All practice questions

Question 1281Question →

Though many astronomers hypothesized that the unusual orbits of trans-Neptunian objects were caused by a massive, undiscovered world known as Planet Nine, a 2021 orbital simulation proposed a different explanation _______ the collective gravitational influence of a vast, dispersed disc of icy debris left over from the solar system's formation.

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

explanation: the

explanation; the

explanation, they cited the

explanation. The

Show answer & explanation

Answer: explanation: the

Answer

explanation: the

The correct answer is the option containing 'explanation: the'. The clause before the blank is independent, and the phrase after the blank is a noun phrase that explains the nature of the alternative explanation. A colon is standard punctuation used to connect an independent clause to a subsequent explanation, list, or quotation.

Step-by-Step Solution

1

Analyze the clause structure of the passage before and after the blank.

The text before the blank is an independent clause ('a 2021 orbital simulation proposed a different explanation'), while the text after the blank is a noun phrase ('the collective gravitational influence of a vast, dispersed disc of icy debris left over from the solar system's formation') acting as explanatory detail.

Identifying the clause boundaries determines what punctuation can legally be used to connect them.

2

Evaluate the punctuation rules for colons and semicolons.

A colon is appropriate here because it is preceded by an independent clause and introduces an explanation or elaboration of that clause. A semicolon is inappropriate because the second part is not an independent clause.

Standard conventions dictate that a colon can introduce explanatory lists or phrases, whereas a semicolon must separate two independent clauses.

Key Concept

Using a colon to connect an independent clause to explanatory details

Question 1282Question →

If $5(2a - 3) - 3(4a - 1) = -(a - 8)$ , what is the value of $a$ ?

-10

-4

-20

-16

Show answer & explanation

Answer: -20

Answer

-20

Distributing the constants on both sides of the equation

5(2a - 3) - 3(4a - 1) = -(a - 8)

yields

10a - 15 - 12a + 3 = -a + 8

. Combining like terms on the left side gives

-2a - 12 = -a + 8

. Adding

2a

to both sides results in

-12 = a + 8

. Subtracting

8

from both sides gives the correct value of

a

, which is

-20

.

Step-by-Step Solution

1

Distribute the coefficients to the terms inside the parentheses on both sides of the equation.

10a - 15 - 12a + 3 = -a + 8

To eliminate parentheses and allow terms to be combined, apply the distributive property:

5(2a - 3) = 10a - 15

,

-3(4a - 1) = -12a + 3

, and

-(a - 8) = -a + 8

.

2

Combine like terms on the left side of the equation.

-2a - 12 = -a + 8

Simplify the left side by combining the variable terms (

10a - 12a = -2a

) and the constant terms (

-15 + 3 = -12

).

3

Isolate the variable

a

on one side of the equation.

a = -20

Add

2a

to both sides of the equation to get

-12 = a + 8

, then subtract

8

from both sides to find that

a = -20

.

Key Concept

Solving linear equations in one variable using the distributive property, combining like terms, and isolating the variable.

Estimated Time:1m 30s

Question 1283Question →

Read the passage below. In the blank, enter the single punctuation mark that correctly completes the sentence by setting off the nonessential element.

Fill in the blanks below

Discovered in a shipwreck off the Greek island of Antikythera, the remains of a complex bronze device have fascinated archaeologists since 1901. This device, a system of interlocking gearwheels dating back to the second century BCEwas used to track astronomical cycles with extraordinary precision. Historians consider it to be the earliest known analog computer, pre-dating similar European technology by more than a millennium.

Show answer & explanation

Answer

A comma (,) must be placed in the blank.

The phrase 'a system of interlocking gearwheels dating back to the second century BCE' is a nonessential appositive modifying the subject 'This device'. Since the nonessential element is opened with a comma after 'device', it must be closed with a matching comma before the main verb 'was used' to prevent asymmetrical punctuation.

Step-by-Step Solution

1

Identify the nonessential element in the sentence.

The phrase 'a system of interlocking gearwheels dating back to the second century BCE' is an appositive phrase that provides nonessential details about the subject 'This device'.

Nonessential elements can be removed from a sentence without changing its core grammatical structure or meaning.

2

Check the punctuation used at the beginning of the nonessential element.

A comma is used immediately after 'This device' to open the nonessential phrase.

To maintain grammatical symmetry, the punctuation mark used to open a nonessential element must match the punctuation mark used to close it.

3

Determine the correct punctuation mark for the blank.

A comma is required in the blank to match the opening comma.

Using another punctuation mark, such as a dash or parenthesis, would create asymmetrical punctuation, which is incorrect.

Key Concept

Punctuation of Nonessential Elements

Question 1284Question →

During the excavation of the ancient Roman villa at Oplontis, a dedicated team of field archaeologists uncovered a complex network of frescoed corridors. In the report, the researchers noted that each of the mural panels, despite showing significant water damage from centuries of subterranean exposure, had retained much of _______ original pigment.

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

their

its

they

its'

Show answer & explanation

Answer: its

Answer

its

The singular possessive pronoun 'its' is correct because it agrees in number with the singular indefinite pronoun antecedent 'each.' The intervening prepositional phrase 'of the mural panels' and the parenthetical element 'despite showing significant water damage...' do not change the number of the subject 'each,' which remains singular.

Step-by-Step Solution

1

Identify the antecedent of the pronoun to be filled in.

The antecedent is the singular indefinite pronoun 'each' in the phrase 'each of the mural panels.'

Pronouns must agree in number with their antecedents. Intervening phrases like 'of the mural panels' do not affect the grammatical number of the subject.

2

Determine the required pronoun case.

The pronoun modifies the noun phrase 'original pigment,' indicating possession, so a possessive pronoun is required.

The sentence describes the pigment belonging to the panels (referred to as 'each'), necessitating the possessive case.

3

Select the option that is both singular and possessive.

The pronoun 'its' is the singular possessive form.

'its' matches the singular antecedent 'each' and correctly denotes possession without using an apostrophe.

Key Concept

Pronoun-antecedent agreement requires pronouns to match their antecedents in number (singular vs. plural). Indefinite pronouns like 'each' are grammatically singular, even when followed by a plural prepositional modifier like 'of the panels.' Additionally, possessive pronouns do not use apostrophes.

Question 1285Question →

For all positive values of $x$ and $y$ , the expression $\frac{(3x^2 y^4)^3}{9x^4 y^7}$ can be written in the form $a x^b y^c$ , where $a$ , $b$ , and $c$ are constants. What is the value of $a + b + c$ ?

Show answer & explanation

Answer: 10

Answer

The value of

a + b + c

is

10

.

To find the value of

a + b + c

, we simplify the given expression using the rules of exponents. First, apply the power of a product rule and power of a power rule to the numerator:

(3x^2 y^4)^3 = 3^3 (x^2)^3 (y^4)^3 = 27x^6y^{12}

. Next, divide this by the denominator:

\frac{27x^6y^{12}}{9x^4y^7}

. Divide the coefficients to get

\frac{27}{9} = 3

, and apply the quotient rule for exponents to the variable terms:

x^{6-4} = x^2

and

y^{12-7} = y^5

. The fully simplified expression is

3x^2y^5

, which matches the form

a x^b y^c

. Comparing coefficients and exponents, we find

a = 3

,

b = 2

, and

c = 5

. Summing these values gives

3 + 2 + 5 = 10

.

Step-by-Step Solution

1

Simplify the numerator of the expression using exponent rules.

27x^6y^{12}

Applying the power of a product rule

(ab)^n = a^n b^n

and the power of a power rule

(a^m)^n = a^{mn}

to

(3x^2 y^4)^3

gives

3^3 \cdot x^{2 \times 3} \cdot y^{4 \times 3} = 27x^6y^{12}

.

2

Divide the simplified numerator by the denominator.

3x^2y^5

Divide the coefficients (

27 \div 9 = 3

) and subtract the exponents of the corresponding variables using the quotient of powers rule

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

(

x^{6-4} = x^2

and

y^{12-7} = y^5

).

3

Identify the values of the constants

a

,

b

, and

c

from the simplified expression

3x^2y^5

.

a = 3

,

b = 2

, and

c = 5

Comparing the simplified expression

3x^2y^5

with the template

a x^b y^c

yields

a = 3

,

b = 2

, and

c = 5

.

4

Calculate the sum of

a

,

b

, and

c

.

10

Adding the identified constant values together:

3 + 2 + 5 = 10

.

Key Concept

Simplifying exponential expressions using exponent rules

Question 1286Question →

Consider the system of linear equations below, where $k$ is a constant:

\begin{aligned} 3x - 2y &= 7 \\ kx + 3y &= 18 \end{aligned}

If the system has a solution $(x, y)$ such that $x + y = 4$ , what is the value of $k$ ?

3

5

9

1

Show answer & explanation

Answer: 5

Answer

5

The correct answer is

5

. To find the value of

k

, we first express

x

in terms of

y

using the given constraint equation:

x = 4 - y

. Next, we substitute this expression into the first equation of the system:

3(4 - y) - 2y = 7

. Distributing and combining like terms yields

12 - 5y = 7

. Solving for

y

gives

y = 1

. Substituting this back into the constraint equation gives

x = 4 - 1 = 3

. Finally, we substitute the solution point

(3, 1)

into the second equation:

k(3) + 3(1) = 18 \implies 3k + 3 = 18 \implies 3k = 15 \implies k = 5

.

Step-by-Step Solution

1

Express one variable in terms of the other using the given constraint equation

x + y = 4

.

x = 4 - y

This allows us to substitute the expression for

x

into the first equation to reduce it to a single variable.

2

Substitute

x = 4 - y

into the first equation

3x - 2y = 7

and solve for

y

.

3(4-y) - 2y = 7 \implies 12 - 3y - 2y = 7 \implies 12 - 5y = 7 \implies -5y = -5 \implies y = 1

By solving this linear equation, we find the unique y-coordinate of the system's solution.

3

Substitute the value of

y = 1

back into the constraint equation to find

x

.

x = 4 - 1 = 3

This determines the x-coordinate of the solution point, giving us the full solution

(3, 1)

.

4

Substitute the point

(3, 1)

into the second equation

kx + 3y = 18

to solve for the constant

k

.

k(3) + 3(1) = 18 \implies 3k + 3 = 18 \implies 3k = 15 \implies k = 5

Since the solution must satisfy all equations in the system, we can solve for

k

using the coordinates we found.

Key Concept

Solving systems of linear equations under linear constraints using algebraic substitution.

Estimated Time:2m 0s

Question 1287Question →

Consider the system of equations below:

\begin{aligned} \frac{1}{3}x + \frac{1}{4}y &= 5 \\ x - y &= 8 \end{aligned}

If

(x, y)

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

y

?

Show answer & explanation

Answer: 4

Answer

The value of

y

is

4

.

Substituting

x = y + 8

from the second equation into the first equation yields

\frac{1}{3}(y + 8) + \frac{1}{4}y = 5

. Multiplying the entire equation by

12

to clear denominators gives

4(y + 8) + 3y = 60

, which simplifies to

7y + 32 = 60

. Solving for

y

gives

7y = 28

, or

y = 4

.

Step-by-Step Solution

1

Express

x

in terms of

y

from the second equation.

x = y + 8

This allows for substitution into the first equation to solve for

y

directly.

2

Substitute

x = y + 8

into the first equation.

_\frac{1}{3}(y + 8) + \frac{1}{4}y = 5

This reduces the system to a single-variable linear equation in terms of

y

.

3

Multiply the entire equation by the least common multiple of the denominators, which is

12

.

4(y + 8) + 3y = 60

This clears the fractional coefficients and simplifies the arithmetic.

4

Distribute the

4

and combine like terms.

7y + 32 = 60

Simplifies the equation to prepare for isolating the variable

y

.

5

Isolate the variable term by subtracting

32

from both sides, then dividing by

7

.

y = 4

This gives the final value of

y

that satisfies the system.

Key Concept

Solving systems of linear equations using substitution and clearing fractional coefficients

Question 1288Question →

A state department of transportation models the relationship between the age of a highway, in years, and its road quality index on a scale from $0$ to $100$ . The index decreases at a constant rate with respect to the age of the highway. At age $4$ years, the highway has a road quality index of $82$ . At age $12$ years, the highway has a road quality index of $66$ . According to the model, after how many years from its construction will the highway's road quality index reach $50$ ?

Show answer & explanation

Answer: 20

Answer

According to the model, the highway's road quality index will reach

50

after

20

years.

Since the relationship is linear, the rate of change is constant. Using the points

(4, 82)

and

(12, 66)

, the slope is

\frac{66 - 82}{12 - 4} = -2

. The linear function can be written in point-slope form as

I - 82 = -2(t - 4)

, which simplifies to

I = -2t + 90

. Setting

I = 50

yields the equation

50 = -2t + 90

. Subtracting

90

from both sides gives

-40 = -2t

, and dividing by

-2

results in

t = 20

. Therefore, the index reaches

50

after

20

years.

Step-by-Step Solution

1

Calculate the slope (rate of change) of the linear function.

The slope is

-2

.

Since the index decreases at a constant rate, the relationship is linear. The slope is calculated as the change in the road quality index divided by the change in the highway's age:

\frac{66 - 82}{12 - 4} = \frac{-16}{8} = -2

.

2

Determine the linear equation relating the road quality index,

I

, and the age,

t

.

I = -2t + 90

Using the point-slope formula with the point

(4, 82)

and slope

-2

:

I - 82 = -2(t - 4) \Rightarrow I - 82 = -2t + 8 \Rightarrow I = -2t + 90

.

3

Solve for the age,

t

, when the index

I

is

50

.

t = 20

Substitute

50

for

I

in the equation:

50 = -2t + 90

. Subtract

90

from both sides to get

-40 = -2t

. Divide by

-2

to find

t = 20

.

Key Concept

Writing and solving linear equations from two points

Question 1289Question →

In conservation biology, the Allee effect describes a phenomenon where a population's density drops below a critical threshold, reducing the reproductive success of individual organisms. Although species with high density easily find mates, underpopulated groups often struggle to locate breeding partners _______ this isolation can lead to a rapid, self-reinforcing decline toward local extinction.

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 correct answer is '; this' because a semicolon is a grammatically standard way to join two independent clauses.

The choice containing the semicolon followed by 'this' is correct because a semicolon is a grammatically standard way to separate two closely related independent clauses. In this sentence, 'underpopulated groups often struggle to locate breeding partners' and 'this isolation can lead to a rapid, self-reinforcing decline toward local extinction' are both independent clauses, and using a semicolon successfully avoids a comma splice or run-on error.

Step-by-Step Solution

1

Analyze the structure of the passage before and after the blank.

The text before the blank contains a subordinate clause followed by a main clause ('underpopulated groups often struggle to locate breeding partners'). The text after the blank is another independent clause ('this isolation can lead to a rapid, self-reinforcing decline toward local extinction').

Identifying the clauses helps determine what type of punctuation or conjunction is needed to link them.

2

Determine the grammatical relationship between the two independent clauses.

The two independent clauses are 'underpopulated groups often struggle to locate breeding partners' and 'this isolation can lead to a rapid, self-reinforcing decline toward local extinction.' The second clause explains the consequence of the first clause.

Because they are both independent clauses, they must be separated by a semicolon, a colon, a period, or a comma followed by a coordinating conjunction.

3

Evaluate the choices based on grammatical correctness and logical meaning.

The choice with '; this' correctly uses a semicolon to link the clauses. The choice with ', this' is a comma splice, the choice with 'this' is a run-on, and the choice with ', but this' is semantically incorrect because 'but' denotes contrast rather than consequence.

Only the correct choice is both grammatically correct and semantically logical.

Key Concept

Clause Boundaries and Linking

Estimated Time:1m 30s

Question 1290Question →

Which of the following expressions is equivalent to $\frac{2x^2 - 3x - 9}{x^2 - 9} - \frac{x}{x+3}$ for all $x > 3$ ?

1

\frac{x^2 + 6x - 9}{x^2 - 9}

\frac{x^2 - 6x - 9}{x^2 - 9}

\frac{3x+3}{x+3}

Show answer & explanation

Answer: $1$

Answer

1

The correct answer is

1

. Factoring the first term gives

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

. Since

x > 3

, we cancel the common non-zero factor of

x-3

to obtain

\frac{2x+3}{x+3}

. Subtracting

\frac{x}{x+3}

from this result yields

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

.

Step-by-Step Solution

1

Factor the numerator and the denominator of the first rational expression.

The numerator

2x^2 - 3x - 9

factors as

(2x+3)(x-3)

and the denominator

x^2 - 9

factors as

(x+3)(x-3)

.

Factoring allows us to identify and divide out common factors to simplify the expression.

2

Simplify the first rational expression by cancelling the common factor.

For all

x > 3

, the factor

x-3

is non-zero, so the expression

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

simplifies to

\frac{2x+3}{x+3}

.

This simplifies the subtraction by reducing the first fraction.

3

Subtract the second expression from the simplified first expression.

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

.

Since the denominators are identical, the numerators can be combined directly.

Key Concept

Simplifying rational expressions by factoring and performing algebraic operations.

Alternative Method

Instead of simplifying the first fraction first, find a common denominator immediately by multiplying the numerator and denominator of the second fraction by

x-3

. This yields:

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

.

Estimated Time:2m 0s

Question 1291Question →

In 1995, astronomers discovered the first exoplanet orbiting a sun-like star, known as 51 Pegasi b. This landmark discovery fundamentally altered our understanding of planetary ______ proved that giant gas planets could exist in extremely close proximity to their parent stars. Prior to this finding, scientists assumed that large planets only formed in the cold, outer regions of solar systems.

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

systems; it

systems, it

systems it

systems, but it

Show answer & explanation

Answer: systems; it

Answer

The choice containing 'systems; it' is correct because it uses a semicolon to properly link two independent clauses without a coordinating conjunction.

The correct option correctly uses a semicolon to separate two independent clauses. The clause preceding the blank ('This landmark discovery fundamentally altered our understanding of planetary systems') and the clause following the blank ('it proved that giant gas planets could exist in extremely close proximity to their parent stars') are both independent. A semicolon is a grammatically valid way to link them.

Step-by-Step Solution

1

Identify the clauses surrounding the blank.

The clause before the blank ('This landmark discovery fundamentally altered our understanding of planetary systems') and the clause after the blank ('it proved that giant gas planets could exist in extremely close proximity to their parent stars') are both independent clauses because they can each stand alone as a complete sentence.

Determining the clause types is necessary to choose the correct punctuation and conjunction rules.

2

Evaluate the choices based on standard punctuation rules for joining two independent clauses.

Two independent clauses can be joined by a semicolon (without a conjunction), a period, or a comma followed by a coordinating conjunction. The choice with 'systems; it' correctly uses a semicolon. The choice with 'systems, it' creates a comma splice. The choice with 'systems it' creates a run-on. The choice with 'systems, but it' incorrectly uses a coordinating conjunction of contrast, which does not fit the cause-and-effect relationship between the clauses.

This step ensures that the grammatical and logical relationship between the clauses is properly maintained.

Key Concept

Joining two independent clauses requires a semicolon, a colon, a period, or a comma combined with a coordinating conjunction.

Question 1292Question →

In the $xy$ -plane, a point with coordinates $(x, y)$ lies in the solution set of the system of inequalities below.

y \leq -x + 7

y \geq 2x - 5

y \geq 0

What is the maximum possible value of $x$ ?

3

4

7

2.5

Show answer & explanation

Answer: 4

Answer

The maximum possible value of

x

is 4.

To find the maximum possible value of

x

in the solution set, we analyze the boundaries of the system. The inequality

y \leq -x + 7

can be rewritten as

x \leq 7 - y

. The inequality

y \geq 2x - 5

can be rewritten as

x \leq 0.5y + 2.5

. For any point in the solution set,

x

must be less than or equal to both

7 - y

and

0.5y + 2.5

. Since

y \geq 0

, the maximum value of

x

will occur where these two upper bounds are equal:

7 - y = 0.5y + 2.5

, which simplifies to

1.5y = 4.5

, or

y = 3

. Substituting

y = 3

back into either equation gives

x = 4

. For any

y > 3

,

x \leq 7 - y < 4

. For any

y < 3

,

x \leq 0.5y + 2.5 < 4

. Thus, the maximum possible value of

x

is 4.

Step-by-Step Solution

1

Express the boundaries of

x

in terms of

y

from the given inequalities.

x \leq 7 - y

and

x \leq 0.5y + 2.5

To find the upper limits on

x

, we solve each inequality for

x

.

2

Find the intersection point of the two boundary lines by setting the expressions equal to each other.

7 - y = 0.5y + 2.5 \implies 1.5y = 4.5 \implies y = 3

. Substituting

y = 3

gives

x = 4

.

The maximum value of

x

occurs at the intersection of the two boundary constraints since one boundary increases with

y

and the other decreases with

y

.

3

Verify that the intersection point

(4, 3)

satisfies all three inequalities.

3 \leq -4 + 7

(true),

3 \geq 2(4) - 5

(true), and

3 \geq 0

(true).

Ensuring the point lies within the solution set confirms it is a valid maximum.

Key Concept

Systems of Linear Inequalities in Two Variables

Question 1293Question →

The following passage is adapted from a historical study of medieval labor. Fill in the blank with the correct form of the noun 'guild' to ensure that the sentence conforms to the conventions of Standard English.

Fill in the blanks below

In her landmark study of the socio-economic dynamics of medieval Europe, historian Sylvia Federici examines how female artisans navigated restrictive labor regulations. Although municipal registries and official documents from the period often minimized the contributions of women, several financial ledgers reveal that female members frequently managed accounts and trained apprentices.

Show answer & explanation

Answer

guilds'

The correct form is the plural possessive noun 'guilds''. The sentence contains the quantifier 'several,' which establishes that the noun must be plural ('guilds'). In addition, because the noun is modifying 'financial ledgers' to show that the ledgers belonged to these groups, it must be in the possessive case. Since 'guilds' is a regular plural noun that ends in -s, its possessive is formed by adding only an apostrophe at the end, resulting in 'guilds''.

Step-by-Step Solution

1

Analyze the word preceding the blank to determine the grammatical number of the noun.

The word 'several' is a plural quantifier, meaning the noun 'guild' must be plural ('guilds').

In English grammar, quantifiers like 'several' must modify plural countable nouns.

2

Analyze the word following the blank to determine if the noun must be possessive.

The noun phrase 'financial ledgers' immediately follows the blank. The ledgers belong to the organizations, meaning the noun must be possessive.

A noun modifying another noun to show ownership or source must be in the possessive case.

3

Apply the appropriate punctuation rule for plural possessive nouns.

For a regular plural noun ending in -s ('guilds'), the possessive form is created by adding an apostrophe after the -s ('guilds'').

Standard English spelling convention dictates that a regular plural noun ending in -s takes only an apostrophe at the end to show possession.

Key Concept

Plural and Possessive Nouns and Pronouns

Question 1294Question →

During the mid-twentieth century, aerospace engineer Mary Golda Ross worked on top-secret military projects at Lockheed Aircraft Corporation. Ross ______ played a pivotal role in drafting design concepts for interplanetary space travel and the Agena rocket program. Her pioneering calculations helped shape the early American space program, though much of her research remains classified.

Which choice completes the text with the most logical and grammatically correct punctuation?

, the first known Native American female engineer,

—the first known Native American female engineer,

, the first known Native American female engineer

the first known Native American female engineer,

Show answer & explanation

Answer: , the first known Native American female engineer,

Answer

The correct option is the one that sets off the nonessential appositive phrase with matching commas: ', the first known Native American female engineer,'

The sentence contains a nonessential appositive phrase, 'the first known Native American female engineer', which provides extra information about Mary Golda Ross. Nonessential elements must be set off from the rest of the sentence using matching punctuation marks at both the beginning and the end of the phrase. The correct choice uses a comma at both the beginning and the end of the appositive phrase, maintaining grammatical symmetry.

Step-by-Step Solution

1

Identify the nonessential element in the sentence.

The phrase 'the first known Native American female engineer' is an appositive that describes Mary Golda Ross but is not essential to the main grammatical structure of the sentence.

Recognizing nonessential elements is necessary to determine if and how they should be punctuated.

2

Determine the punctuation rule for nonessential elements.

Nonessential elements must be set off by matching punctuation marks—either a pair of commas, a pair of em dashes, or a pair of parentheses.

This establishes the grammatical requirement for symmetrical punctuation.

3

Evaluate the choices for punctuation symmetry.

Only the option starting and ending with a comma successfully isolates the nonessential clause with matching punctuation marks.

This identifies the correct choice and eliminates options that use mismatched or incomplete punctuation.

Key Concept

Punctuation of Nonessential Elements

Estimated Time:1m 0s

Question 1295Question →

A shipping company uses a cargo plane to transport two types of cargo containers: Type A and Type B. Let $x$ represent the number of Type A containers, and let $y$ represent the number of Type B containers. The constraints on the shipment are represented by the system of inequalities below:

\begin{aligned} x + y &\leq 45 \\ 3x + 5y &\leq 165 \end{aligned}

If the plane must carry at least $10$ Type A containers, what is the maximum number of Type B containers the plane can transport?

25

27

35

39

Show answer & explanation

Answer: 27

Answer

27

The correct answer of 27 is found by substituting the minimum value of

x

, which is 10, into the second inequality constraint. This yields

3(10) + 5y \leq 165

, which simplifies to

5y \leq 135

, or

y \leq 27

. Since the coordinate pair (10, 27) also satisfies the first inequality constraint (

10 + 27 = 37 \leq 45

), 27 is the maximum possible value for

y

.

Step-by-Step Solution

1

Identify the given constraints and the variable to maximize.

We are given the system of inequalities

x + y \leq 45

and

3x + 5y \leq 165

, with

x \geq 10

, and we want to find the maximum possible value of

y

.

This establishes the mathematical model and boundaries for the problem.

2

Express

y

in terms of

x

for both inequalities to see which constraint is more restrictive.

From the first inequality:

y \leq 45 - x

. From the second inequality:

3x + 5y \leq 165 \implies 5y \leq 165 - 3x \implies y \leq 33 - 0.6x

.

This allows us to analyze

y

as a function of

x

under both constraints.

3

Substitute the minimum possible value of

x

, which is 10, into both boundary expressions for

y

.

Under the first constraint:

y \leq 45 - 10 = 35

. Under the second constraint:

y \leq 33 - 0.6(10) = 27

.

Since both boundary lines have negative slopes, the maximum value of

y

will occur when

x

is at its minimum value of 10.

4

Determine the most restrictive upper bound for

y

when

x = 10

.

The value of

y

must satisfy both

y \leq 35

and

y \leq 27

, so the maximum possible value is 27.

A solution to a system of inequalities must satisfy all inequalities in the system simultaneously.

Key Concept

Solving systems of linear inequalities in context by evaluating boundary conditions and identifying constraints.

Question 1296Question →

In the $xy$ -plane, a point $(x, y)$ is in the solution set of the system of inequalities below:

x - 2y \leq -6

x + y \leq 11

y \leq 5

What is the maximum possible value of $x$ ?

Show answer & explanation

Answer: 4

Answer

The maximum possible value of

x

is

4

.

The maximum possible value of

x

is

4

. For all points in the solution set where

y \leq 5

, the active upper bound is

x \leq 2y - 6

. Since

2y - 6

increases as

y

increases, the maximum value of

x

occurs at the maximum boundary

y = 5

, giving

x = 2(5) - 6 = 4

.

Step-by-Step Solution

1

Express

x

in terms of

y

using the first two inequalities.

x \leq 2y - 6

and

x \leq 11 - y

To isolate the variable

x

and analyze how its upper bound is constrained by

y

.

2

Apply the third inequality constraint,

y \leq 5

, to find the limits on these upper bounds.

2y - 6 \leq 4

and

11 - y \geq 6

Since

y

cannot exceed

5

, the value of

2y - 6

is maximized when

y = 5

, and

11 - y

is minimized when

y = 5

.

3

Compare the two upper bounds to determine the active constraint for the domain

y \leq 5

.

Since

2y - 6 \leq 4

and

11 - y \geq 6

, the active bound is

x \leq 2y - 6

.

For any point to satisfy the system,

x

must be less than or equal to both bounds, meaning it is restricted by the smaller of the two bounds.

4

Calculate the maximum value of

x

at the boundary point.

x = 4

at the point

(4, 5)

The function

2y - 6

is increasing with respect to

y

, so its maximum value occurs at the largest possible value of

y

, which is

5

.

Key Concept

Finding the maximum value of a coordinate within a system of linear inequalities.

Question 1297Question →

For all $x > 2$ , which of the following is equivalent to the expression $\frac{6x^3 - 7x^2 - 16x + 12}{2x^2 - 3x - 2}$ ?

3x + 1 - \frac{7}{2x + 1}

3x + 1 - \frac{5}{2x + 1}

3x - 1 - \frac{5}{2x + 1}

3x - 8 + \frac{14}{2x - 1}

Show answer & explanation

Answer: 3x + 1 - \frac{7}{2x + 1}

Answer

The correct answer is the expression that simplifies to

3x + 1 - \frac{7}{2x + 1}

.

The correct answer is found by factoring the quadratic denominator into

(2x + 1)(x - 2)

. Testing the root

x = 2

in the numerator reveals that

(x - 2)

is also a factor of the numerator, allowing the expression to be rewritten as

(x - 2)(6x^2 + 5x - 6)

. Canceling the common factor

(x - 2)

leaves

\frac{6x^2 + 5x - 6}{2x + 1}

. Performing polynomial long division on this remaining term yields a quotient of

3x + 1

and a remainder of

-7

, which is expressed as the correct answer.

Step-by-Step Solution

1

Factor the denominator of the rational expression.

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

Identifying the factors of the denominator helps reveal potential common factors in the numerator.

2

Factor the numerator

6x^3 - 7x^2 - 16x + 12

by testing

x = 2

as a possible root.

Since

6(2)^3 - 7(2)^2 - 16(2) + 12 = 48 - 28 - 32 + 12 = 0

,

(x - 2)

is a factor. Dividing the cubic polynomial by

(x - 2)

yields

6x^3 - 7x^2 - 16x + 12 = (x - 2)(6x^2 + 5x - 6)

.

Finding the common linear factor allows us to simplify the rational expression.

3

Simplify the rational expression by canceling the common factor

(x - 2)

.

For

x > 2

,

\frac{(x - 2)(6x^2 + 5x - 6)}{(x - 2)(2x + 1)} = \frac{6x^2 + 5x - 6}{2x + 1}

.

Since

x > 2

,

x - 2 \neq 0

, which mathematically permits dividing out the common factor.

4

Perform polynomial long division on

\frac{6x^2 + 5x - 6}{2x + 1}

.

6x^2 + 5x - 6 = (3x + 1)(2x + 1) - 7

.

Dividing the quadratic expression by the linear expression separates the polynomial into a quotient and a rational remainder.

5

Rewrite the final expression with the quotient and remainder.

3x + 1 - \frac{7}{2x + 1}

Expressing the result in the standard quotient-remainder form matches the target expression.

Key Concept

Simplification and division of rational algebraic expressions

Question 1298Question →

In the $xy$ -plane, a line with a positive slope $m$ passes through the points $(2, 5)$ and $(m, 13)$ . Which of the following is the $y$ -intercept of this line?

(0, 3)

(0, 9)

(0, -3)

(0, 1)

Show answer & explanation

Answer: $(0, -3)$

Answer

(0, -3)

The correct answer is the point

(0, -3)

. The slope of the line passing through

(2, 5)

and

(m, 13)

is expressed as

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

. Multiplying by

m - 2

yields the quadratic equation

m^2 - 2m - 8 = 0

. Factoring this equation gives

(m - 4)(m + 2) = 0

. Since the slope

m

is given as positive, we select

m = 4

. Using the point-slope form with

m = 4

and the point

(2, 5)

yields the equation

y - 5 = 4(x - 2)

, which simplifies to

y = 4x - 3

. The

y

-intercept of this line is found by setting

x = 0

, giving

y = -3

, or the coordinate point

(0, -3)

.

Step-by-Step Solution

1

Use the slope formula to express the slope of the line passing through

(2, 5)

and

(m, 13)

in terms of

m

.

m = \frac{13 - 5}{m - 2} \implies m = \frac{8}{m - 2}

The slope of a line is defined as the ratio of the change in

y

to the change in

x

between any two points on the line.

2

Solve the equation

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

for

m

by converting it into a quadratic equation.

m(m - 2) = 8 \implies m^2 - 2m - 8 = 0 \implies (m - 4)(m + 2) = 0

, which gives

m = 4

or

m = -2

. Since the slope is positive,

m = 4

.

Multiplying both sides by the denominator clears the fraction and allows us to solve for the unknown parameter

m

using factoring.

3

Determine the equation of the line using the point-slope form with

m = 4

and the point

(2, 5)

.

y - 5 = 4(x - 2) \implies y = 4x - 8 + 5 \implies y = 4x - 3

The point-slope form provides a direct way to write the linear equation using a known point and the calculated slope.

4

Find the

y

-intercept by evaluating the line equation at

x = 0

.

y = 4(0) - 3 \implies y = -3

, which corresponds to the point

(0, -3)

.

The

y

-intercept of any line is the point where the line crosses the

y

-axis, which occurs at

x = 0

.

Key Concept

Linear Equations in Two Variables

Alternative Method

Alternatively, once the slope

m = 4

is found, we can write the equation of the line in slope-intercept form

y = 4x + b

. Substituting the coordinates of the point

(2, 5)

into the equation gives

5 = 4(2) + b \implies 5 = 8 + b

. Solving for

b

gives

b = -3

. Since

b

is the

y

-intercept, the point is

(0, -3)

.

Estimated Time:2m 30s

Question 1299Question →

In the $xy$ -plane, a point $(x, y)$ lies in the solution set of the system of inequalities below.

$y \leq -x + 8$
$y \leq 2x + 2$

What is the maximum possible value of $y$ ?

Show answer & explanation

Answer: 6

Answer

The maximum possible value of

y

is 6.

The solution set to the system of inequalities is the region in the coordinate plane that lies on or below both boundary lines,

y = -x + 8

and

y = 2x + 2

. The maximum

y

-value in this region occurs at the intersection of the two lines. Solving the system of equations by setting

-x + 8 = 2x + 2

gives

3x = 6

, which simplifies to

x = 2

. Substituting

x = 2

back into either equation yields

y = 6

. Therefore, the maximum possible value of

y

is 6.

Step-by-Step Solution

1

Set the two boundary equations equal to each other to find the

x

-coordinate of the intersection point.

-x + 8 = 2x + 2 \implies 3x = 6 \implies x = 2

Since the solution region is bounded from above by both inequalities, the maximum value of

y

must occur at the intersection of the two boundary lines.

2

Substitute the

x

-value back into one of the equations to find the corresponding

y

-value.

y = 2(2) + 2 = 6

This determines the

y

-coordinate of the intersection point, which represents the maximum height of the shaded region.

Key Concept

Systems of Linear Inequalities in Two Variables

Estimated Time:45s

Question 1300Question →

If the expression $\frac{3x^3 - 8x^2 + kx - 6}{3x - 2}$ is equivalent to $x^2 - 2x + 3$ for all $x \neq \frac{2}{3}$ , where $k$ is a constant, what is the value of $k$ ?

Show answer & explanation

Answer: 13

Answer

13

By multiplying both sides of the equation by

3x - 2

, the rational expression simplifies to a polynomial identity:

3x^3 - 8x^2 + kx - 6 = (x^2 - 2x + 3)(3x - 2)

. Expanding the right side yields

3x^3 - 8x^2 + 13x - 6

. Since the two polynomials are equivalent, their corresponding coefficients must be equal, meaning the coefficient of the linear term,

k

, must be equal to

13

.

Step-by-Step Solution

1

Multiply both sides of the equivalence by the denominator

(3x - 2)

3x^3 - 8x^2 + kx - 6 = (x^2 - 2x + 3)(3x - 2)

To clear the fraction and align the polynomial expressions for coefficient comparison.

2

Expand the right side of the equation using the distributive property

3x^3 - 8x^2 + 13x - 6

To obtain the expanded form of the polynomial so that we can identify the coefficients of each term.

3

Equate the corresponding coefficients of the linear

x

terms on both sides of the equation

k = 13

Since the two expressions are equivalent for all values of

x

, their coefficients for each corresponding power of

x

must be equal.

Key Concept

Equivalent Algebraic Expressions

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer