All practice questions

Question 1201Question →

In the $xy$ -plane, the graph of the linear function $f$ passes through the point $(2, 5)$ and has a $y$ -intercept of $(0, b)$ , where $b$ is a constant. The function $g$ is defined by $g(x) = f(x) - 4$ . If the $x$ -intercept of the graph of $g$ is $(6, 0)$ , what is the value of $b$ ?

10

\frac{19}{2}

\frac{11}{2}

\frac{11}{4}

Show answer & explanation

Answer: $\frac{11}{2}$

Answer

The correct value of

b

is

\frac{11}{2}

.

The correct answer is the value

\frac{11}{2}

. By writing

f(x) = mx + b

and using the point

(2, 5)

, we get

m = \frac{5-b}{2}

. Applying the definition of

g(x) = f(x) - 4

gives

g(x) = mx + b - 4

. Since the graph of

g

has an

x

-intercept of

(6, 0)

, substituting

x = 6

and

g(6) = 0

gives

6m + b - 4 = 0

. Substituting the expression for

m

yields

3(5-b) + b - 4 = 0

, which simplifies to

11 - 2b = 0

, and thus

b = \frac{11}{2}

.

Step-by-Step Solution

1

Write the linear function

f(x)

in slope-intercept form using the given

y

-intercept

(0, b)

.

f(x) = mx + b

, where

m

is the slope of the line.

This sets up the general equation of the line with unknown parameters

m

and

b

.

2

Substitute the coordinates of the point

(2, 5)

into the equation for

f(x)

to express

m

in terms of

b

.

5 = 2m + b

, which simplifies to

m = \frac{5 - b}{2}

.

Since the point lies on the graph of

f

, its coordinates must satisfy the equation.

3

Define

g(x)

using the relationship

g(x) = f(x) - 4

and apply the

x

-intercept

(6, 0)

.

g(x) = mx + b - 4

. Since the

x

-intercept is

(6, 0)

, we have

g(6) = 0

, which gives

6m + b - 4 = 0

.

The

x

-intercept is the point where the output of the function is zero.

4

Substitute the expression for

m

from Step 2 into the equation from Step 3 and solve for

b

.

6\left(\frac{5-b}{2}\right) + b - 4 = 0 \Rightarrow 3(5-b) + b - 4 = 0 \Rightarrow 15 - 3b + b - 4 = 0 \Rightarrow 11 - 2b = 0 \Rightarrow b = \frac{11}{2}

.

This solves the system of equations to determine the value of the constant

b

.

Key Concept

Linear Functions and Graphs

Question 1202Question →

Which punctuation mark correctly completes the passage by joining the independent clause to the explanatory details that follow?

Fill in the blanks below

In 1992, astronomers Aleksander Wolszczan and Dale Frail announced a discovery that challenged long-held assumptions about the cosmos three terrestrial planets, later named Poltergeist, Phobetor, and Draugr, orbiting a rapidly spinning neutron star. Prior to this finding, scientists believed that the violent supernova explosions marking the death of massive stars would inevitably destroy any orbiting bodies. The detection of these pulsar planets proved that planetary systems could survive, or even form in the aftermath of, such cataclysmic stellar events.

Show answer & explanation

Answer

A colon (:) or a dash (—) is correct because it links an independent clause to a list or explanatory detail at the end of a sentence.

The correct punctuation is a colon (:) or a dash (—). The text preceding the blank is an independent clause, and the text following it is a noun phrase that provides explanatory details about the 'discovery'. Standard English conventions dictate that a colon or a single dash can be used to set off explanatory lists or details following an independent clause. A colon or dash is particularly necessary here because the explanatory phrase contains internal commas, meaning a comma would result in punctuation confusion.

Step-by-Step Solution

1

Analyze the grammatical structure of the clause preceding the blank.

The clause 'In 1992, astronomers Aleksander Wolszczan and Dale Frail announced a discovery that challenged long-held assumptions about the cosmos' is independent and can stand alone as a complete sentence.

This establishes that the preceding clause can be followed by a colon or dash.

2

Analyze the structure of the phrase following the blank.

The phrase 'three terrestrial planets, later named Poltergeist, Phobetor, and Draugr, orbiting a rapidly spinning neutron star' is a noun phrase functioning as an appositive or explanatory detail, not an independent clause.

Since it is not an independent clause, a semicolon cannot be used to join them.

3

Select the correct punctuation mark based on punctuation rules.

A colon or a dash is appropriate here because both can connect an independent clause to an explanatory phrase or list. A comma is incorrect because the following phrase contains internal commas ('later named Poltergeist, Phobetor, and Draugr'), so adding another comma would create structural confusion.

This confirms that a colon (:) or a dash (—) is the only grammatically sound choice.

Key Concept

Colons and dashes can join an independent clause to explanatory details, lists, or appositives at the end of a sentence, especially when those details contain internal punctuation.

Question 1203Question →

The passage below contains a blank. Which punctuation mark correctly sets off the nonessential element in the sentence?

Fill in the blanks below

In 1953, botanist Katherine Esau published her landmark book *Plant Anatomy*, which remains a definitive text on plant biology. Prior to Esau's work, the field focused primarily on static descriptions of plant tissues. Esau, who was awarded the National Medal of Science in 1989 revolutionized the study of plant anatomy by integrating microscopic structural details with dynamic developmental processes.

Show answer & explanation

Answer

comma (or ,)

The correct answer is a comma. The relative clause 'who was awarded the National Medal of Science in 1989' is nonessential and must be separated from the rest of the sentence. Since a comma is used immediately after 'Esau' to introduce the clause, a matching comma must be placed after '1989' to close the clause before the main verb 'revolutionized'.

Step-by-Step Solution

1

Identify the nonessential clause in the sentence.

The phrase 'who was awarded the National Medal of Science in 1989' is a nonessential relative clause modifying the proper noun 'Esau'.

This clause provides descriptive information that is not essential to identifying the subject or understanding the core meaning of the main clause.

2

Identify the punctuation mark used to introduce the clause.

A comma is placed immediately after 'Esau' to open the clause.

Nonessential elements must be set off with matching pairs of punctuation to maintain grammatical symmetry.

3

Select the matching punctuation mark to close the clause.

A comma is required in the blank after '1989' to close the clause before the verb 'revolutionized'.

Using any other punctuation mark (like a dash or parenthesis) or omitting the punctuation entirely would result in asymmetrical punctuation, which is incorrect in Standard English.

Key Concept

Punctuation of Nonessential Elements

Question 1204Question →

Which expression is equivalent to $3(2x - 5) - (x - 4)$ ?

5x - 19

5x - 11

5x - 9

5x - 1

Show answer & explanation

Answer: $5x - 11$

Answer

5x - 11

To find the equivalent expression, apply the distributive property to both parts of the expression:

3(2x - 5)

expands to

6x - 15

, and

-(x - 4)

expands to

-x + 4

. Combining the variable terms gives

6x - x = 5x

. Combining the constant terms gives

-15 + 4 = -11

. Thus, the simplified equivalent expression is

5x - 11

.

Step-by-Step Solution

1

Distribute the number

3

to each term inside the first set of parentheses.

3(2x - 5) = 6x - 15

To eliminate the first set of parentheses by applying the distributive property.

2

Distribute the negative sign (equivalent to multiplying by

-1

) to each term inside the second set of parentheses.

-(x - 4) = -x + 4

To eliminate the second set of parentheses, ensuring the negative sign is applied to both

x

and

-4

.

3

Combine the like terms (the variable terms and the constant terms).

(6x - x) + (-15 + 4) = 5x - 11

To simplify the expression into its final equivalent form.

Key Concept

Equivalent Algebraic Expressions

Question 1205Question →

The table below shows some values of the linear function $h$ .

$x$	$h(x)$
$2$	$k - 4$
$5$	$k + 8$
$8$	$2k + 2$

If $k$ is a constant, what is the value of $k$ ?

Show answer & explanation

Answer: 18

Answer

The value of the constant

k

is

18

.

Since the function

h

is linear, its rate of change (slope) is constant. Calculating the slope using the first two coordinate pairs

(2, k - 4)

and

(5, k + 8)

gives

\frac{(k + 8) - (k - 4)}{5 - 2} = \frac{12}{3} = 4

. Using the next two coordinate pairs

(5, k + 8)

and

(8, 2k + 2)

gives the slope as

\frac{(2k + 2) - (k + 8)}{8 - 5} = \frac{k - 6}{3}

. Setting these two slope values equal to each other gives the equation

\frac{k - 6}{3} = 4

. Multiplying both sides by

3

results in

k - 6 = 12

, and adding

6

to both sides yields

k = 18

.

Step-by-Step Solution

1

Identify that the rate of change (slope) of a linear function is constant between any two points.

The slope calculated from the first two points must equal the slope calculated from the second and third points.

This relationship allows us to set up an algebraic equation to solve for the unknown constant

k

.

2

Calculate the slope using the first two points:

(2, k - 4)

and

(5, k + 8)

.

Slope =

\frac{(k + 8) - (k - 4)}{5 - 2} = \frac{k + 8 - k + 4}{3} = \frac{12}{3} = 4

.

This simplifies to a constant numerical value of

4

for the slope of the function.

3

Calculate the slope using the second and third points:

(5, k + 8)

and

(8, 2k + 2)

.

Slope =

\frac{(2k + 2) - (k + 8)}{8 - 5} = \frac{2k + 2 - k - 8}{3} = \frac{k - 6}{3}

.

This provides a second expression for the slope in terms of the variable

k

.

4

Equate the two slope expressions and solve for

k

.

\frac{k - 6}{3} = 4 \implies k - 6 = 12 \implies k = 18

.

Setting the two expressions equal and solving isolating

k

gives the correct value of

18

.

Key Concept

A linear function has a constant rate of change (slope) between any two points on its graph.

Question 1206Question →

A local delivery service charges a flat fee of $12$ dollars plus $1.50$ dollars per mile to deliver a package. If a customer wants to spend no more than $30$ dollars for a package delivery, what is the maximum number of miles the delivery service can travel?

Show answer & explanation

Answer: 12

Answer

The maximum number of miles the delivery service can travel is 12.

The total delivery cost is the sum of the flat fee (

12

dollars) and the rate per mile (

1.50

dollars multiplied by

m

miles), which is represented by

1.50m + 12

. Since the total cost cannot exceed

30

dollars, the inequality is

1.50m + 12 \leq 30

. Subtracting

12

from both sides gives

1.50m \leq 18

. Dividing both sides by

1.50

yields

m \leq 12

. Thus, the maximum distance the delivery service can travel is

12

miles.

Step-by-Step Solution

1

Set up the inequality representing the delivery cost constraint.

1.50m + 12 \leq 30

The total cost of the delivery is the flat fee of

12

dollars plus

1.50

dollars per mile,

m

, which must be less than or equal to the budget of

30

dollars.

2

Subtract 12 from both sides of the inequality.

1.50m \leq 18

To isolate the variable term, subtract the constant flat fee from both sides of the inequality.

3

Divide both sides of the inequality by 1.50.

m \leq 12

Dividing by the per-mile rate calculates the maximum distance constraint on the variable

m

.

Key Concept

Setting up and solving a one-variable linear inequality to determine a maximum boundary value in context.

Question 1207Question →

The following passage is adapted from an archaeological study of ancient trade networks. Based on the context of the passage, what is the correct form of the noun 'vessel' that should fill the blank?

Fill in the blanks below

To construct a precise chronology of the region's ancient trade networks, archaeologists analyzed the chemical composition of several newly excavated clay vessels. By identifying the unique trace elements in the clay, the researchers concluded that the pottery's distinct mineral signature could reliably trace each origin to a single, localized kiln in the river valley.

Show answer & explanation

Answer

vessel's

The correct answer is the singular possessive form 'vessel's'. The word is preceded by the singular determiner 'each', which requires a singular noun rather than a plural one. Furthermore, the noun modifies 'origin', meaning it must be in the possessive case to show the relationship between the vessel and its origin.

Step-by-Step Solution

1

Identify the syntactic role of the word in the blank.

The blank precedes the noun 'origin' and must modify it to indicate relationship or association, which requires a possessive form.

Nouns modifying other nouns to show possession or relationship must be in the possessive case.

2

Determine the number (singular or plural) of the noun based on surrounding modifiers.

The determiner 'each' precedes the blank and specifies a singular noun.

The word 'each' is grammatically singular and must modify a singular noun.

3

Combine the singular number and possessive case to form the correct word.

The singular possessive form of 'vessel' is 'vessel's'.

Singular nouns are made possessive by adding an apostrophe followed by the letter 's'.

Key Concept

Plural and Possessive Nouns

Question 1208Question →

In 1418, the builders of the Florence Cathedral faced a seemingly impossible engineering challenge: constructing a massive dome without any temporary wooden scaffolding. The architect Filippo Brunelleschi ______ devised an ingenious solution that utilized a herringbone brick pattern to distribute weight. This self-supporting design allowed the dome to be built without collapsing inward during construction, forever altering the course of Renaissance architecture.

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

—who had studied the ancient construction techniques of the Roman Pantheon,

who had studied the ancient construction techniques of the Roman Pantheon,

, who had studied the ancient construction techniques of the Roman Pantheon,

, who had studied the ancient construction techniques of the Roman Pantheon

Show answer & explanation

Answer: , who had studied the ancient construction techniques of the Roman Pantheon,

Answer

The phrase that begins and ends with a comma: ', who had studied the ancient construction techniques of the Roman Pantheon,'

The relative clause 'who had studied the ancient construction techniques of the Roman Pantheon' provides extra, nonrestrictive information about Filippo Brunelleschi. In Standard Written English, nonessential elements must be set off from the rest of the sentence using matching punctuation marks on both sides. The correct choice employs a pair of commas to appropriately isolate the clause. Other options fail because they create asymmetrical punctuation by mixing different marks or omitting one of the boundary markers.

Step-by-Step Solution

1

Identify the boundary of the relative clause in the sentence.

The relative clause is 'who had studied the ancient construction techniques of the Roman Pantheon'.

Determining where the clause begins and ends is necessary to apply the punctuation rules.

2

Determine if the relative clause is essential or nonessential to the sentence's meaning.

The relative clause is nonessential (nonrestrictive) because it modifies the specific proper noun 'Filippo Brunelleschi' and its removal does not change the core meaning of the sentence.

Nonessential elements must be set off with punctuation, whereas essential elements are not punctuated.

3

Apply the matching punctuation rule to set off the nonessential relative clause.

The clause must start with a comma and end with a comma (or start and end with em dashes/parentheses) to maintain grammatical symmetry.

Standard English conventions require matching punctuation marks on both sides of a nonessential element.

Key Concept

Punctuation of Nonessential Elements

Estimated Time:1m 0s

Question 1209Question →

A graphic designer is exporting images for a website. There are two types of images: standard images and high-resolution images. Each standard image, $x$ , requires 3 megabytes (MB) of storage and takes 2 seconds to upload. Each high-resolution image, $y$ , requires 8 MB of storage and takes 5 seconds to upload. The designer must limit the total storage of the exported images to at most 120 MB and the total upload time to at least 60 seconds. Which of the following systems of inequalities models this situation?

3x + 2y \leq 120

8x + 5y \geq 60

3x + 8y \geq 120

2x + 5y \leq 60

8x + 3y \leq 120

5x + 2y \geq 60

3x + 8y \leq 120

2x + 5y \geq 60

Show answer & explanation

Answer: $3x + 8y \leq 120$
$2x + 5y \geq 60$

Answer

The system of inequalities is

3x + 8y \leq 120

and

2x + 5y \geq 60

.

The correct system models the limits correctly. The storage constraint is represented by

3x + 8y \leq 120

since 'at most' corresponds to a less-than-or-equal-to sign. The upload time constraint is represented by

2x + 5y \geq 60

since 'at least' corresponds to a greater-than-or-equal-to sign.

Step-by-Step Solution

1

Set up the inequality for the storage constraint.

3x + 8y \leq 120

Each standard image uses 3 megabytes and each high-resolution image uses 8 megabytes. The total storage must be at most (less than or equal to) 120 megabytes.

2

Set up the inequality for the upload time constraint.

2x + 5y \geq 60

Each standard image takes 2 seconds and each high-resolution image takes 5 seconds to upload. The total upload time must be at least (greater than or equal to) 60 seconds.

3

Combine both inequalities into a system.

The system containing

3x + 8y \leq 120

and

2x + 5y \geq 60

.

Both constraints must be satisfied simultaneously.

Key Concept

Translating verbal constraints into systems of linear inequalities in two variables.

Question 1210Question →

In the $xy$ -plane, a system of two linear equations has no solutions. One of the equations in the system is $4x - 6y = 15$ . The graph of the second equation is a line that passes through the points $(3, k)$ and $(9, 7)$ , where $k$ is a constant. What is the value of $k$ ?

Show answer & explanation

Answer: 3

Answer

The value of

k

is

3

.

For a system of linear equations to have no solutions, the lines representing the equations must be parallel, which means they have equal slopes but different y-intercepts. The first equation,

4x - 6y = 15

, can be rewritten in slope-intercept form as

y = \frac{2}{3}x - 2.5

, showing its slope is

\frac{2}{3}

. The slope of the second line, passing through

(3, k)

and

(9, 7)

, is given by

\frac{7 - k}{9 - 3} = \frac{7 - k}{6}

. Setting the two slopes equal yields

\frac{2}{3} = \frac{7 - k}{6}

. Multiplying both sides by

6

gives

4 = 7 - k

, which solves to

k = 3

. Substituting

k = 3

back into the second line gives

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

. Since the slopes are equal and the y-intercepts (

-2.5

and

1

) are different, the lines are parallel and distinct, confirming there are no solutions.

Step-by-Step Solution

1

Find the slope of the first line by converting the equation to slope-intercept form.

Slope is

\frac{2}{3}

and y-intercept is

-2.5

.

Converting

4x - 6y = 15

to

y = \frac{2}{3}x - 2.5

reveals the slope of the first line.

2

Express the slope of the second line using the coordinates of the two points on the line.

Slope expression is

\frac{7 - k}{6}

.

Applying the slope formula

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

to

(3, k)

and

(9, 7)

defines the slope in terms of

k

.

3

Equate the two slopes and solve for the unknown parameter.

k = 3

Since the system has no solutions, the lines must be parallel and have equal slopes. Setting

\frac{2}{3} = \frac{7 - k}{6}

and solving gives

k = 3

.

Key Concept

For a system of two linear equations to have no solutions, the lines representing the equations must be parallel, which requires them to have the same slope but different y-intercepts.

Estimated Time:2m 0s

Question 1211Question →

If $3x - 2(y - 5) = 12$ and $x = 8$ , what is the value of $y$ ?

1

6

11

10

Show answer & explanation

Answer: 11

Answer

11

The correct answer is 11. Substituting

x = 8

into the equation gives

3(8) - 2(y - 5) = 12

, which simplifies to

24 - 2y + 10 = 12

. Combining the constants on the left side yields

34 - 2y = 12

. Subtracting 34 from both sides gives

-2y = -22

. Finally, dividing both sides by

-2

yields

y = 11

.

Step-by-Step Solution

1

Substitute

x = 8

into the given equation.

3(8) - 2(y - 5) = 12

, which simplifies to

24 - 2(y - 5) = 12

.

This isolates the variable

y

by replacing the variable

x

with its given value.

2

Distribute

-2

to the terms inside the parentheses.

24 - 2y + 10 = 12

.

Multiplying

-2

by

y

gives

-2y

, and multiplying

-2

by

-5

gives

+10

because the product of two negative numbers is positive.

3

Combine the constant terms on the left side of the equation and subtract the result from both sides.

34 - 2y = 12 \Rightarrow -2y = -22

.

Combining

24

and

10

gives

34

. Subtracting

34

from both sides isolates the variable term

-2y

.

4

Divide both sides by

-2

to solve for

y

.

y = 11

.

Dividing both sides of the equation by the coefficient of the variable term isolates the variable.

Key Concept

Solving linear equations in two variables by substitution and isolation.

Estimated Time:45s

Question 1212Question →

Which of the following represents all possible values of $x$ that satisfy the inequality $-2(x - 5) > 6$ ?

x < -8

x < 2

x > 2

x < 8

Show answer & explanation

Answer: $x < 2$

Answer

The inequality is satisfied when

x < 2

.

Dividing both sides of the inequality

-2(x - 5) > 6

by

-2

yields

x - 5 < -3

, where the inequality sign is reversed because of division by a negative number. Adding

5

to both sides of

x - 5 < -3

results in

x < 2

.

Step-by-Step Solution

1

Divide both sides of the inequality

-2(x - 5) > 6

by

-2

.

x - 5 < -3

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

2

Add

5

to both sides of the inequality to isolate

x

.

x < 2

To solve for

x

, eliminate the constant term

-5

on the left side of the inequality.

Key Concept

Solving linear inequalities in one variable, including reversing the inequality sign when multiplying or dividing by a negative number.

Question 1213Question →

A system of inequalities is shown below.

\begin{aligned} y &\leq -3x + 15 \\ y &\leq 2x + 2 \end{aligned}

If

(x, y)

is a solution to the system, what is the maximum possible integer value of

y

?

6

7

8

15

Show answer & explanation

Answer: 7

Answer

7

The maximum value of

y

in the system occurs at the intersection of the boundary lines

y = -3x + 15

and

y = 2x + 2

. Solving for the intersection gives

x = 2.6

and

y = 7.2

. Since any point in the solution set must satisfy

y \leq 7.2

, the largest possible integer value for

y

is 7. We can confirm this is achievable because when

y = 7

,

x

can be any value in the range

[2.5, 2.67]

(such as

x=2.6

), which satisfies both inequalities.

Step-by-Step Solution

1

Find the intersection point of the boundary lines.

Set the two equations equal to find the

x

-coordinate:

-3x + 15 = 2x + 2 \implies 5x = 13 \implies x = 2.6

.

The maximum value of

y

in a system bounded from above by two lines with opposing slopes occurs at their point of intersection.

2

Calculate the corresponding maximum

y

-value.

Substitute

x = 2.6

into either boundary equation:

y = 2(2.6) + 2 = 7.2

.

This determines the absolute upper bound of

y

for any coordinate pair in the solution set.

3

Determine the maximum integer value of

y

.

Since

y \leq 7.2

, the largest integer value that

y

can take is 7.

The question asks specifically for the maximum integer value of

y

, and since

7 \leq 7.2

, a solution exists at this

y

-value.

4

Verify that a solution exists for

y = 7

.

For

y = 7

, the system requires

7 \leq -3x + 15 \implies x \leq 2.67

and

7 \leq 2x + 2 \implies x \geq 2.5

. The interval

[2.5, 2.67]

is non-empty, so valid solutions (e.g.,

(2.6, 7)

) exist.

This confirms that

y = 7

is attainable within the feasible region.

Key Concept

Optimization of variables within systems of linear inequalities

Question 1214Question →

A graphic designer allocates their weekly work time between designing layouts, which takes $d$ hours, and client consultations, which takes $c$ hours. The designer works a maximum of 35 hours per week. Additionally, the designer spends at least twice as much time designing layouts as they do in client consultations. Which of the following systems of inequalities represents this situation?

\begin{aligned} d + c &\leq 35 \\ d &\geq 2c \end{aligned}

\begin{aligned} d + c &\geq 35 \\ d &\leq 2c \end{aligned}

\begin{aligned} d + c &\leq 35 \\ c &\geq 2d \end{aligned}

\begin{aligned} d + c &\geq 35 \\ c &\leq 2d \end{aligned}

Show answer & explanation

Answer: $\begin{aligned} d + c &\leq 35 \\ d &\geq 2c \end{aligned}$

Answer

The system with the inequalities

d + c \leq 35

and

d \geq 2c

The system containing the inequalities

d + c \leq 35

and

d \geq 2c

is correct. The constraint 'works a maximum of 35 hours per week' means the sum of the hours,

d + c

, must be less than or equal to 35 (

d + c \leq 35

). The constraint 'spends at least twice as much time designing layouts as they do in client consultations' means that design hours,

d

, must be greater than or equal to twice the consultation hours,

c

(

d \geq 2c

).

Step-by-Step Solution

1

Identify the inequality representing the limit on total weekly hours.

d + c \leq 35

The phrase 'works a maximum of 35 hours per week' means the total hours spent on designing (

d

) and consulting (

c

) cannot exceed 35.

2

Identify the inequality representing the relative relationship between designing and consulting hours.

d \geq 2c

The phrase 'at least twice as much time designing layouts as they do in client consultations' means the design hours (

d

) must be greater than or equal to two times the consultation hours (

2c

).

3

Combine the individual inequalities into a system.

The system consists of

d + c \leq 35

and

d \geq 2c

.

Both conditions must be satisfied simultaneously to represent the designer's weekly schedule constraint.

Key Concept

Systems of Linear Inequalities in Two Variables

Estimated Time:1m 0s

Question 1215Question →

A closed triangular region in the coordinate plane is defined by the following system of linear inequalities:

\begin{aligned} y &\geq 2x - 4 \\ y &\leq -x + 8 \\ x &\geq 1 \end{aligned}

What is the maximum possible value of the expression $2x + y$ for any point $(x, y)$ that lies within or on the boundary of this region?

Show answer & explanation

Answer: 12

Answer

The maximum possible value of the expression

2x + y

is 12.

To find the maximum possible value of the expression

2x + y

subject to the given system of inequalities, we first identify the boundary lines and find the vertices of the bounded triangular region in the coordinate plane. The boundary lines are

y = 2x - 4

,

y = -x + 8

, and

x = 1

. The intersection of

y = 2x - 4

and

y = -x + 8

occurs at

x = 4, y = 4

, which gives vertex

(4, 4)

. The intersection of

y = 2x - 4

and

x = 1

occurs at

(1, -2)

. The intersection of

y = -x + 8

and

x = 1

occurs at

(1, 7)

. Evaluating the linear expression

2x + y

at these three vertices gives

2(4) + 4 = 12

,

2(1) - 2 = 0

, and

2(1) + 7 = 9

. By the corner point theorem, the maximum value of a linear function on a closed polygonal region occurs at one of the vertices. Comparing the values, the maximum possible value is 12.

Step-by-Step Solution

1

Find the vertex formed by the intersection of the boundary lines

y = 2x - 4

and

y = -x + 8

.

Vertex

A(4, 4)

Setting the two equations equal:

2x - 4 = -x + 8 \implies 3x = 12 \implies x = 4

. Substituting

x = 4

back into either equation yields

y = 4

.

2

Find the vertex formed by the intersection of the boundary line

y = 2x - 4

and the vertical line

x = 1

.

Vertex

B(1, -2)

Substituting

x = 1

into

y = 2x - 4

gives

y = 2(1) - 4 = -2

.

3

Find the vertex formed by the intersection of the boundary line

y = -x + 8

and the vertical line

x = 1

.

Vertex

C(1, 7)

Substituting

x = 1

into

y = -x + 8

gives

y = -1 + 8 = 7

.

4

Evaluate the expression

2x + y

at each of the three vertices.

At

A(4, 4)

, the value is

12

; at

B(1, -2)

, the value is

0

; at

C(1, 7)

, the value is

9

.

According to the Corner Point Theorem of linear programming, the maximum or minimum of a linear objective function on a closed bounded region must occur at one of the vertices.

5

Identify the maximum value from the evaluated points.

The maximum value is 12.

Comparing the values 12, 0, and 9 shows that 12 is the largest value.

Key Concept

Linear Programming and Systems of Inequalities

Question 1216Question →

An online streaming service offers the two monthly subscription plans described in the table below:

Plan	Monthly fee	Cost per premium movie rental
Plan A	$12	$1.50
Plan B	$27 (includes first 4 rentals)	$0.75 (for each rental after the first 4)

If a user rented $m$ premium movies in a month, where $m > 4$ , and the total cost for both plans would be the same, what is the value of $m$ ?

Show answer & explanation

Answer: 16

Answer

The value of

m

that results in the same total monthly cost for both plans is 16.

The correct answer of 16 represents the exact number of premium movie rentals where the total monthly cost for both plans is equal to $36. Any other number of movie rentals will result in different costs for the two plans.

Step-by-Step Solution

1

Define the cost equation for Plan A

\text{Cost}_A = 12 + 1.50m

Plan A charges a flat

12 monthly fee plus

1.50 for each of the

m

movies rented.

2

Define the cost equation for Plan B

\text{Cost}_B = 27 + 0.75(m - 4)

Plan B charges a

27 monthly fee that covers the first 4 movies, and

0.75 for each of the

m - 4

additional movies rented because

m > 4

.

3

Equate the two cost expressions and solve for

m

12 + 1.50m = 27 + 0.75(m - 4) \implies 12 + 1.50m = 24 + 0.75m \implies 0.75m = 12 \implies m = 16

To find when the costs are identical, set the two algebraic expressions equal to each other and isolate the variable

m

.

Key Concept

Setting up and solving linear equations in one variable from context

Alternative Method

Instead of setting up full equations, we can look at the cost difference at

m = 4

. At

4

movies, Plan A costs

12 + 1.50(4) = 18

dollars, and Plan B costs

27

dollars (since

4

movies are included). The price difference is

27 - 18 = 9

dollars. For each movie rented beyond

4

, the cost of Plan A increases by

1.50

dollars while Plan B only increases by

0.75

dollars. The rate of change difference is

1.50 - 0.75 = 0.75

dollars per movie. To bridge the initial

9

dollar difference, the user needs to rent

\frac{9}{0.75} = 12

more movies. Thus, the total number of movies is

4 + 12 = 16

.

Estimated Time:1m 30s

Question 1217Question →

Many desert plants have evolved specialized root systems that are uniquely adapted to survive in extremely arid environments. The saguaro cactus, for instance, grows a shallow network of roots that spreads horizontally from its ______ this extensive structure allows the cactus to absorb water rapidly from even the briefest rainfall events.

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

base;

base,

base

base, but

Show answer & explanation

Answer: base;

Answer

base;

The option ending with a semicolon correctly links two independent clauses without a coordinating conjunction. Since both 'The saguaro cactus... grows a shallow network of roots that spreads horizontally from its base' and 'this extensive structure allows...' can stand alone as complete sentences, a semicolon is appropriate to separate them.

Step-by-Step Solution

1

Analyze the structure of the two clauses surrounding the blank.

The first clause ('The saguaro cactus... grows a shallow network of roots that spreads horizontally from its base') and the second clause ('this extensive structure allows...') are both independent clauses.

Identifying the clause types is necessary to determine the appropriate punctuation or coordinating conjunction needed to connect them.

2

Determine the logical relationship between the two clauses.

The second clause explains the benefit or function of the structure mentioned in the first clause. The relationship is explanatory, not contrastive.

This helps rule out inappropriate coordinating conjunctions like 'but'.

3

Select the correct punctuation to link the independent clauses.

A semicolon is the standard way to link two independent clauses when no coordinating conjunction is used.

It avoids both a comma splice and a run-on sentence while maintaining grammatical correctness.

Key Concept

Semicolons are used to connect two closely related independent clauses that are not joined by a coordinating conjunction.

Question 1218Question →

The table below shows some values for a linear function $f$ , where $k$ is a positive constant.

$x$	$f(x)$
$0$	$3$
$k$	$2k + 6$
$4k$	$27$

What is the slope of the graph of $y = f(x)$ in the $xy$ -plane?

2

-4

4

\frac{1}{4}

Show answer & explanation

Answer: 4

Answer

The slope of the graph of

f

is

4

.

The correct answer is

4

. A linear function can be written in the form

f(x) = mx + b

, where

m

is the slope and

b

is the

y

-intercept. Since the

y

-intercept of the graph of

f

is

(0, 3)

, we have

b = 3

, so the function is

f(x) = mx + 3

. Using the given points, we can set up the equations

f(k) = mk + 3 = 2k + 6

and

f(4k) = 4mk + 3 = 27

. Solving this system yields

mk = 6

and

2k = 3

, which gives

k = 1.5

. Substituting

k = 1.5

into

mk = 6

gives the slope

m = 4

.

Step-by-Step Solution

1

Express the linear function

f(x)

using the slope-intercept form and the given

y

-intercept.

f(x) = mx + 3

, where

m

is the slope and the

y

-intercept is

(0, 3)

.

A linear function has the general form

f(x) = mx + b

. The table indicates that when

x = 0

,

f(x) = 3

, which gives the

y

-intercept

(0, 3)

and establishes

b = 3

.

2

Use the point

(k, 2k + 6)

from the table to write an equation involving

m

and

k

.

mk + 3 = 2k + 6 \implies mk = 2k + 3

.

Substituting

x = k

into the function expression gives

f(k) = mk + 3

. Setting this equal to the table value

2k + 6

allows us to express

mk

in terms of

k

.

3

Use the point

(4k, 27)

from the table to write another equation involving

m

and

k

, and solve for the product

mk

.

4mk + 3 = 27 \implies 4mk = 24 \implies mk = 6

.

Substituting

x = 4k

into the function expression gives

f(4k) = 4mk + 3

. Setting this equal to the table value

27

allows us to solve directly for the numerical value of

mk

.

4

Substitute the value of

mk

into the equation from Step 2 to solve for the constant

k

.

6 = 2k + 3 \implies 2k = 3 \implies k = 1.5

.

By replacing

mk

with

6

in the equation

mk = 2k + 3

, we get a single-variable linear equation that we can solve for

k

.

5

Solve for the slope

m

using the value of

k

and the product

mk

.

m(1.5) = 6 \implies m = 4

.

Since

mk = 6

and

k = 1.5

, dividing the product

6

by

1.5

yields the slope

m

.

Key Concept

Using coordinate points and intercepts to determine the slope of a linear function represented in a table.

Question 1219Question →

Research in cognitive psychology has increasingly focused on the phenomenon of cognitive fatigue, which occurs when prolonged mental exertion depletes executive resources. While brief rests can temporarily restore attention, more sustained recovery requires complete detachment from task-related demands ______ individuals who engage in active, non-cognitive leisure activities during breaks show significantly greater cognitive replenishment than those who simply remain idle.

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

, indeed,

, but

; indeed,

indeed

Show answer & explanation

Answer: ; indeed,

Answer

The choice with a semicolon followed by the transitional adverb 'indeed' and a comma ('; indeed,').

The passage contains two independent clauses: the first ends with 'demands' and the second begins with 'individuals'. The correct choice uses a semicolon to separate the two independent clauses, followed by the transitional adverb 'indeed' to show that the second clause elaborates on or supports the assertion of the first clause.

Step-by-Step Solution

1

Analyze the structure of the passage before and after the blank to identify the clause boundaries.

The text before the blank is an independent clause, and the text after the blank is also an independent clause.

Identifying the clause boundaries determines whether we need to link dependent or independent clauses.

2

Determine the appropriate grammatical punctuation and coordinating elements needed to join two independent clauses.

Two independent clauses must be joined by a period, a semicolon, a colon, or a comma coupled with a coordinating conjunction. Linking them with a comma alone creates a comma splice, while linking them with no punctuation creates a run-on sentence.

This narrows down the choices to grammatically correct linking methods.

3

Evaluate the semantic relationship between the two independent clauses to select the correct transition or coordinator.

The second clause reinforces and provides specific evidence for the claim in the first clause. Therefore, an elaborative transitional adverb like 'indeed' is appropriate, whereas a contrastive coordinator like 'but' is logically incorrect.

This ensures that the chosen option is both grammatically correct and semantically logical.

Key Concept

Clause Boundaries and Linking

Estimated Time:1m 0s

Question 1220Question →

A system of linear equations is given by

\begin{aligned} kx - 4y &= 12 \\ 3x + ky &= 5 \end{aligned}

where

k

is a constant. If the system has a unique solution

(x, y)

such that

x > 0

and

y < 0

, how many possible integer values of

k

exist?

7

8

9

10

Show answer & explanation

Answer: 9

Answer

9

To find the number of integer values of

k

for which the system has a solution with

x > 0

and

y < 0

, we first express

x

and

y

in terms of

k

. Eliminating

y

by multiplying the first equation by

k

and the second by

4

and adding them yields

(k^2 + 12)x = 12k + 20

, or

x = \frac{12k+20}{k^2+12}

. Similarly, eliminating

x

yields

y = \frac{5k-36}{k^2+12}

. Since the denominator

k^2 + 12

is strictly positive for all real

k

, the sign of

x

and

y

depends solely on their numerators. For

x > 0

, we require

12k + 20 > 0

, which gives

k > -\frac{5}{3} \approx -1.67

. For

y < 0

, we require

5k - 36 < 0

, which gives

k < \frac{36}{5} = 7.2

. Combining these constraints gives the interval

-\frac{5}{3} < k < \frac{36}{5}

. The integers in this interval are

-1, 0, 1, 2, 3, 4, 5, 6,

and

7

, which is a total of 9 integers.

Step-by-Step Solution

1

Solve the system of equations for

x

in terms of

k

by eliminating

y

.

x = \frac{12k + 20}{k^2 + 12}

Multiply the first equation by

k

and the second by 4, then add them:

k(kx - 4y) + 4(3x + ky) = 12k + 20 \implies (k^2 + 12)x = 12k + 20

.

2

Solve the system of equations for

y

in terms of

k

by eliminating

x

.

y = \frac{5k - 36}{k^2 + 12}

Multiply the first equation by 3 and the second by

k

, then subtract the first from the second:

k(3x + ky) - 3(kx - 4y) = 5k - 36 \implies (k^2 + 12)y = 5k - 36

.

3

Apply the condition

x > 0

to find a constraint on

k

.

k > -\frac{5}{3}

Since the denominator

k^2 + 12

is positive for all real values of

k

, the expression for

x

is positive if and only if its numerator is positive:

12k + 20 > 0 \implies k > -\frac{20}{12} = -\frac{5}{3}

.

4

Apply the condition

y < 0

to find another constraint on

k

.

k < \frac{36}{5}

Similarly, since

k^2 + 12 > 0

, the expression for

y

is negative if and only if its numerator is negative:

5k - 36 < 0 \implies k < \frac{36}{5}

.

5

Combine the constraints and count the number of integer values of

k

in the resulting interval.

9 integer values

The combined inequality is

-\frac{5}{3} < k < \frac{36}{5}

, which simplifies to approximately

-1.67 < k < 7.2

. The integers in this range are

-1, 0, 1, 2, 3, 4, 5, 6, 7

, giving a total of 9 integers.

Key Concept

Solving systems of linear equations with parameters under inequality constraints

Answer

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