All practice questions

Question 1161Question →

In the $xy$ -plane, the graphs of two linear functions, $f$ and $g$ , are perpendicular lines that intersect at the point $(12, k)$ , where $k$ is a positive constant. If the $y$ -intercept of the graph of $f$ is $(0, 24)$ and the $y$ -intercept of the graph of $g$ is $(0, -6)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 18

Answer

18

To find the value of

k

, we determine the slopes of the two lines. The line representing

f

passes through

(0, 24)

and

(12, k)

, so its slope is

m_f = \frac{k - 24}{12}

. The line representing

g

passes through

(0, -6)

and

(12, k)

, so its slope is

m_g = \frac{k + 6}{12}

. Since the two lines are perpendicular, the product of their slopes is

-1

. This gives the equation

\left(\frac{k - 24}{12}\right)\left(\frac{k + 6}{12}\right) = -1

. Multiplying both sides by

144

yields

(k - 24)(k + 6) = -144

. Expanding the left side gives

k^2 - 18k - 144 = -144

. Adding

144

to both sides results in

k^2 - 18k = 0

, which factors as

k(k - 18) = 0

. Since

k

is a positive constant,

k

must be

18

.

Step-by-Step Solution

1

Determine the slopes of the lines

f

and

g

in terms of

k

.

The slope of

f

is

m_f = \frac{k - 24}{12 - 0} = \frac{k - 24}{12}

. The slope of

g

is

m_g = \frac{k - (-6)}{12 - 0} = \frac{k + 6}{12}

.

We use the slope formula

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

for each line passing through their respective

y

-intercepts and their intersection point

(12, k)

.

2

Set up an equation using the perpendicular condition.

m_f \cdot m_g = -1 \implies \left(\frac{k - 24}{12}\right)\left(\frac{k + 6}{12}\right) = -1 \implies \frac{(k - 24)(k + 6)}{144} = -1

Since the graphs of

f

and

g

are perpendicular, the product of their slopes must equal

-1

.

3

Solve the quadratic equation for

k

.

(k - 24)(k + 6) = -144 \implies k^2 - 18k - 144 = -144 \implies k^2 - 18k = 0 \implies k(k - 18) = 0

. Since

k

must be positive,

k = 18

.

Multiplying by

144

, expanding, and factoring the quadratic equation gives the possible values

k = 0

and

k = 18

. The problem states

k

is positive, so

k = 18

.

Key Concept

Perpendicular lines have slopes that are negative reciprocals of each other, meaning their product is

-1

.

Question 1162Question →

In the $xy$ -plane, a line with a negative slope passes through the point $(4, 3)$ and intersects the positive $x$ -axis at $(a, 0)$ and the positive $y$ -axis at $(0, b)$ . If the area of the triangle formed by this line and the coordinate axes is $24$ , what is the value of $b$ ?

Show answer & explanation

Answer: 6

Answer

The value of

b

is 6.

The correct value is 6. By representing the linear equation in intercept form as

\frac{x}{a} + \frac{y}{b} = 1

and substituting the given point

(4, 3)

, we obtain

\frac{4}{a} + \frac{3}{b} = 1

. Using the area of the triangle,

\frac{1}{2}ab = 24

, we can write

a = \frac{48}{b}

. Substituting this into the intercept equation yields

\frac{b}{12} + \frac{3}{b} = 1

. Multiplying by

12b

leads to the quadratic equation

b^2 - 12b + 36 = 0

, which factors as

(b-6)^2 = 0

, giving the unique solution

b = 6

.

Step-by-Step Solution

1

Express the line in intercept form

\frac{x}{a} + \frac{y}{b} = 1

Since the intercepts are

(a, 0)

and

(0, b)

with

a > 0

and

b > 0

, the intercept form of a linear equation is the most direct representation.

2

Substitute the given point

(4, 3)

into the intercept form

\frac{4}{a} + \frac{3}{b} = 1

The line passes through

(4, 3)

, so these coordinates must satisfy the equation of the line.

3

Express

a

in terms of

b

using the area of the triangle

a = \frac{48}{b}

The area of the right triangle with base

a

and height

b

is

\frac{1}{2}ab = 24

, which gives

ab = 48

.

4

Substitute

a = \frac{48}{b}

into the equation from Step 2

\frac{b}{12} + \frac{3}{b} = 1

This reduces the equation to a single variable

b

.

5

Clear denominators and write the equation in standard quadratic form

b^2 - 12b + 36 = 0

Multiplying both sides by

12b

allows us to form a quadratic equation.

6

Factor the quadratic equation to solve for

b

b = 6

The quadratic expression is a perfect square trinomial,

(b-6)^2 = 0

, which yields

b = 6

.

Key Concept

Using the intercept form of a linear equation and geometric properties of linear graphs to solve for unknown parameters.

Question 1163Question →

Which form of the noun in parentheses correctly completes the passage?

Fill in the blanks below

To protect their vulnerable nests from sudden tidal flooding, saltmarsh sparrows construct intricate, woven cup structures suspended within the dense cordgrass of coastal estuaries. However, these (bird) nesting efforts are increasingly threatened by sea-level rise and coastal development, both of which degrade the delicate marsh ecosystems that these avian species rely on for survival.

Show answer & explanation

Answer

birds'

The correct spelling 'birds'' functions as a plural possessive noun. The sentence refers to the nesting efforts of multiple saltmarsh sparrows, as indicated by the plural subject 'saltmarsh sparrows' earlier in the passage and the plural demonstrative determiner 'these' immediately preceding the blank. Because the nesting efforts belong to the birds, a possessive form is required. For a regular plural noun ending in -s, the possessive is formed by adding a single apostrophe at the end.

Step-by-Step Solution

1

Analyze the sentence context to determine if the noun 'bird' should be singular or plural.

The noun must be plural ('birds').

The sentence begins by discussing 'saltmarsh sparrows' (plural) and later uses the plural demonstrative determiner 'these' to modify the noun in question, indicating that we are referring to multiple birds.

2

Determine if the noun needs to be possessive or non-possessive.

The noun must be possessive ('birds'').

The noun modifies the noun phrase 'nesting efforts,' which are the efforts belonging to the birds, thereby requiring the possessive case.

3

Apply the correct punctuation rule for a plural possessive noun.

The final form is 'birds''.

For plural nouns ending in -s, the possessive is formed by adding an apostrophe at the end of the word after the -s.

Key Concept

Plural and possessive nouns are used to indicate number (singular vs. plural) and ownership or modification (possessive vs. non-possessive). A plural possessive noun ending in -s requires only an apostrophe at the end.

Question 1164Question →

Initiated in 2007 by the African Union, the Great Green Wall was originally conceived as a simple corridor of trees designed to combat desertification in the Sahel region. However, the initiative [blank] now emphasizes a broader range of sustainable land management practices to support local communities.

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

, which spans the entire width of the continent—

—which spans the entire width of the continent,

—which spans the entire width of the continent—

which spans the entire width of the continent,

Show answer & explanation

Answer: —which spans the entire width of the continent—

Answer

The relative clause 'which spans the entire width of the continent' is a nonessential element and must be set off with matching punctuation. The only choice that does this is the one using matching em dashes on both sides: '—which spans the entire width of the continent—'.

The relative clause 'which spans the entire width of the continent' is a nonessential element that provides extra information about 'the initiative.' In Standard English, nonessential elements must be set off from the rest of the sentence using a pair of matching punctuation marks, such as two em dashes. The correct choice properly uses an em dash before and after the clause.

Step-by-Step Solution

1

Identify the nonessential element in the sentence.

The clause 'which spans the entire width of the continent' is nonessential because the main sentence 'the initiative now emphasizes a broader range of sustainable land management practices' remains grammatically complete and retains its core meaning without it.

Determining if a clause is nonessential helps decide how it must be punctuated.

2

Apply the rule for punctuation of nonessential elements.

Nonessential elements must be set off from the main clause by matching punctuation marks: either two commas, two em dashes, or parentheses.

Using mismatched punctuation (such as a comma on one side and an em dash on the other) is grammatically incorrect in Standard English.

3

Evaluate the choices to find the one with matching punctuation marks.

The choice with matching em dashes ('—which spans the entire width of the continent—') is correct, while the others use asymmetrical punctuation combinations.

This ensures grammatical symmetry and correctness.

Key Concept

Punctuation of Nonessential Elements

Question 1165Question →

If $\frac{3}{5}(2x - 4) - \frac{1}{2}(x - 3) = \frac{6}{5}$ , what is the value of $5x - 4$ ?

Show answer & explanation

Answer: 11

Answer

11

Evaluating the linear equation by clearing the fractions with a common denominator of 10 gives

6(2x - 4) - 5(x - 3) = 12

. Distributing terms yields

12x - 24 - 5x + 15 = 12

. Combining like terms results in

7x - 9 = 12

. Solving for

x

gives

x = 3

. Finally, substituting

x = 3

into the target expression

5x - 4

yields

5(3) - 4 = 11

.

Step-by-Step Solution

1

Multiply both sides of the equation by 10 to clear the denominators.

6(2x - 4) - 5(x - 3) = 12

The least common denominator of 5 and 2 is 10. Multiplying both sides by 10 simplifies the equation by removing the fractions.

2

Distribute the constants 6 and -5 into the parentheses.

12x - 24 - 5x + 15 = 12

Applying the distributive property removes the parentheses. Be careful to distribute the negative sign for the second term:

-5 \cdot (x - 3) = -5x + 15

.

3

Combine like terms on the left side of the equation.

7x - 9 = 12

Grouping the

x

terms (

12x - 5x = 7x

) and the constant terms (

-24 + 15 = -9

) simplifies the equation.

4

Isolate the variable term

7x

and solve for

x

.

x = 3

Adding 9 to both sides gives

7x = 21

. Dividing both sides by 7 yields

x = 3

.

5

Substitute the value of

x

into the expression

5x - 4

.

11

The question asks for the value of

5x - 4

, not just

x

, so we evaluate

5(3) - 4 = 11

.

Key Concept

Solving linear equations in one variable by clearing fractions, distributing terms, and isolating the variable.

Question 1166Question →

In her analysis of the social structures of the early Harappan civilization, archaeologist Leela Sen argues that the absence of monumental royal palaces does not indicate a lack of centralized authority ______ the uniform grid layout of cities like Mohenjo-daro suggests a highly organized administrative system governed by a ruling class of merchants or priests.

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

authority; the

authority, the

authority the

authority, but the

Show answer & explanation

Answer: authority; the

Answer

authority; the

The correct option correctly uses a semicolon to separate two independent clauses. The first independent clause ends with the word 'authority,' and the second independent clause begins with the word 'the.' A semicolon is an appropriate punctuation mark to link two closely related independent clauses when no coordinating conjunction is present.

Step-by-Step Solution

1

Identify the structure of the two clauses surrounding the blank.

The first clause ('In her analysis... does not indicate a lack of centralized authority') is an independent clause. The second clause ('the uniform grid layout... suggests a highly organized administrative system...') is also an independent clause.

Determining whether the clauses are independent or dependent is necessary to apply the correct punctuation and linking rules.

2

Evaluate the grammatical validity of the linking options.

Two independent clauses must be separated by a period, a semicolon, a colon, or a comma paired with a coordinating conjunction. Connecting them with only a comma creates a comma splice, and connecting them with no punctuation creates a run-on.

This rules out grammatically incorrect connection structures.

3

Analyze the logical relationship between the two clauses to determine if a coordinator is appropriate.

The first clause asserts that a lack of palaces does not mean a lack of authority. The second clause provides evidence of that authority (the urban grid layout). Thus, the second clause supports the first; a contrasting coordinator like 'but' is logically incorrect.

This ensures the chosen linking method is both grammatically and semantically correct.

Key Concept

Clause Boundaries and Linking

Question 1167Question →

Linguists studying the creole languages of the Caribbean have identified complex grammatical structures that cannot be traced back to the languages of European colonizers. For many years, researchers assumed these structures were simplified versions of colonial speech ________ these features instead stem from a combination of African substrate languages and universal cognitive principles of language acquisition.

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

; however,

, however,

however

; although

Show answer & explanation

Answer: ; however,

Answer

; however,

The correct choice must successfully connect two independent clauses. The clause preceding the blank ('For many years, researchers assumed these structures were simplified versions of colonial speech') and the clause following the blank ('these features instead stem from...') are both independent. Separating them with a semicolon, followed by the transition word 'however' and a comma, is a grammatically correct way to link independent clauses while expressing contrast.

Step-by-Step Solution

1

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

The clause before the blank ('For many years, researchers assumed these structures were simplified versions of colonial speech') is an independent clause. The clause after the blank ('these features instead stem from a combination of African substrate languages and universal cognitive principles of language acquisition') is also an independent clause.

Identifying whether the clauses are independent or dependent determines the type of punctuation required to link them.

2

Evaluate the relationship between the two clauses.

The second clause contrasts with the first clause (challenging the assumption that creole structures are merely simplified versions of colonial speech).

Understanding the logical connection helps select the correct transition word or punctuation style.

3

Test the punctuation and linking options for grammatical correctness.

A semicolon followed by 'however' and a comma correctly joins two independent clauses and establishes a contrast. A comma alone causes a comma splice, no punctuation causes a run-on, and using the subordinator 'although' after a semicolon incorrectly joins a dependent clause.

Applying standard English conventions ensures the final sentence is grammatically correct and coherent.

Key Concept

Clause Boundaries and Linking

Question 1168Question →

During the late Devonian period, early tetrapods transitioned from aquatic environments to terrestrial habitats, a shift that required significant anatomical adaptations. To survive on land, these organisms evolved by developing lungs for breathing air, transforming their muscular fins into sturdy limbs for support, and _______

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

to modify their skeletal systems for terrestrial locomotion

modifying their skeletal systems for terrestrial locomotion

they modified their skeletal systems for terrestrial locomotion

the modification of their skeletal systems for terrestrial locomotion

Show answer & explanation

Answer: modifying their skeletal systems for terrestrial locomotion

Answer

modifying their skeletal systems for terrestrial locomotion

The sentence presents a list of three anatomical adaptations, introduced by the preposition 'by'. The first two items in the list are gerund phrases: 'developing lungs for breathing air' and 'transforming their muscular fins into sturdy limbs for support'. To maintain parallel structure, the third and final item must also be a gerund phrase. The choice starting with 'modifying' correctly provides this parallel structure.

Step-by-Step Solution

1

Analyze the structure of the list in the sentence.

The sentence contains a list of actions introduced by the preposition 'by': 'developing lungs...' and 'transforming their muscular fins...'.

Understanding the grammatical form of the existing items is necessary to determine what is needed to maintain parallelism.

2

Identify the part of speech and pattern of the list items.

Both existing list items begin with gerunds ('developing' and 'transforming').

Parallel structure requires that items in a list or comparison share the same grammatical form.

3

Evaluate the answer choices for a matching gerund phrase.

The option beginning with 'modifying' is the only choice that starts with a gerund, making it grammatically parallel to the rest of the list.

Selecting the option with the matching gerund phrase satisfies the rules of parallel structure.

Key Concept

Parallel Structure

Question 1169Question →

Deep-sea hydrothermal vents support highly complex and productive biological communities in the complete absence of solar energy. In these aphotic marine environments, chemosynthetic bacteria form the foundation of the local trophic structure ________ they oxidize dissolved hydrogen sulfide to synthesize organic compounds, thereby providing essential nutrients for specialized species like giant tubeworms.

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

; they

, they

they

although they

Show answer & explanation

Answer: ; they

Answer

The option containing '; they' is correct because it properly connects the two independent clauses using a semicolon.

The correct option correctly uses a semicolon to link two independent clauses: the first describes chemosynthetic bacteria forming the foundation of the trophic structure, and the second explains how they do so by oxidizing hydrogen sulfide.

Step-by-Step Solution

1

Identify the clauses surrounding the blank.

The first clause is 'In these aphotic marine environments, chemosynthetic bacteria form the foundation of the local trophic structure' (independent). The second clause is 'they oxidize dissolved hydrogen sulfide to synthesize organic compounds...' (independent).

Determining whether the clauses are independent or dependent is necessary to apply the correct punctuation rules.

2

Determine the appropriate punctuation or conjunction to link two independent clauses.

Two independent clauses must be joined by a period, a semicolon, or a comma coupled with a coordinating conjunction (FANBOYS).

Connecting two independent clauses with only a comma or with no punctuation leads to grammatical errors like comma splices and run-ons.

3

Evaluate the choices based on grammatical correctness and semantic relationship.

The semicolon successfully links the two clauses. A comma alone creates a comma splice, omitting punctuation creates a run-on, and using a contrast subordinator introduces an illogical relationship.

Choosing the grammatically correct option that preserves the logical connection between the clauses ensures sentence flow and standard English conventions.

Key Concept

Connecting Independent Clauses

Question 1170Question →

A linear model is used to estimate the height of a plant, in centimeters, based on the number of weeks since it was planted. According to the model, the plant's height is $5$ centimeters at week $2$ , and its height is $8$ centimeters at week $8$ . Which of the following is the predicted height of the plant, in centimeters, at week $16$ ?

14

29

12

47

Show answer & explanation

Answer: $12$

Answer

The predicted height of the plant at week 16 is 12 centimeters.

The correct answer is 12. To find the predicted height, we determine the linear model equation. The slope is calculated as

m = \frac{8 - 5}{8 - 2} = \frac{3}{6} = 0.5

. Substituting the point

(2, 5)

into the slope-intercept form

y = mx + b

gives

5 = 0.5(2) + b

, which simplifies to

b = 4

. The linear model is

y = 0.5x + 4

. Substituting

16

for

x

yields

y = 0.5(16) + 4 = 8 + 4 = 12

.

Step-by-Step Solution

1

Identify two data points from the given information to represent the linear relationship.

The coordinates are

(2, 5)

and

(8, 8)

where the x-coordinate represents the week and the y-coordinate represents the height of the plant in centimeters.

A linear relationship is uniquely determined by two points, which allows us to find the slope and equation of the line.

2

Calculate the slope of the line, which represents the constant growth rate of the plant.

The slope

m

is

\frac{8 - 5}{8 - 2} = \frac{3}{6} = 0.5

centimeters per week.

The slope represents the constant rate of change of the plant's height per week.

3

Determine the y-intercept of the linear model using one of the coordinates.

Using the point

(2, 5)

and the equation

y = mx + b

, we get

5 = 0.5(2) + b

, which simplifies to

b = 4

.

Finding the y-intercept allows us to write the complete equation for the linear function representing the plant's growth.

4

Substitute the target week into the linear function to predict the height of the plant.

For week 16,

y = 0.5(16) + 4 = 8 + 4 = 12

centimeters.

Evaluating the function at

x = 16

yields the predicted height of the plant at that specific time.

Key Concept

Finding and evaluating linear functions using two given coordinate points.

Estimated Time:1m 30s

Question 1171Question →

For a linear function $f$ , the equation $f(x + 3) - f(x - 1) = 16$ is true for all real numbers $x$ . If the graph of $y = f(x)$ in the $xy$ -plane passes through the point $(2, 5)$ , what is the $x$ -intercept of the graph of $f$ ?

-\frac{3}{4}

\frac{3}{4}

\frac{11}{8}

-18

Show answer & explanation

Answer: $\frac{3}{4}$

Answer

The

x

-intercept of the graph of

f

is

\frac{3}{4}

.

By writing the linear function as

f(x) = mx + b

, we can evaluate

f(x+3) - f(x-1)

as

(mx + 3m + b) - (mx - m + b) = 4m

. Setting this equal to the given value of

16

yields a slope of

m = 4

. Using the point

(2, 5)

in

f(2) = 4(2) + b = 5

gives the

y

-intercept

b = -3

, so the function is

f(x) = 4x - 3

. The

x

-intercept is found by setting

f(x) = 0

, which yields

x = \frac{3}{4}

.

Step-by-Step Solution

1

Represent the linear function in slope-intercept form

f(x) = mx + b

and substitute the expressions

x+3

and

x-1

into the function.

f(x+3) = m(x+3) + b

and

f(x-1) = m(x-1) + b

.

This allows us to evaluate the given functional equation using the general parameters of a linear function.

2

Calculate the difference

f(x+3) - f(x-1)

and set it equal to

16

to find the slope

m

.

(mx + 3m + b) - (mx - m + b) = 4m = 16

, which simplifies to

m = 4

.

The difference between two values of a linear function is directly proportional to the difference in their inputs, where the constant of proportionality is the slope.

3

Substitute the point

(2, 5)

and the slope

m = 4

into

f(x) = 4x + b

to find the

y

-intercept

b

.

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

.

A linear function is completely defined once both its slope and a point on its graph are known.

4

Find the

x

-intercept of the graph by setting

f(x) = 0

and solving for

x

.

4x - 3 = 0 \implies x = \frac{3}{4}

.

The

x

-intercept of a function's graph is the input value at which the function's output equals zero.

Key Concept

Determining the slope, equation, and intercepts of a linear function from functional relationships and coordinate points.

Estimated Time:2m 0s

Question 1172Question →

\begin{cases} y \geq -2x + 4 \\ y < \frac{1}{2}x + 2 \end{cases}

Which of the following ordered pairs $(x, y)$ is a solution to the system of inequalities above?

(1, 2)

(0, -1)

(2, 3)

(0, 3)

Show answer & explanation

Answer: $(1, 2)$

Answer

The ordered pair

(1, 2)

is a solution to the system of inequalities.

Substituting the coordinates of the point stating

(1, 2)

into the system of inequalities produces two true statements:

2 \geq -2(1) + 4

simplifies to

2 \geq 2

, which is true; and

2 < \frac{1}{2}(1) + 2

simplifies to

2 < 2.5

, which is also true. Therefore, the ordered pair

(1, 2)

is a solution to the system.

Step-by-Step Solution

1

Substitute the coordinates of each option into the first inequality,

y \geq -2x + 4

.

For

(1, 2)

,

2 \geq 2

(True). For

(0, -1)

,

-1 \geq 4

(False). For

(2, 3)

,

3 \geq 0

(True). For

(0, 3)

,

3 \geq 4

(False).

To determine which ordered pairs satisfy the first inequality.

2

Substitute the coordinates of the remaining candidate pairs,

(1, 2)

and

(2, 3)

, into the second inequality,

y < \frac{1}{2}x + 2

.

For

(1, 2)

,

2 < 2.5

(True). For

(2, 3)

,

3 < 3

(False).

To determine which of the remaining candidates also satisfies the second inequality.

3

Identify the ordered pair that satisfies both inequalities in the system.

Only

(1, 2)

satisfies both inequalities.

An ordered pair is a solution to a system of inequalities if and only if it satisfies all inequalities in the system.

Key Concept

Verifying solutions to a system of linear inequalities by substitution.

Question 1173Question →

A system of linear equations is shown below.

\begin{aligned} 5x - 2y &= 14 \\ 2x + y &= 11 \end{aligned}

If

(x, y)

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

x

?

3

36

4

2

Show answer & explanation

Answer: 4

Answer

4

The correct answer is

4

. Isolating

y

in the second equation gives

y = 11 - 2x

. Substituting this into the first equation yields

5x - 2(11 - 2x) = 14

. Distributing the

-2

results in

5x - 22 + 4x = 14

. Combining like terms simplifies the equation to

9x - 22 = 14

. Adding

22

to both sides gives

9x = 36

, and dividing by

9

yields

x = 4

.

Step-by-Step Solution

1

Isolate

y

in the second equation.

y = 11 - 2x

This allows for substitution into the first equation.

2

Substitute the expression for

y

into the first equation.

5x - 2(11 - 2x) = 14

This creates an equation with only one variable,

x

.

3

Distribute the

-2

through the parentheses and combine like terms.

9x - 22 = 14

Distributing

-2

to

-2x

yields

+4x

, and combining

5x + 4x

gives

9x

.

4

Add

22

to both sides and divide by

9

.

x = 4

Isolating

x

gives the final value of the variable.

Key Concept

Solving systems of linear equations using the substitution method.

Question 1174Question →

Which of the following ordered pairs $(x, y)$ is a solution to the system of inequalities below?

\begin{aligned} y &\geq x + 1 \\ y &> -2x + 4 \end{aligned}

(1, 3)

(3, 1)

(0, 2)

(-1, 5)

Show answer & explanation

Answer: $(1, 3)$

Answer

The ordered pair

(1, 3)

The ordered pair

(1, 3)

is the correct answer because substituting

x = 1

and

y = 3

into the system of inequalities results in two true statements:

3 \geq 1 + 1

simplifies to

3 \geq 2

(true), and

3 > -2(1) + 4

simplifies to

3 > 2

(true).

Step-by-Step Solution

1

Substitute the coordinates of the ordered pair

(1, 3)

into the first inequality,

y \geq x + 1

.

3 \geq 1 + 1 \Rightarrow 3 \geq 2

, which is true.

To be a solution to the system, the ordered pair must satisfy the first inequality.

2

Substitute the coordinates of the ordered pair

(1, 3)

into the second inequality,

y > -2x + 4

.

3 > -2(1) + 4 \Rightarrow 3 > 2

, which is true.

The ordered pair must also satisfy the second inequality to be a solution to the system.

3

Verify that the other ordered pairs do not satisfy both inequalities.

None of the other options satisfy both inequalities:

(3, 1)

fails the first inequality,

(0, 2)

fails the second inequality, and

(-1, 5)

fails the second inequality.

Ensuring that there is exactly one correct solution.

Key Concept

A solution to a system of linear inequalities in two variables is an ordered pair

(x, y)

that makes all inequalities in the system true.

Alternative Method

Alternatively, you can graph the two inequalities on a coordinate plane. The region representing the solution to the system is the overlapping shaded area of both inequalities. By plotting each point, you can visually identify which point lies within this double-shaded region.

Estimated Time:1m 0s

Question 1175Question →

The naturalist John Muir was deeply committed to preserving the wilderness of the American West. In his writings, he argued that nature served a spiritual _______ it was a cathedral that offered people sanctuary from the noise of the industrial world.

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

purpose: it

purpose, it

purpose; because it

purpose it

Show answer & explanation

Answer: purpose: it

Answer

purpose: it

The correct option uses a colon to connect the independent clause introducing the concept of a spiritual purpose with the second clause that explains what that purpose is (likening nature to a cathedral). Under Standard English conventions, a colon is appropriate to introduce an explanation or elaboration when the preceding clause is independent.

Step-by-Step Solution

1

Determine the grammatical structure of the clauses surrounding the blank.

The clause before the blank ('In his writings... spiritual purpose') is independent. The clause after the blank ('it was a cathedral...') is also independent.

Identifying the independence of clauses helps determine the appropriate punctuation conventions.

2

Analyze the logical relationship between the two clauses.

The second clause directly explains and describes the 'spiritual purpose' mentioned in the first clause.

A colon is used to link an independent clause to another clause that explains, illustrates, or describes the concept in the first clause.

Key Concept

Colons linking independent clauses to explanatory clauses

Question 1176Question →

In the system of equations below,

c

is a constant.

\begin{aligned} 2x + y &= 15 \\ cx - 2y &= 6 \end{aligned}

If the solution

(x, y)

to the system of equations above lies on the line

y = 3x

in the

xy

-plane, what is the value of the constant

c

?

Show answer & explanation

Answer: 8

Answer

The value of the constant

c

is 8.

Since the solution

(x, y)

lies on the line

y = 3x

, we can substitute

y = 3x

into the first equation,

2x + y = 15

, to get

2x + 3x = 15

, which simplifies to

5x = 15

, yielding

x = 3

. Substituting

x = 3

back into

y = 3x

gives

y = 9

. Thus, the intersection point of the system is

(3, 9)

. Substituting

x = 3

and

y = 9

into the second equation,

cx - 2y = 6

, gives

c(3) - 2(9) = 6

, or

3c - 18 = 6

. Adding 18 to both sides gives

3c = 24

, which simplifies to

c = 8

.

Step-by-Step Solution

1

Substitute the equation of the line

y = 3x

into the first equation of the system,

2x + y = 15

.

2x + 3x = 15

, which simplifies to

5x = 15

. Solving for

x

gives

x = 3

.

Since the solution to the system lies on the line

y = 3x

, the coordinates of the solution must satisfy all three equations.

2

Calculate the corresponding

y

-value using the relation

y = 3x

.

y = 3(3) = 9

.

The

y

-coordinate is three times the

x

-coordinate for any point on the line

y = 3x

.

3

Substitute the point

(3, 9)

into the second equation of the system,

cx - 2y = 6

.

c(3) - 2(9) = 6

, which simplifies to

3c - 18 = 6

. Adding 18 to both sides gives

3c = 24

, so

c = 8

.

Since

(3, 9)

is the solution to the system, it must satisfy the second equation as well.

Key Concept

Solving systems of linear equations using substitution and finding parameter values based on geometric constraints.

Question 1177Question →

Which of the following inequalities is equivalent to $-3(x - 4) < 15$ ?

x > -1

x < -1

x > -9

x < -9

Show answer & explanation

Answer: $x > -1$

Answer

The inequality

x > -1

The correct answer is obtained by dividing both sides of the inequality

-3(x - 4) < 15

by

-3

, which reverses the inequality symbol to yield

x - 4 > -5

. Adding

4

to both sides isolates the variable, resulting in

x > -1

. Alternatively, distributing the

-3

first yields

-3x + 12 < 15

, which simplifies to

-3x < 3

, and dividing by

-3

with a sign flip yields

x > -1

.

Step-by-Step Solution

1

Divide both sides of the inequality

-3(x - 4) < 15

by

-3

.

x - 4 > -5

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

2

Add

4

to both sides of the inequality to isolate

x

.

x > -1

Adding a constant to both sides preserves the inequality and isolates the variable

x

.

Key Concept

Solving linear inequalities in one variable involving distribution and negative coefficients.

Question 1178Question →

Young sunflowers exhibit a behavior known as heliotropism; they track the movement of the sun from east to west during the day. At night, the stems bend in the opposite direction _______ return to face the east in anticipation of dawn.

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

; they

, they

they

although they

Show answer & explanation

Answer: ; they

Answer

; they

The correct option correctly uses a semicolon to link two independent clauses: the clause describing the stems bending in the opposite direction and the clause describing their return to face the east. A semicolon is a standard way to link two closely related independent clauses without any coordinating conjunction.

Step-by-Step Solution

1

Analyze the structure of the two clauses surrounding the blank space.

The clause before the blank ('At night, the stems bend in the opposite direction') and the clause after the blank ('they return to face the east in anticipation of dawn') are both independent clauses.

Identifying independent clauses helps determine the correct punctuation and linking words needed to join them.

2

Apply the punctuation rules for joining independent clauses.

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

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

3

Evaluate the choices to find the grammatically correct link.

The option with a semicolon followed by the pronoun 'they' correctly connects the two independent clauses. The comma-only option is a comma splice, the option with no punctuation is a run-on, and the option with 'although' is an incorrect subordinator.

This identifies the correct choice according to Standard English conventions.

Key Concept

Linking independent clauses with appropriate punctuation

Estimated Time:45s

Question 1179Question →

A water tank is being filled at a constant rate. The volume of water in the tank, in gallons, is a linear function of the time, in minutes, since the filling process began. The volume of water in the tank was $24$ gallons after $3$ minutes of filling, and it was $40$ gallons after $7$ minutes of filling. What was the initial volume of water, in gallons, in the tank before the filling process began?

Show answer & explanation

Answer: 12

Answer

The initial volume of water in the tank was 12 gallons.

To find the initial volume of water, we must determine the y-intercept of the linear relationship between the volume and time. First, find the rate of change (slope) using the formula m = \frac{V_2 - V_1}{t_2 - t_1}. Substituting the given values gives m = \frac{40 - 24}{7 - 3} = \frac{16}{4} = 4. Using the slope-intercept form V(t) = mt + b, substitute one of the points, such as (3, 24), to solve for b: 24 = 4(3) + b, which simplifies to b = 12. Thus, the initial volume of water is 12 gallons.

Step-by-Step Solution

1

Calculate the slope (rate of change) of the volume with respect to time.

4

The slope is the change in volume divided by the change in time: \frac{40 - 24}{7 - 3} = \frac{16}{4} = 4.

2

Write the linear function V(t) = mt + b using the calculated slope, and substitute one of the given points to solve for b.

b = 12

Using the point (3, 24) and the slope m = 4, we substitute into V(t) = 4t + b to get 24 = 4(3) + b, which simplifies to 24 = 12 + b, meaning b = 12.

3

Identify the initial volume of water in the tank, which is the value of V(t) when t = 0.

12

When t = 0, V(0) = 4(0) + 12 = 12, which represents the initial volume.

Key Concept

Determining the y-intercept (initial value) of a linear function given two points.

Question 1180Question →

In the system of equations below,

k

is a positive integer that is a multiple of 9.

\begin{aligned} 3x + 8y &= 213 \\ x - 3y &= k \end{aligned}

If the system has a solution

(x, y)

such that

x

and

y

are both positive integers, what is the value of

k

?

Show answer & explanation

Answer: 54

Answer

54

Substituting

x = 3y + k

into the first equation yields

17y + 3k = 213

. Solving for

k

gives

k = 71 - \frac{17}{3}y

. For

k

to be a positive integer,

y

must be a positive multiple of 3 less than 12.5. Testing the possible values for

y

(3, 6, 9, 12) yields the possible values of

k

as 54, 37, 20, and 3. Since

k

must be a multiple of 9, the correct value is 54.

Step-by-Step Solution

1

Express

x

in terms of

y

and

k

using the second equation.

x = 3y + k

Isolating

x

allows for easy substitution into the first equation to eliminate one of the variables.

2

Substitute the expression for

x

into the first equation.

17y + 3k = 213

Substituting

x = 3y + k

into

3x + 8y = 213

yields

3(3y + k) + 8y = 213

, which simplifies to

9y + 3k + 8y = 213

and then to

17y + 3k = 213

.

3

Solve for

k

in terms of

y

and analyze the divisibility constraint.

k = 71 - \frac{17}{3}y

, where

y

is a multiple of 3

For

k

to be an integer, the fractional term

\frac{17}{3}y

must resolve to an integer, requiring

y

to be divisible by 3.

4

Apply the constraint that

k

must be a positive integer (

k > 0

).

y < 12.5

, meaning

y

can be 3, 6, 9, or 12

Setting

71 - \frac{17}{3}y > 0

yields

17y < 213

, or

y < 12.53

. The positive integer multiples of 3 in this range are 3, 6, 9, and 12.

5

Find the corresponding values of

k

and select the one that is a multiple of 9.

The possible values for

k

are 54, 37, 20, and 3. The only multiple of 9 is 54.

Testing the possible values of

y

gives:

y=3 \rightarrow k=54

,

y=6 \rightarrow k=37

,

y=9 \rightarrow k=20

, and

y=12 \rightarrow k=3

. Among these, 54 is the only multiple of 9.

Key Concept

Systems of Linear Equations with Integer Constraints

Estimated Time:3m 0s

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

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