + 16

To find the final simplified polynomial of the form

ax + b

.

5

Compare the simplified expression to the standard form to find the value of

a

a = 16

The constant

a

represents the coefficient of the linear term

x

, which is 16.

Key Concept

Simplifying algebraic expressions by expanding binomial products and combining like terms.

Question 1346Question →

Let $c$ be a constant such that $c < 0$ . If $c(x - 3) \leq 4(x + c)$ , which of the following inequalities must be true?

x \leq \frac{7c}{c - 4}

x \geq \frac{c}{c - 4}

x \geq \frac{7c}{c - 4}

x \geq \frac{7c}{c + 4}

Show answer & explanation

Answer: $x \geq \frac{7c}{c - 4}$

Answer

x \geq \frac{7c}{c - 4}

The correct answer is obtained by first distributing terms on both sides to get

cx - 3c \leq 4x + 4c

. Rearranging the terms to isolate the variable

x

on one side yields

(c - 4)x \leq 7c

. Since

c

is a negative constant (

c < 0

), the coefficient

(c - 4)

must also be negative. Dividing both sides of the inequality by this negative coefficient requires reversing the direction of the inequality sign, which yields the final result.

Step-by-Step Solution

1

Distribute the constants on both sides of the inequality.

cx - 3c \leq 4x + 4c

This simplifies the parentheses so that variable terms can be grouped.

2

Group all terms with

x

on the left side and terms with

c

on the right side.

cx - 4x \leq 7c

Isolating the variable terms on one side makes it possible to factor and solve for

x

.

3

Factor out

x

on the left side of the inequality.

(c - 4)x \leq 7c

This expresses the left side as a product of

x

and a single coefficient.

4

Analyze the sign of the coefficient

(c - 4)

given that

c < 0

.

c - 4 < 0

Since

c

is less than

0

, subtracting

4

from

c

must result in a value less than

-4

, which is strictly negative.

5

Divide both sides of the inequality by

(c - 4)

and flip the inequality symbol.

x \geq \frac{7c}{c - 4}

Dividing an inequality by a negative number requires reversing the direction of the inequality sign.

Key Concept

Solving linear inequalities by isolating the variable, and correctly reversing the inequality direction when multiplying or dividing by a negative variable parameter.

Question 1347Question →

A software company offers a Basic subscription for $\$15$ per month and a Premium subscription for $\$30$ per month. In April, the company had a total of $1,200$ active subscriptions. In May, the number of Basic subscriptions increased by $20\%$ , and the number of Premium subscriptions increased by $5\%$ compared to April. If the company's total monthly revenue increased by $\$2,700$ from April to May, how many Premium subscriptions did the company have in April?

Show answer & explanation

Answer: 600

Answer

600

To find the number of Premium subscriptions in April, we can set up a system of linear equations. Let

B

represent the number of Basic subscriptions in April and

P

represent the number of Premium subscriptions in April. Since the total number of subscriptions is

1,200

, we have

B + P = 1,200

. The revenue increase from April to May is

\$2,700

. The increase in Basic subscriptions is

20\%

, which contributes

15 \times 0.20B = 3B

dollars to the revenue increase. The increase in Premium subscriptions is

5\%

, which contributes

30 \times 0.05P = 1.5P

dollars to the revenue increase. Thus,

3B + 1.5P = 2,700

. Substituting

B = 1,200 - P

into this equation yields

3(1,200 - P) + 1.5P = 2,700

. Solving for

P

gives

3,600 - 1.5P = 2,700

, which simplifies to

1.5P = 900

, or

P = 600

.

Step-by-Step Solution

1

Define variables and write the first equation based on the total number of subscriptions in April.

B + P = 1,200

, where

B

is the number of Basic subscriptions and

P

is the number of Premium subscriptions.

To represent the relationship between the two types of subscriptions in April.

2

Write the second equation representing the change in monthly revenue from April to May.

3B + 1.5P = 2,700

The change in revenue is the sum of the increase in revenue from each subscription type:

15(0.20B) + 30(0.05P) = 2,700

.

3

Solve the system of equations by substituting

B = 1,200 - P

into the revenue equation.

3(1,200 - P) + 1.5P = 2,700 \implies 3,600 - 1.5P = 2,700

Substitution eliminates the variable

B

, allowing us to solve for

P

directly.

4

Isolate the variable

P

to find the number of Premium subscriptions in April.

-1.5P = -900 \implies P = 600

Dividing the revenue difference by the coefficient solves for the value of

P

.

Key Concept

Solving systems of linear equations in real-life contexts involving percentage changes.

Question 1348Question →

An environmental scientist is monitoring the water level of a reservoir during a dry season. The water level, $L(d)$ , in meters, can be modeled by a linear function of the number of days, $d$ , since the start of the dry season. On day 12, the water level was 30 meters, and on day 20, the water level was 26 meters. If the water level continues to decrease at this constant rate, on which day will the water level be exactly 18 meters?

Day 18

Day 24

Day 36

Day 40

Show answer & explanation

Answer: Day 36

Answer

Day 36

To find the day when the water level is exactly 18 meters, we first find the constant rate of change (slope) using the given points

(12, 30)

and

(20, 26)

. The slope is

m = \frac{26 - 30}{20 - 12} = -0.5

meters per day. Using the point-slope form with

(12, 30)

, we get

L(d) - 30 = -0.5(d - 12)

, which simplifies to

L(d) = -0.5d + 36

. Substituting 18 for

L(d)

gives

18 = -0.5d + 36

. Subtracting 36 from both sides results in

-18 = -0.5d

, and dividing by

-0.5

yields

d = 36

. Thus, the correct answer is Day 36.

Step-by-Step Solution

1

Calculate the constant rate of change (slope,

m

) using the points

(12, 30)

and

(20, 26)

.

m = \frac{26 - 30}{20 - 12} = \frac{-4}{8} = -0.5

meters per day.

A linear function has a constant rate of change, which is determined by the slope formula:

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

.

2

Write the equation of the linear model in point-slope form and solve for the standard form.

L(d) - 30 = -0.5(d - 12) \Rightarrow L(d) = -0.5d + 36

.

Using the point-slope formula

y - y_1 = m(x - x_1)

with the point

(12, 30)

and slope

-0.5

defines the function for any day

d

.

3

Substitute

L(d) = 18

into the equation and solve for

d

.

18 = -0.5d + 36 \Rightarrow -18 = -0.5d \Rightarrow d = 36

.

Setting the dependent variable to the target water level of 18 meters yields the corresponding day.

Key Concept

Determining and evaluating linear equations from two given coordinate points

Alternative Method

Instead of constructing the entire equation, you can analyze the constant rate of change directly. The water level decreases by

4

meters (from 30 to 26) over

8

days (from day 12 to day 20), which means the rate of decrease is

0.5

meters per day. To drop from 26 meters (on day 20) to 18 meters requires a further decrease of

26 - 18 = 8

meters. At a rate of

0.5

meters per day, dropping 8 meters will take

\frac{8}{0.5} = 16

days. Adding these 16 days to day 20 yields day 36.

Estimated Time:1m 30s

Question 1349Question →

A shipping company offers two types of delivery services: standard and express. The total shipping cost, in dollars, for a package sent via standard service is modeled by the function $C(w) = 1.25w + b_s$ , where $w$ is the weight of the package, in pounds, and $b_s$ is a constant representing the flat handling fee. The total shipping cost, in dollars, for a package sent via express service is modeled by the function $E(k) = m_e k + b_e$ , where $k$ is the weight of the package, in kilograms, and $b_e$ is a constant representing the flat handling fee.

The rate of change of the express shipping cost with respect to the package's weight, in dollars per kilogram, is $2.4$ times the rate of change of the standard shipping cost with respect to the package's weight, in dollars per pound. The flat handling fee for the express service is $4.50$ dollars more than the flat handling fee for the standard service. If it costs $58.50$ dollars to ship a package weighing $26.4$ pounds using the express service, what is the cost, in dollars, to ship a package weighing $20$ pounds using the standard service? (Assume $1\text{ kilogram} = 2.2\text{ pounds}$ .)

Show answer & explanation

Answer: 43

Answer

The cost to ship a package weighing 20 pounds using the standard service is 43 dollars.

The correct answer is obtained by first calculating the rate of change of the express shipping cost (

2.4 \times 1.25 = 3.00

dollars per kilogram). Next, the weight of the package is converted to kilograms (

26.4 / 2.2 = 12

kilograms) to match the express cost function's variable. Using the given cost of

58.50

dollars for the express shipment, the express handling fee is determined to be

58.50 - 3(12) = 22.50

dollars. The standard handling fee is then found by subtracting

4.50

dollars from the express fee (

22.50 - 4.50 = 18.00

dollars). Finally, the cost of a

20

-pound standard package is computed as

1.25(20) + 18.00 = 43.00

dollars.

Step-by-Step Solution

1

Determine the rate of change for the express service (

m_e

).

m_e = 3.00

dollars per kilogram

The rate of change of the standard shipping cost is

1.25

dollars per pound. Since the rate of change for the express shipping cost is

2.4

times this rate,

m_e = 2.4 \times 1.25 = 3.00

dollars per kilogram.

2

Convert the weight of the express package from pounds to kilograms.

k = 12

kilograms

The weight of the package is given as

26.4

pounds. Using the conversion

1\text{ kilogram} = 2.2\text{ pounds}

, the weight in kilograms is

\frac{26.4}{2.2} = 12

kilograms.

3

Find the express flat handling fee (

b_e

) using the given cost of the express shipment.

b_e = 22.50

dollars

We are given that the cost of shipping a

12

-kilogram package using the express service is

58.50

dollars. Substituting these values into the express cost function:

3(12) + b_e = 58.50 \Rightarrow 36 + b_e = 58.50 \Rightarrow b_e = 22.50

dollars.

4

Find the standard flat handling fee (

b_s

).

b_s = 18.00

dollars

The express handling fee is

4.50

dollars more than the standard handling fee:

b_e = b_s + 4.50 \Rightarrow 22.50 = b_s + 4.50 \Rightarrow b_s = 18.00

dollars.

5

Calculate the cost to ship a

20

-pound package using the standard service.

43.00

dollars

Using the standard cost function

C(w) = 1.25w + b_s

with

w = 20

and

b_s = 18.00

:

C(20) = 1.25(20) + 18.00 = 25.00 + 18.00 = 43.00

dollars.

Key Concept

Interpreting slope, y-intercept, and rates of change of linear functions in a real-world context with unit conversions.

Question 1350Question →

In a study of cognitive processing and memory consolidation, psychologist Dr. Elena Rostova analyzed how sleep deprivation affects learning outcomes. Her research concluded that a full night of sleep not only strengthens the neural connections formed during the day but also ______

Which choice completes the passage so that it conforms to the conventions of Standard English?

integrating new information with existing knowledge

to integrate new information with existing knowledge

integrates new information with existing knowledge

the integration of new information with existing knowledge

Show answer & explanation

Answer: integrates new information with existing knowledge

Answer

integrates new information with existing knowledge

The correct answer completes the passage with the present-tense verb 'integrates', which is parallel to the present-tense verb 'strengthens' that follows 'not only'.

Step-by-Step Solution

1

Identify the grammatical structure of the sentence.

The sentence uses the correlative conjunction 'not only... but also...', with the first part followed by the singular, present-tense verb 'strengthens'.

This establishes that the second part of the conjunction must also be followed by a singular, present-tense verb to maintain parallel structure.

2

Evaluate the grammatical form of each answer choice.

Only one choice uses a singular, present-tense verb ('integrates') to complete the parallel structure.

This maintains grammatical consistency across both parts of the correlative conjunction.

Key Concept

Parallel Structure

Question 1351Question →

In molecular biology, histone acetylation plays a crucial role in regulating gene expression by modifying chromatin structure. When acetyl groups attach to histones, the DNA wraps more loosely around these ______ transcription factors can then access the genetic sequence and initiate the process of transcription. Consequently, this modification is vital for cell differentiation.

Which choice completes the text so that it conforms to the conventions of Standard English?

proteins;

proteins,

proteins

proteins, although

Show answer & explanation

Answer: proteins;

Answer

The choice ending with a semicolon ('proteins;') is the correct answer because it successfully links two independent clauses without a coordinating conjunction.

The correct answer is the option ending with a semicolon. The sentence contains two independent clauses: the first begins with the subordinate clause 'When acetyl groups attach to histones' followed by the main clause 'the DNA wraps more loosely around these proteins,' and the second clause is 'transcription factors can then access the genetic sequence...' A semicolon is a standard and correct way to link two closely related independent clauses.

Step-by-Step Solution

1

Identify the clause boundaries in the sentence containing the blank.

The sentence contains two distinct clauses: the first begins with 'When acetyl groups attach...' and ends at the blank, and the second begins with 'transcription factors can then...'

Understanding where one clause ends and the next begins is necessary to determine the correct punctuation or conjunction.

2

Analyze the grammatical structure of both clauses.

The first clause ('When acetyl groups attach to histones, the DNA wraps more loosely around these proteins') is an independent clause because it contains a subject ('the DNA') and a verb ('wraps') and can stand alone as a sentence. The second clause ('transcription factors can then access the genetic sequence and initiate the process of transcription') is also an independent clause with its own subject ('transcription factors') and verbs ('can access' and 'initiate').

Two independent clauses must be separated by a period, a semicolon, or a comma combined with a coordinating conjunction.

3

Evaluate the options to find the correct punctuation or conjunction that follows standard grammar rules.

The option with a semicolon ('proteins;') is the only grammatically correct way to join these two independent clauses. A comma alone ('proteins,') creates a comma splice; omitting punctuation entirely ('proteins') creates a run-on sentence; and using 'although' ('proteins, although') is logically incorrect because it incorrectly signals a contrast instead of a direct consequence.

Applying these rules identifies the grammatically correct option.

Key Concept

Linking Independent Clauses

Question 1352Question →

Consider the system of equations below.

\begin{aligned} 3(x + y) - 2(x - y) &= 20 \\ 2(x + y) + 3(x - y) &= 22 \end{aligned}

If

(x, y)

is the solution to the system of equations, what is the value of

x

?

Show answer & explanation

Answer: 5

Answer

The value of

x

is

5

.

Expanding and simplifying the given system yields

x + 5y = 20

and

5x - y = 22

. Multiplying the second equation by 5 results in

25x - 5y = 110

. Adding this to the first equation eliminates

y

, leaving

26x = 130

. Dividing both sides by 26 gives

x = 5

.

Step-by-Step Solution

1

Expand the algebraic expressions in both equations to rewrite the system in standard form.

x + 5y = 20

and

5x - y = 22

Expanding the first equation gives

3x + 3y - 2x + 2y = 20

, which simplifies to

x + 5y = 20

. Expanding the second equation gives

2x + 2y + 3x - 3y = 22

, which simplifies to

5x - y = 22

.

2

Multiply the second equation by 5 to align the coefficients of the y-terms.

25x - 5y = 110

Multiplying

5x - y = 22

by 5 allows the subtraction of

y

to cancel with the addition of

5y

in the first equation.

3

Add the first equation to the modified second equation to eliminate the y-variable and solve for x.

26x = 130

, which simplifies to

x = 5

.

Adding

(x + 5y) + (25x - 5y) = 20 + 110

yields

26x = 130

. Dividing by 26 gives the final value of

x

.

Key Concept

Solving systems of linear equations using expansion and elimination.

Alternative Method

Define substitution variables

u = x + y

and

v = x - y

. The system simplifies to

3u - 2v = 20

and

2u + 3v = 22

. Multiplying the first equation by 3 and the second by 2 gives

9u - 6v = 60

and

4u + 6v = 44

. Adding these equations yields

13u = 104 \implies u = 8

. Substituting

u = 8

back in gives

16 + 3v = 22 \implies 3v = 6 \implies v = 2

. Now, solve the system

x + y = 8

and

x - y = 2

. Adding these two equations gives

2x = 10 \implies x = 5

.

Estimated Time:1m 30s

Question 1353Question →

A manufacturer produces standard chairs and deluxe chairs. Each standard chair requires $2$ hours of assembly and $1$ hour of finishing. Each deluxe chair requires $3$ hours of assembly and $2$ hours of finishing. The manufacturer has a maximum of $240$ hours available for assembly and a maximum of $150$ hours available for finishing each day. Additionally, the manufacturer must produce at least $10$ deluxe chairs daily. If $x$ represents the number of standard chairs produced daily and $y$ represents the number of deluxe chairs produced daily, which of the following systems of inequalities best represents this situation?

\begin{aligned} 2x + 3y &\geq 240 \\ x + 2y &\geq 150 \\ y &\geq 10 \end{aligned}

\begin{aligned} 3x + 2y &\leq 240 \\ 2x + y &\leq 150 \\ y &\geq 10 \end{aligned}

\begin{aligned} 2x + 3y &\leq 240 \\ x + 2y &\leq 150 \\ y &\geq 10 \end{aligned}

\begin{aligned} 2x + 3y &\leq 240 \\ x + 2y &\leq 150 \\ y &\leq 10 \end{aligned}

Show answer & explanation

Answer: $\begin{aligned} 2x + 3y &\leq 240 \\ x + 2y &\leq 150 \\ y &\geq 10 \end{aligned}$

Answer

The system containing

2x + 3y \leq 240

,

x + 2y \leq 150

, and

y \geq 10

.

The correct system represents each constraint accurately. The assembly time requirement is

2x + 3y \leq 240

because standard chairs consume

2

hours each, deluxe chairs consume

3

hours each, and the total cannot exceed

240

. The finishing time requirement is

x + 2y \leq 150

since standard chairs consume

1

hour each, deluxe chairs consume

2

hours each, and the total cannot exceed

150

. Finally, the requirement to produce at least

10

deluxe chairs means

y \geq 10

.

Step-by-Step Solution

1

Set up the inequality for assembly time.

2x + 3y \leq 240

Each standard chair (

x

) needs

2

hours and each deluxe chair (

y

) needs

3

hours, with a maximum limit of

240

hours.

2

Set up the inequality for finishing time.

x + 2y \leq 150

Each standard chair (

x

) needs

1

hour and each deluxe chair (

y

) needs

2

hours, with a maximum limit of

150

hours.

3

Set up the inequality for the minimum production limit of deluxe chairs.

y \geq 10

The manufacturer must produce at least

10

deluxe chairs, meaning the quantity must be greater than or equal to

10

.

4

Combine the individual inequalities into a single system.

The final system is

2x + 3y \leq 240

,

x + 2y \leq 150

, and

y \geq 10

.

All three conditions must be satisfied simultaneously.

Key Concept

Systems of Linear Inequalities in Two Variables

Question 1354Question →

The passage below is from an essay about ancient Roman engineering. Which form of the verb 'remove' completes the passage so that it conforms to the conventions of Standard English?

Fill in the blanks below

In his analysis of ancient Roman infrastructure, historian Marcus Vance argues that aqueducts were designed not only to transport freshwater to growing cities but also to waste from public baths. These stone channels represented a major leap in civil engineering, vastly transforming urban sanitation and improving public health across the empire.

Show answer & explanation

Answer

remove

The sentence employs the correlative conjunction 'not only... but also...' to connect two actions. The first action is expressed using the infinitive construction 'to transport'. To maintain parallel structure, the second action must also use an infinitive construction. Because the word 'to' precedes the blank in the sentence ('but also to {{blank_1}}'), the blank must be filled with the base verb 'remove' to form the parallel infinitive 'to remove'.

Step-by-Step Solution

1

Identify the grammatical construction linking the elements in the sentence.

The sentence uses the correlative conjunction 'not only... but also...' to link two parallel actions.

Correlative conjunctions require the phrases they connect to be grammatically parallel.

2

Determine the grammatical form of the first element in the parallel construction.

The first element is the infinitive phrase 'to transport freshwater'.

This establishes that the second element must also use an infinitive verb form.

3

Select the form of the verb 'remove' that completes the second infinitive phrase parallel to 'to transport'.

Since the preposition 'to' is already present before the blank ('but also to {{blank_1}}'), the blank must be completed with the base form of the verb: 'remove'.

This creates the parallel construction 'not only to transport... but also to remove...'.

Key Concept

Parallel Structure with Correlative Conjunctions

Estimated Time:1m 0s

Question 1355Question →

A worker at a distribution center packages boxes at a constant rate. The total number of boxes, $B$ , the worker has packaged $h$ hours after starting their shift can be modeled by the equation $B = 12h + 15$ . According to the model, how many boxes were already packaged at the start of the worker's shift?

Show answer & explanation

Answer: 15

Answer

At the start of the worker's shift,

15

boxes were already packaged.

In the linear model

B = 12h + 15

, the term

15

is the constant term (y-intercept), which represents the value of

B

when

h = 0

. In this context,

h = 0

represents the start of the worker's shift. Therefore,

15

boxes were already packaged at the start of the shift.

Step-by-Step Solution

1

Identify the value of

h

that represents the start of the shift.

h = 0

The variable

h

represents the number of hours since the shift started, so the start of the shift corresponds to

0

hours.

2

Substitute

h = 0

into the given equation to find the value of

B

.

B = 15

Evaluating the equation at

h = 0

gives the initial number of packaged boxes, which is represented by the constant term of the linear equation.

Key Concept

Interpreting the y-intercept of a linear relationship in context

Question 1356Question →

In the $xy$ -plane, the graph of the linear equation $ax + by = c$ , where $a$ , $b$ , and $c$ are constants, has a slope of $\frac{2}{3}$ and passes through the point $(6, 5)$ . If $a + b = 5$ , what is the value of $c$ ?

Show answer & explanation

Answer: 15

Answer

15

Rearranging the linear equation

ax + by = c

into slope-intercept form

y = -\frac{a}{b}x + \frac{c}{b}

shows the slope is

-\frac{a}{b}

. Setting

-\frac{a}{b} = \frac{2}{3}

gives

2b = -3a

, or

b = -1.5a

. Substituting this into

a + b = 5

yields

a - 1.5a = 5

, which simplifies to

-0.5a = 5

, so

a = -10

. This means

b = -1.5(-10) = 15

. Substituting the values of

a = -10

and

b = 15

along with the point

(6, 5)

into the equation

ax + by = c

yields

(-10)(6) + (15)(5) = 15

, so the value of

c

is

15

.

Step-by-Step Solution

1

Rewrite the standard form equation

ax + by = c

in slope-intercept form.

y = -\frac{a}{b}x + \frac{c}{b}

This allows the identification of the slope of the line in terms of the coefficients

a

and

b

.

2

Set the slope expression equal to the given slope of

\frac{2}{3}

and solve for

b

in terms of

a

.

b = -1.5a

The slope of the line is

-\frac{a}{b}

, so

-\frac{a}{b} = \frac{2}{3} \implies 2b = -3a \implies b = -1.5a

.

3

Substitute

b = -1.5a

into the given equation

a + b = 5

and solve for

a

, then find

b

.

a = -10

and

b = 15

Substituting gives

a - 1.5a = 5 \implies -0.5a = 5 \implies a = -10

. Substituting

a = -10

back into the relationship gives

b = -1.5(-10) = 15

.

4

Substitute

a = -10

,

b = 15

, and the point

(6, 5)

into the equation

ax + by = c

and solve for

c

.

c = 15

(-10)(6) + (15)(5) = c \implies -60 + 75 = 15 \implies c = 15

.

Key Concept

Linear Equations in Two Variables

Question 1357Question →

A hiker begins a climb at an elevation of $1,200$ feet above sea level and climbs at a constant rate of $350$ feet per hour. The hiker's elevation, $E$ , in feet, $t$ hours after beginning the climb is given by the equation $E = 350t + 1,200$ . Which of the following is the best interpretation of $350$ in this context?

The starting elevation, in feet, of the hiker

The increase in the hiker's elevation, in feet, for each hour of climbing

The number of hours it takes the hiker to climb one foot

The total elevation, in feet, of the hiker after 350 hours of climbing

Show answer & explanation

Answer: The increase in the hiker's elevation, in feet, for each hour of climbing

Answer

The increase in the hiker's elevation, in feet, for each hour of climbing

The coefficient of

t

, which is

350

, represents the rate of change of the hiker's elevation with respect to time. Since elevation is measured in feet and time is measured in hours, this rate is

350

feet per hour. Therefore, the value

350

represents the increase in the hiker's elevation, in feet, for each hour of climbing.

Step-by-Step Solution

1

Identify the structure of the linear equation.

The equation

E = 350t + 1,200

is in the slope-intercept form

y = mx + b

, where

m = 350

is the slope and

b = 1,200

is the

y

-intercept.

Linear equations in context have constant rates of change (slope) and starting values (

y

-intercept).

2

Determine the meaning of the slope in context.

The slope

350

represents the change in the dependent variable

E

(elevation in feet) per unit change in the independent variable

t

(time in hours).

The unit of the slope is the unit of the dependent variable divided by the unit of the independent variable, which is feet per hour.

3

Interpret the positive sign of the slope.

Since

350

is positive, the elevation increases by

350

feet for each hour of climbing.

A positive slope indicates a constant increase over time.

Key Concept

Interpreting slope in a linear context

Question 1358Question →

For all positive real numbers $a$ and $b$ such that $a \neq b$ , which of the following is equivalent to the expression $\frac{a^{3/2} - b^{3/2}}{\sqrt{a} - \sqrt{b}}$ ?

a + \sqrt{ab} + b

a + b

a - \sqrt{ab} + b

a + 2\sqrt{ab} + b

Show answer & explanation

Answer: $a + \sqrt{ab} + b$

Answer

a + \sqrt{ab} + b

The correct answer is obtained by expressing the numerator as a difference of cubes:

(\sqrt{a})^3 - (\sqrt{b})^3

. Factoring this expression gives

(\sqrt{a} - \sqrt{b})(a + \sqrt{ab} + b)

. Since

a \neq b

, dividing by the denominator

\sqrt{a} - \sqrt{b}

simplifies the expression to

a + \sqrt{ab} + b

.

Step-by-Step Solution

1

Rewrite the terms in the numerator using square roots to reveal a difference of cubes pattern.

a^{3/2} = (\sqrt{a})^3

and

b^{3/2} = (\sqrt{b})^3

, so the numerator is

(\sqrt{a})^3 - (\sqrt{b})^3

.

This allows us to factor the numerator using the algebraic identity for the difference of two cubes.

2

Factor the numerator using the difference of cubes formula:

u^3 - v^3 = (u - v)(u^2 + uv + v^2)

.

(\sqrt{a})^3 - (\sqrt{b})^3 = (\sqrt{a} - \sqrt{b})(a + \sqrt{ab} + b)

where

u = \sqrt{a}

and

v = \sqrt{b}

.

Factoring allows us to identify common factors shared with the denominator.

3

Substitute the factored expression back into the fraction and cancel the common factor of

\sqrt{a} - \sqrt{b}

.

\frac{(\sqrt{a} - \sqrt{b})(a + \sqrt{ab} + b)}{\sqrt{a} - \sqrt{b}} = a + \sqrt{ab} + b

.

Since

a \neq b

,

\sqrt{a} - \sqrt{b} \neq 0

, which makes it mathematically valid to divide by this term.

Key Concept

Equivalent algebraic expressions involving fractional exponents and difference of cubes factoring

Alternative Method

Let

a = 4

and

b = 1

. Substitute these values into the original expression:

\frac{4^{3/2} - 1^{3/2}}{\sqrt{4} - \sqrt{1}} = \frac{8 - 1}{2 - 1} = 7

. Now substitute these same values into each option to see which one evaluates to 7. The correct option evaluates to

4 + \sqrt{4(1)} + 1 = 7

.

Estimated Time:2m 0s

Question 1359Question →

If the solution to the inequality $a(2 - 3x) - 4(x + 3) \ge 12$ , where $a$ is a constant, is $x \le -1$ , what is the value of $a$ ?

Show answer & explanation

Answer: 4

Answer

The value of

a

is

4

.

To solve the inequality

a(2 - 3x) - 4(x + 3) \ge 12

, we first expand it to get

2a - 3ax - 4x - 12 \ge 12

. Grouping the

x

terms gives

(-3a - 4)x \ge 24 - 2a

. Since the inequality's solution is

x \le -1

, the direction of the inequality must flip, which means the coefficient of

x

, namely

-3a - 4

, must be negative. Dividing both sides by this coefficient gives the boundary value of the inequality as

\frac{24 - 2a}{-3a - 4}

. Setting this boundary equal to

-1

yields

24 - 2a = 3a + 4

, which simplifies to

5a = 20

, or

a = 4

. Since

a = 4

makes the coefficient

-3(4) - 4 = -16

negative, the solution holds.

Step-by-Step Solution

1

Expand the inequality using the distributive property.

2a - 3ax - 4x - 12 \ge 12

This allows us to separate and group the terms containing the variable

x

and the constant terms.

2

Group like terms and isolate the variable terms on the left-hand side.

(-3a - 4)x \ge 24 - 2a

By combining the coefficients of

x

and adding

12 - 2a

to both sides, we prepare the inequality to solve for

x

.

3

Determine the effect of dividing by the variable's coefficient.

Since the given solution is

x \le -1

, the inequality sign flipped from

\ge

to

\le

. Therefore, the coefficient

-3a - 4

must be negative.

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

4

Set the boundary value of the solution equal to

-1

and solve for

a

.

a = 4

Setting the boundary of the inequality

\frac{24 - 2a}{-3a - 4}

equal to

-1

allows us to find the specific constant

a

that produces this solution set.

Key Concept

Solving linear inequalities in one variable involving parameters and sign flips.

Question 1360Question →

For all $x > 1$ , which of the following is equivalent to the expression $\frac{2x^2 - 5x - 3}{2x + 1} - (x - 4)$ ?

-7

2x - 7

1

7

Show answer & explanation

Answer: 1

Answer

1

The correct answer is

1

. Factoring the numerator of the rational expression gives

2x^2 - 5x - 3 = (2x + 1)(x - 3)

. Since

x > 1

, the denominator

2x + 1

is non-zero, allowing the expression to be simplified to

x - 3

. Subtracting

(x - 4)

and distributing the negative sign to both terms inside the parentheses yields

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

.

Step-by-Step Solution

1

Factor the quadratic expression in the numerator of the rational term.

2x^2 - 5x - 3 = (2x + 1)(x - 3)

This allows for the identification of common factors that can be simplified with the denominator.

2

Simplify the rational expression by canceling the common factor in the numerator and denominator.

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

Since

x > 1

, the term

2x + 1

is positive and non-zero, making the division valid.

3

Subtract the linear expression from the simplified rational expression, distributing the negative sign to both terms inside the parentheses.

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

This performs the final subtraction and simplifies the expression to its equivalent constant form.

Key Concept

Simplifying rational expressions by factoring and performing operations on equivalent expressions

Estimated Time:1m 15s

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