All practice questions

Question 1301Question →

In a study of the migratory monarch butterfly (*Danaus plexippus*), entomologist Dr. Sarah Carter determined that navigating thousands of miles requires not only sensing the Earth's magnetic field but also utilizing the position of the Sun. According to Carter, the butterflies integrate these cues to maintain a consistent heading, ensuring they can locate their overwintering sites, survive harsh weather conditions, and _______

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

establishing new colonies in the spring

the establishment of new colonies in the spring

establish new colonies in the spring

to establish new colonies in the spring

Show answer & explanation

Answer: establish new colonies in the spring

Answer

establish new colonies in the spring

The correct choice uses the base verb form 'establish' to match the base verbs 'locate' and 'survive'. The auxiliary verb 'can' governs all three items in the list: they can 'locate', 'survive', and 'establish'. Using any other verb form disrupts this parallel structure.

Step-by-Step Solution

1

Identify the list of items in the sentence that must be parallel.

The list follows the phrase 'ensuring they can' and includes 'locate their overwintering sites' and 'survive harsh weather conditions'.

Recognizing the structure of the list is essential to determining the grammatical form needed for the final item.

2

Analyze the grammatical structure of the existing list items.

'locate' and 'survive' are both base form verbs that follow the modal auxiliary verb 'can'.

To maintain parallel structure, the final item in the list must also begin with a base form verb.

3

Evaluate the choices to find the one that matches the base verb structure.

The phrase starting with 'establish' is the only option that uses the base form of the verb, completing the parallel sequence: 'locate...', 'survive...', and 'establish...'.

This maintains the grammatical consistency required by the conventions of Standard English.

Key Concept

Parallel Structure

Estimated Time:1m 0s

Question 13020% of students answered this correctlyQuestion →

In the system of linear equations below, $c$ is a constant.

\begin{aligned} 2x + 3y &= 12 \\ cx - y &= 6 \end{aligned}

If the system has a unique solution $(x, y)$ such that $x > 0$ and $y < 0$ , which of the following could be the value of $c$ ?

-2

-1

\frac{1}{2}

2

Show answer & explanation

Answer: $\frac{1}{2}$

Answer

The option containing the value

\frac{1}{2}

is correct.

The correct answer is the option containing the value

\frac{1}{2}

. Solving the system of linear equations in terms of the constant

c

gives

x = \frac{30}{2+3c}

and

y = \frac{12c-12}{2+3c}

. For the solution

(x, y)

to lie in the fourth quadrant, we require

x > 0

and

y < 0

. The condition

x > 0

is satisfied when

2+3c > 0

, which simplifies to

c > -\frac{2}{3}

. Using this result, the condition

y < 0

requires the numerator of

y

to be negative, so

12c - 12 < 0

, which simplifies to

c < 1

. Combining these inequalities yields the interval

-\frac{2}{3} < c < 1

. The only value among the given options that falls within this interval is

\frac{1}{2}

.

Step-by-Step Solution

1

Express

y

in terms of

x

and the constant

c

using the second equation.

y = cx - 6

This allows for substitution into the first equation to solve for

x

.

2

Substitute the expression for

y

into the first equation and solve for

x

.

2x + 3(cx - 6) = 12 \implies (2 + 3c)x - 18 = 12 \implies (2 + 3c)x = 30 \implies x = \frac{30}{2 + 3c}

This isolates the variable

x

in terms of the constant

c

.

3

Substitute the expression for

x

back into the equation for

y

to express

y

in terms of

c

.

y = c\left(\frac{30}{2 + 3c}\right) - 6 = \frac{30c - 6(2 + 3c)}{2 + 3c} = \frac{12c - 12}{2 + 3c}

This isolates the variable

y

in terms of the constant

c

.

4

Apply the condition

x > 0

to find the constraint on

c

.

Since

x = \frac{30}{2 + 3c} > 0

and the numerator is positive, the denominator must also be positive:

2 + 3c > 0 \implies c > -\frac{2}{3}

.

This determines the lower bound for the constant

c

.

5

Apply the condition

y < 0

to find the constraint on

c

.

Since

y = \frac{12c - 12}{2 + 3c} < 0

and the denominator

2 + 3c

is positive, the numerator must be negative:

12c - 12 < 0 \implies c < 1

.

This determines the upper bound for the constant

c

.

6

Combine the inequalities to find the complete range for

c

and identify the matching option.

-\frac{2}{3} < c < 1

. The only value among the options that lies in this interval is

\frac{1}{2}

.

This identifies the correct option based on the mathematical constraints.

Key Concept

Solving systems of linear equations with parameters and applying quadrant boundary constraints.

Question 1303Question →

Consider the system of inequalities shown below:

\begin{aligned} y &\geq 2x + 1 \\ y &< -x + 6 \end{aligned}

Which coordinate pair $(x, y)$ is a solution to the system?

(2, 3)

(1, 4)

(1, 5)

(2, 5)

Show answer & explanation

Answer: $(1, 4)$

Answer

The coordinate pair

(1, 4)

is a solution to the system of inequalities.

The coordinate pair

(1, 4)

is the correct answer because substituting

x = 1

and

y = 4

into both inequalities yields true statements:

4 \geq 2(1) + 1 \Rightarrow 4 \geq 3

is true, and

4 < -1 + 6 \Rightarrow 4 < 5

is true.

Step-by-Step Solution

1

Substitute the coordinates of the correct pair,

(1, 4)

, into the first inequality,

y \geq 2x + 1

.

4 \geq 2(1) + 1 \Rightarrow 4 \geq 3

, which is true.

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

2

Substitute the same coordinates,

(1, 4)

, into the second inequality,

y < -x + 6

.

4 < -1 + 6 \Rightarrow 4 < 5

, which is true.

The point must also satisfy the second inequality to be a solution to the entire system.

3

Conclude that since

(1, 4)

satisfies both inequalities, it is a solution to the system.

The point

(1, 4)

is in the solution set.

A coordinate pair is a solution to a system of inequalities if and only if it satisfies every inequality in the system.

Key Concept

Testing coordinate pairs in a system of linear inequalities by substitution.

Estimated Time:45s

Question 1304Question →

Although the newly formed committee of environmental advocates has proposed several ambitious initiatives to reduce urban carbon emissions, _______ must still secure the substantial funding required for implementation. The group's primary goal is to install energy-efficient solar panels on municipal buildings throughout the city, but without additional financial backing from local businesses, these plans cannot move forward.

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

they

it

them

these

Show answer & explanation

Answer: it

Answer

The correct pronoun is 'it' because it agrees in number with the singular collective noun 'committee.'

The pronoun 'it' is correct because the antecedent is the singular collective noun 'committee.' Since the committee acts as a single unit in proposing and securing funding, it requires a singular pronoun. Additionally, because the pronoun functions as the subject of the verb phrase 'must still secure,' the subject pronoun 'it' is the correct case.

Step-by-Step Solution

1

Identify the antecedent of the pronoun to be placed in the blank.

The antecedent is the collective noun 'committee.'

Determining the correct antecedent is necessary to establish pronoun-antecedent agreement.

2

Determine the grammatical number (singular or plural) of the antecedent.

The noun 'committee' is a singular collective noun that functions as a single unit here, as confirmed by the singular verb 'has proposed' in the preceding clause.

The pronoun must agree in number with the antecedent, ignoring the intervening plural noun 'advocates' in the prepositional phrase 'of environmental advocates.'

3

Identify the grammatical case required by the pronoun's function in the clause.

The pronoun functions as the subject of the verb phrase 'must still secure,' meaning a subject-case pronoun is required.

Choosing the correct pronoun requires matching both number and case constraints.

4

Select the option that is singular and in the subject case.

The pronoun 'it' meets these criteria.

The pronoun 'it' is singular and fits the subject position of the clause.

Key Concept

Pronoun-Antecedent Agreement in Number

Estimated Time:45s

Question 1305Question →

In sociology, the concept of the "third place" refers to social surroundings that exist independently of the two primary environments of home and work. These informal gathering spaces—such as cafes, parks, and libraries—offer individuals opportunities to connect with their local communities. While some urban planners have overlooked their social utility, research suggests that they are vital to civic [blank] they cultivate a sense of shared identity and support public dialogue.

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

life; they

life, they

life they

life, but they

Show answer & explanation

Answer: life; they

Answer

life; they

The text contains two independent clauses: "research suggests that they are vital to civic life" and "they cultivate a sense of shared identity and support public dialogue." A semicolon is a grammatically correct way to link two independent clauses without a coordinating conjunction, making "life; they" the correct choice.

Step-by-Step Solution

1

Analyze the sentence structure before and after the blank.

The clause before the blank ("research suggests that they are vital to civic life") is independent, and the clause after the blank ("they cultivate a sense of shared identity and support public dialogue") is also independent.

Identifying clause types is necessary to determine how they can be grammatically linked.

2

Evaluate the punctuation and conjunctions needed to link two independent clauses.

Two independent clauses can be joined by a semicolon, a period, or a comma followed by a coordinating conjunction.

This helps rule out options that create run-on sentences or comma splices.

3

Check the logical relationship between the two clauses to verify conjunction usage.

The second clause explains why the gathering spaces are vital to civic life, so a contrasting coordinator like "but" is inappropriate.

This ensures the correct transition or punctuation is selected based on context.

Key Concept

Clause Boundaries and Linking

Estimated Time:45s

Question 1306Question →

To reconstruct past climate patterns, glaciologists Dr. Clara Vance and Dr. Liam Chen analyzed ice cores retrieved from high-altitude glaciers in the Andes. The _______ findings, which detailed a rapid increase in local temperatures during the mid-Holocene, provided crucial evidence that challenged existing models of prehistoric climate stability and prompted a reevaluation of tropical glacier sensitivity.

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

glaciologists

glaciologist's

glaciologist

glaciologists'

Show answer & explanation

Answer: glaciologists'

Answer

glaciologists'

The correct answer is the plural possessive form 'glaciologists''. The sentence establishes that there are two scientists, Dr. Clara Vance and Dr. Liam Chen, making the noun plural. Because these scientists own or produced the 'findings' mentioned directly after the blank, the noun must also be possessive. For a plural noun ending in -s, possessive case is indicated by placing an apostrophe at the very end of the word.

Step-by-Step Solution

1

Analyze the context to determine the noun's number (singular or plural).

The text identifies two specific individuals: Dr. Clara Vance and Dr. Liam Chen. Therefore, the noun referring to them ('glaciologists') must be plural.

Identifying the number prevents choosing a singular noun form.

2

Analyze the syntax to determine the noun's case (possessive or non-possessive).

The blank modifies the noun 'findings,' indicating that the findings belong to the scientists. Therefore, the noun must be possessive.

Identifying the case prevents choosing a non-possessive noun form.

3

Combine number and case to select the correct punctuation.

A plural possessive noun that ends in -s is formed by adding an apostrophe after the -s (glaciologists').

This complies with the conventions of Standard English.

Key Concept

Plural and Possessive Nouns

Estimated Time:1m 0s

Question 1307Question →

A nutritionist is designing a diet plan using two food types, Food A and Food B. The table below shows the amount of protein and carbohydrates, in grams, per gram of each food type:

Food Type	Protein (g)	Carbohydrates (g)
Food A	$0.12$	$0.08$
Food B	$0.18$	$0.05$

The nutritionist wants to combine the two foods to obtain a mixture that has a total mass of $250$ grams and contains exactly $36$ grams of protein. How many grams of carbohydrates will be in this mixture?

Show answer & explanation

Answer: 17

Answer

The total amount of carbohydrates in the mixture is 17 grams.

To find the total amount of carbohydrates in the mixture, we first define

x

as the mass in grams of Food A and

y

as the mass in grams of Food B. Since the total mass of the mixture is

250

grams, we write the equation

x + y = 250

. Since the total protein in the mixture must be

36

grams, and Food A and Food B contain

0.12

grams and

0.18

grams of protein per gram respectively, we write the equation

0.12x + 0.18y = 36

. Multiplying the first equation by

12

yields

12x + 12y = 3000

, and multiplying the second equation by

100

yields

12x + 18y = 3600

. Subtracting the first new equation from the second gives

6y = 600

, which simplifies to

y = 100

. Substituting this back into the first equation gives

x + 100 = 250

, so

x = 150

. Using these values, the total amount of carbohydrates is

0.08(150) + 0.05(100) = 12 + 5 = 17

grams.

Step-by-Step Solution

1

Define variables for the mass of each food type.

Let

x

be the mass, in grams, of Food A, and let

y

be the mass, in grams, of Food B.

To represent the unknown quantities in the system of equations.

2

Set up the system of linear equations.

x + y = 250

(total mass equation) and

0.12x + 0.18y = 36

(total protein equation).

The mixture has a total mass of

250

grams and must contain exactly

36

grams of protein.

3

Solve the system of equations for

x

and

y

.

Multiply the first equation by

12

to get

12x + 12y = 3000

. Multiply the second equation by

100

to get

12x + 18y = 3600

. Subtract the first modified equation from the second to get

6y = 600

, which gives

y = 100

. Substitute

y = 100

back into

x + y = 250

to get

x = 150

.

To find the specific masses of Food A and Food B required for the mixture.

4

Calculate the total mass of carbohydrates in the mixture.

Total carbohydrates

= 0.08x + 0.05y = 0.08(150) + 0.05(100) = 12 + 5 = 17

grams.

The question asks for the total grams of carbohydrates in the final mixture.

Key Concept

Systems of Linear Equations

Question 1308Question →

An empty water tank is being filled by two hoses. Hose A fills the tank at a constant rate of 12 gallons per minute. Hose B fills the tank at a constant rate of 15 gallons per minute. Hose A is turned on first, and 5 minutes later, Hose B is also turned on. If both hoses run until the tank contains a total of 330 gallons of water, for how many minutes was Hose A running?

Show answer & explanation

Answer: 15

Answer

15

Let

m

be the number of minutes Hose A was running. Since Hose B was turned on 5 minutes after Hose A, Hose B was running for

m - 5

minutes. The volume of water contributed by Hose A is

12m

gallons, and the volume of water contributed by Hose B is

15(m - 5)

gallons. The sum of these volumes is 330 gallons, which gives the equation

12m + 15(m - 5) = 330

. Distributing 15 gives

12m + 15m - 75 = 330

. Combining like terms yields

27m - 75 = 330

. Adding 75 to both sides gives

27m = 405

. Dividing both sides by 27 yields

m = 15

. Thus, Hose A was running for 15 minutes.

Step-by-Step Solution

1

Define variables for the running time of each hose.

Let Hose A run for

m

minutes. Then Hose B runs for

m - 5

minutes.

Hose B started 5 minutes after Hose A, so it ran for 5 fewer minutes.

2

Set up a linear equation for the total gallons of water.

12m + 15(m - 5) = 330

The sum of the water from both hoses must equal 330 gallons.

3

Simplify and solve the linear equation.

27m = 405

, which gives

m = 15

.

Distribute, combine like terms, and isolate the variable

m

.

Key Concept

Setting up and solving a multi-step linear equation in one variable from a real-world context.

Question 1309Question →

A public library is purchasing new books for its collection. The library plans to buy paperback books for $6 each and hardcover books for$ 20 each. The library has a budget of at most $1,200 for this purchase, wants to buy at most 120 total books, and must purchase at least 45 hardcover books. What is the maximum number of paperback books the library can purchase?

Show answer & explanation

Answer: 50

Answer

The maximum number of paperback books the library can purchase is 50.

The correct maximum number of paperback books is 50. By expressing the budget constraint as

p \le 200 - \frac{10}{3}h

, we see that the number of paperback books is maximized when the number of hardcover books,

h

, is minimized. Since the library must buy at least 45 hardcover books, we substitute

h = 45

into the inequality to get

p \le 50

. This combination also satisfies the total book constraint because

50 + 45 = 95

, which is less than or equal to 120.

Step-by-Step Solution

1

Set up the system of inequalities representing the library's constraints.

Let

p

be the number of paperbacks and

h

be the number of hardcovers. The constraints are

6p + 20h \le 1200

,

p + h \le 120

, and

h \ge 45

.

This translates the word problem into mathematical inequalities.

2

Isolate the variable

p

in the budget inequality to express it in terms of

h

.

p \le 200 - \frac{10}{3}h

This shows how the number of paperbacks depends on the number of hardcovers purchased.

3

Maximize

p

by using the minimum possible value of

h

.

p \le 200 - \frac{10}{3}(45) = 50

To maximize

p

, we must minimize

h

because buying more hardcovers decreases the remaining budget for paperbacks. The minimum value for

h

is 45.

4

Verify if the solution satisfies the remaining total books constraint.

50 + 45 = 95 \le 120

, which is true.

This ensures the solution is feasible and does not violate any other constraints.

Key Concept

Systems of Linear Inequalities in Two Variables

Question 1310Question →

For a constant $k$ , the graph of the linear equation $kx - 4y = 15$ in the $xy$ -plane contains the point $(3, 3)$ . What is the slope of the line?

Show answer & explanation

Answer: 2.25

Answer

2.25

Substituting the coordinates of the point

(3, 3)

into the given equation yields

k(3) - 4(3) = 15

, which simplifies to

3k - 12 = 15

. Adding

12

to both sides gives

3k = 27

, and dividing by

3

gives

k = 9

. Substituting

k = 9

back into the original equation gives

9x - 4y = 15

. To find the slope of the line, the equation can be written in slope-intercept form,

y = mx + b

, by isolating

y

. Subtracting

9x

from both sides gives

-4y = -9x + 15

. Dividing all terms by

-4

yields

y = \frac{9}{4}x - \frac{15}{4}

, or

y = 2.25x - 3.75

. Therefore, the slope of the line is

\frac{9}{4}

, or

2.25

.

Step-by-Step Solution

1

Substitute the point

(3, 3)

into the equation

kx - 4y = 15

3k - 12 = 15

Since the graph of the equation contains the point

(3, 3)

, substituting

x = 3

and

y = 3

must satisfy the equation.

2

Solve for the constant

k

k = 9

Adding

12

to both sides of the equation gives

3k = 27

. Dividing both sides by

3

yields

k = 9

.

3

Substitute

k = 9

into the equation and rewrite it in slope-intercept form (

y = mx + b

)

y = \frac{9}{4}x - \frac{15}{4}

Substituting

k = 9

gives

9x - 4y = 15

. Subtracting

9x

from both sides gives

-4y = -9x + 15

. Dividing both sides by

-4

isolates

y

.

4

Identify the slope from the slope-intercept form

Slope =

2.25

(or

\frac{9}{4}

)

In the slope-intercept form

y = mx + b

, the coefficient

m

represents the slope of the line.

Key Concept

Determining the slope of a line from a linear equation by first finding an unknown constant using a given point on the line.

Question 1311Question →

Which of the following is equivalent to the expression $\frac{2x^2 - 7x - 4}{x^2 - 16} \cdot \frac{x^2 + 4x}{2x + 1}$ for all values of $x$ where the expression is defined?

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

x

x^2

\frac{x(x+4)}{x-4}

Show answer & explanation

Answer: $x$

Answer

x

The correct answer is

x

because factoring all numerators and denominators of the given product results in

\frac{(2x + 1)(x - 4)}{(x - 4)(x + 4)} \cdot \frac{x(x + 4)}{2x + 1}

. After cancelling the common factors of

(2x + 1)

,

(x - 4)

, and

(x + 4)

, only

x

remains.

Step-by-Step Solution

1

Factor the quadratic expression in the numerator of the first fraction.

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

Finding the factors of the quadratic trinomial allows us to look for common terms to cancel later.

2

Factor the denominator of the first fraction using the difference of squares identity.

x^2 - 16 = (x - 4)(x + 4)

Simplifying the quadratic binomial in the denominator exposes common factors.

3

Factor the numerator of the second fraction by extracting the greatest common factor.

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

Exhibiting the shared variable

x

helps identify terms that will cancel with the denominator.

4

Substitute the factored forms back into the original expression and cancel the common factors.

\frac{(2x + 1)(x - 4)}{(x - 4)(x + 4)} \cdot \frac{x(x + 4)}{2x + 1} = x

The terms

(2x + 1)

,

(x - 4)

, and

(x + 4)

appear in both the numerator and the denominator, leaving only the factor

x

.

Key Concept

Simplifying rational expressions by factoring and cancelling common factors

Estimated Time:1m 30s

Question 1312Question →

In sociolinguistic studies of bilingualism, researchers analyze how speakers transition between languages within a single conversation. For decades, early academic theories incorrectly characterized this code-switching behavior as a sign of mental fatigue or lexical deficiencies______ contemporary linguists recognize that the practice is actually a highly structured, rule-governed phenomenon requiring advanced cognitive control and linguistic competence.

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

, however,

; however,

however

because

Show answer & explanation

Answer: ; however,

Answer

The semicolon followed by the conjunctive adverb 'however' and a comma

The transition links two independent clauses. A semicolon followed by a conjunctive adverb and a comma is the standard way to connect two independent clauses when expressing a relationship of contrast.

Step-by-Step Solution

1

Identify the clause boundary and determine the grammatical structure of the clauses on either side of the blank.

The text before the blank is an independent clause ('For decades... deficiencies'), and the text after the blank is also an independent clause ('contemporary... competence').

Identifying the clause types is necessary to choose the correct punctuation and linking words.

2

Analyze the logical relationship between the two independent clauses.

The first clause describes an outdated, negative view of code-switching, while the second clause presents a modern, positive view. This indicates a relationship of contrast.

The correct transition word must match the logical flow of contrast between the ideas.

3

Select the option that correctly links the two independent clauses while expressing the contrasting relationship.

A semicolon followed by the conjunctive adverb 'however' and a comma is the grammatically correct way to join these two independent clauses.

This configuration avoids run-on sentences and comma splices, conforming to Standard English conventions.

Key Concept

Clause Boundaries and Linking

Estimated Time:1m 0s

Question 1313Question →

Which of the following expressions is equivalent to $3(x^2 - 2x) - 2(x^2 - 5x)$ for all values of $x$ ?

x^2 - 16x

x^2 + 4x

x^2 - 4x

5x^2 - 16x

Show answer & explanation

Answer: $x^2 + 4x$

Answer

x^2 + 4x

To simplify the given expression, distribute the coefficients outside the parentheses to each term inside. Distributing

3

to

(x^2 - 2x)

yields

3x^2 - 6x

. Distributing

-2

to

(x^2 - 5x)

yields

-2x^2 + 10x

. Grouping and combining the like terms gives

(3x^2 - 2x^2) + (-6x + 10x)

, which simplifies to

x^2 + 4x

.

Step-by-Step Solution

1

Distribute the number

3

to the terms in the first set of parentheses.

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

To eliminate the parentheses by multiplying each term inside by the outer coefficient.

2

Distribute the number

-2

to the terms in the second set of parentheses.

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

To eliminate the second set of parentheses, remembering that multiplying two negative numbers yields a positive product:

-2 \times -5x = 10x

.

3

Combine like terms by grouping the quadratic terms and the linear terms together.

(3x^2 - 2x^2) + (-6x + 10x) = x^2 + 4x

To simplify the expression by adding or subtracting the coefficients of terms with the same variable power.

Key Concept

Equivalent Algebraic Expressions

Estimated Time:45s

Question 1314Question →

A beverage company produces a fruit juice blend using apple juice and cranberry juice. Let $x$ represent the volume, in liters, of apple juice, and let $y$ represent the volume, in liters, of cranberry juice used in one batch of the blend. The production constraints for each batch are modeled by the system of inequalities below:

x + y \geq 100

y \geq \frac{1}{3}x

y \leq -\frac{1}{2}x + 75

What is the maximum possible volume, in liters, of cranberry juice that can be used in a single batch?

Show answer & explanation

Answer: 50

Answer

50

To find the maximum possible value of

y

, we examine the bounds of the feasible region. Rewriting the inequalities in terms of

x

gives

x \geq 100 - y

and

x \leq 150 - 2y

. Since

x

must satisfy both inequalities simultaneously, the lower bound must be less than or equal to the upper bound, meaning

100 - y \leq 150 - 2y

. Solving this inequality for

y

gives

y \leq 50

. The point

(50, 50)

also satisfies the remaining constraint

y \geq \frac{1}{3}x

, confirming that 50 is the maximum possible value.

Step-by-Step Solution

1

Isolate

x

in the first inequality:

x + y \geq 100

becomes

x \geq 100 - y

.

x \geq 100 - y

This establishes a lower bound for the volume of apple juice in terms of the volume of cranberry juice.

2

Isolate

x

in the third inequality:

y \leq -\frac{1}{2}x + 75

can be rewritten by subtracting 75 from both sides to get

y - 75 \leq -\frac{1}{2}x

, then multiplying by

-2

and reversing the inequality sign to get

x \leq 150 - 2y

.

x \leq 150 - 2y

This establishes an upper bound for the volume of apple juice in terms of the volume of cranberry juice.

3

Combine the lower and upper bounds of

x

:

100 - y \leq x \leq 150 - 2y

. For a valid solution to exist, the lower bound must be less than or equal to the upper bound.

100 - y \leq 150 - 2y

This allows us to solve directly for the constraint on the cranberry juice volume

y

.

4

Solve the inequality

100 - y \leq 150 - 2y

by adding

2y

to both sides and subtracting 100 from both sides.

y \leq 50

This determines that the maximum possible value for

y

is 50.

5

Substitute

y = 50

back into the compound inequality to find

x

:

100 - 50 \leq x \leq 150 - 2(50) \implies 50 \leq x \leq 50

, which means

x = 50

. Check if the point

(50, 50)

satisfies the remaining inequality

y \geq \frac{1}{3}x

:

50 \geq \frac{1}{3}(50) \approx 16.67

, which is true.

The point

(50, 50)

is a valid solution to the system.

This confirms that 50 is a achievable value within the system of inequalities.

Key Concept

Finding the boundary limits and optimization points within a system of linear inequalities in two variables.

Question 1315Question →

A hot air balloon is at an altitude of $h$ meters. The balloon begins to descend at a constant rate. After $3$ minutes, the altitude of the balloon is $540$ meters. After $8$ minutes, the altitude of the balloon is $390$ meters. If the altitude of the balloon is modeled by a linear function of time, what was the initial altitude of the balloon, in meters?

Show answer & explanation

Answer: 630

Answer

The initial altitude of the balloon was 630 meters.

To find the initial altitude, we model the balloon's descent as a linear equation of the form

h(t) = mt + b

, where

h(t)

is the altitude at time

t

,

m

is the constant rate of change, and

b

is the initial altitude. Using the given points

(3, 540)

and

(8, 390)

, we find the slope

m = \frac{390 - 540}{8 - 3} = -30

meters per minute. Substituting

m = -30

and the point

(3, 540)

into

h(t) = mt + b

gives

540 = -30(3) + b

. Solving for

b

yields

b = 540 + 90 = 630

meters.

Step-by-Step Solution

1

Identify two data points from the problem context

The coordinate points are

(3, 540)

and

(8, 390)

To define the linear relationship, we need at least two coordinate points representing (time, altitude)

2

Calculate the slope (

m

) of the linear function

m = \frac{390 - 540}{8 - 3} = \frac{-150}{5} = -30

The rate of change represents the speed at which the balloon descends each minute

3

Solve for the vertical intercept (

b

) using the slope-intercept form

y = mx + b

540 = -30(3) + b \implies b = 630

The initial altitude corresponds to the altitude at time

t = 0

, which is the vertical intercept of the linear function

Key Concept

Linear Functions and Graphs

Question 1316Question →

A system of two equations is shown below.

y = x^2 - 4

y = x - 2

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

(2, 0)

(0, -2)

(-2, 0)

(1, -3)

Show answer & explanation

Answer: $(2, 0)$

Answer

The ordered pair

(2, 0)

is the correct solution to the system.

The ordered pair

(2, 0)

is the correct solution because substituting these values into both equations in the system results in true statements. For the quadratic equation,

0 = 2^2 - 4

simplifies to

0 = 0

. For the linear equation,

0 = 2 - 2

simplifies to

0 = 0

.

Step-by-Step Solution

1

Substitute the

x

-value and

y

-value from the candidate solution

(2, 0)

into the first equation,

y = x^2 - 4

.

0 = 2^2 - 4 \implies 0 = 4 - 4 \implies 0 = 0

, which is a true statement.

An ordered pair must satisfy the first equation to be a candidate solution for the system.

2

Substitute the

x

-value and

y

-value from the candidate solution

(2, 0)

into the second equation,

y = x - 2

.

0 = 2 - 2 \implies 0 = 0

, which is also a true statement.

An ordered pair must satisfy all equations in the system simultaneously to be a valid solution.

Key Concept

Verifying a solution to a nonlinear system of equations by substituting the coordinate values into both equations.

Question 1317Question →

If the expression $\frac{6x^3 + 7x^2 - 14x + 19}{2x + 5}$ is equivalent to $ax^2 + bx + c + \frac{k}{2x + 5}$ for all $x \neq -2.5$ , where $a$ , $b$ , $c$ , and $k$ are constants, what is the value of $a - b + c + k$ ?

Show answer & explanation

Answer: 14

Answer

14

Dividing the numerator

6x^3 + 7x^2 - 14x + 19

by the denominator

2x + 5

using polynomial division yields a quotient of

3x^2 - 4x + 3

and a remainder of

4

. The equivalent expression is

3x^2 - 4x + 3 + \frac{4}{2x + 5}

. Comparing this with the form

ax^2 + bx + c + \frac{k}{2x + 5}

gives the values

a = 3

,

b = -4

,

c = 3

, and

k = 4

. Substituting these values into

a - b + c + k

gives

3 - (-4) + 3 + 4 = 14

.

Step-by-Step Solution

1

Perform the first step of polynomial long division by dividing

6x^3

by

2x

.

Quotient term:

3x^2

; Remainder:

-8x^2 - 14x + 19

To eliminate the highest-degree term of the numerator.

2

Perform the second step of division by dividing

-8x^2

by

2x

.

Quotient term:

-4x

; Remainder:

6x + 19

To find the next term of the quotient.

3

Perform the third step of division by dividing

6x

by

2x

.

Quotient term:

3

; Remainder:

4

To find the constant term of the quotient and the final remainder.

4

Compare the resulting expression

3x^2 - 4x + 3 + \frac{4}{2x + 5}

with the given form to identify the constants

a

,

b

,

c

, and

k

.

a = 3

,

b = -4

,

c = 3

,

k = 4

To map the coefficients of equivalent algebraic expressions.

5

Evaluate the expression

a - b + c + k

using the identified values.

3 - (-4) + 3 + 4 = 14

To obtain the final numeric value requested.

Key Concept

Equivalent Algebraic Expressions via Polynomial Long Division

Question 1318Question →

In the $xy$ -plane, the region $R$ consists of all points $(x, y)$ that satisfy the system of inequalities below:

\begin{aligned} 3x - 4y &\geq -12 \\ y &\geq 2x - 7 \\ x + 2y &\geq 2 \end{aligned}

If $(x, y)$ is a point in region $R$ , what is the maximum possible value of the expression $x - 2y$ ?

2.0

4.4

5.0

-5.2

Show answer & explanation

Answer: 4.4

Answer

4.4

The maximum value of a linear expression over a closed, bounded polygonal region must occur at one of the vertices of that region. By solving the systems of equations for each pair of boundary lines, the vertices of the triangular region are found to be

(8, 9)

,

(3.2, -0.6)

, and

(-1.6, 1.8)

. Evaluating the expression

x - 2y

at each vertex gives

-10

at

(8, 9)

,

4.4

at

(3.2, -0.6)

, and

-5.2

at

(-1.6, 1.8)

. Comparing these values,

4.4

is the maximum possible value.

Step-by-Step Solution

1

Find the intersection points of the boundary equations by solving the three systems of two linear equations.

System 1 (

3x - 4y = -12

and

y = 2x - 7

) yields

(8, 9)

.
System 2 (

y = 2x - 7

and

x + 2y = 2

) yields

(3.2, -0.6)

.
System 3 (

3x - 4y = -12

and

x + 2y = 2

) yields

(-1.6, 1.8)

.

The boundaries of the region are defined by these lines, so the vertices of the region occur at their intersections.

2

Verify that the region defined by the inequalities is the interior of the triangle formed by these vertices by testing a point inside, such as

(2, 2)

.

The point

(2, 2)

satisfies all three inequalities:

3(2) - 4(2) = -2 \geq -12

(True)

2 \geq 2(2) - 7 = -3

(True)

2 + 2(2) = 6 \geq 2

(True)

This confirms that the region is bounded and the vertices are the extreme points of the closed triangular region.

3

Evaluate the expression

x - 2y

at each of the three vertices.

At

(8, 9)

:

8 - 2(9) = -10

At

(3.2, -0.6)

:

3.2 - 2(-0.6) = 3.2 + 1.2 = 4.4

At

(-1.6, 1.8)

:

-1.6 - 2(1.8) = -1.6 - 3.6 = -5.2

A linear expression defined over a closed, bounded polygonal region reaches its maximum and minimum values at the vertices of that region.

4

Compare the evaluated values to identify the maximum value.

The maximum value is

4.4

.

Among the three candidate values (

-10

,

4.4

, and

-5.2

),

4.4

is the greatest.

Key Concept

Linear programming vertex method for systems of linear inequalities.

Estimated Time:3m 0s

Question 1319Question →

For all positive values of $x$ , which of the following expressions is equivalent to $(8x^6)^{\frac{1}{3}}$ ?

2x^3

\frac{8}{3}x^2

2x^2

8x^2

Show answer & explanation

Answer: $2x^2$

Answer

The expression

2x^2

To find an equivalent expression, apply the exponent of

\frac{1}{3}

to each factor inside the parentheses. The expression

(8x^6)^{\frac{1}{3}}

becomes

8^{\frac{1}{3}} \cdot (x^6)^{\frac{1}{3}}

. Since

8^{\frac{1}{3}} = \sqrt[3]{8} = 2

, and

(x^6)^{\frac{1}{3}} = x^{6 \cdot \frac{1}{3}} = x^2

, the simplified expression is

2x^2

.

Step-by-Step Solution

1

Apply the power of a product rule,

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

, to distribute the exponent of

\frac{1}{3}

to both the coefficient and the variable term.

(8x^6)^{\frac{1}{3}} = 8^{\frac{1}{3}} \cdot (x^6)^{\frac{1}{3}}

This separates the numerical coefficient and the variable part to simplify them individually.

2

Simplify the numerical coefficient by evaluating the cube root of

8

.

8^{\frac{1}{3}} = 2

The exponent of

\frac{1}{3}

represents the cube root, and

2^3 = 8

.

3

Simplify the variable term using the power of a power rule,

(x^a)^b = x^{a \cdot b}

.

(x^6)^{\frac{1}{3}} = x^{6 \cdot \frac{1}{3}} = x^2

Multiplying the exponents simplifies the expression.

4

Multiply the simplified coefficient and variable term together.

2x^2

This gives the final simplified equivalent expression.

Key Concept

Simplifying algebraic expressions with fractional exponents using exponent rules.

Question 1320Question →

Although a colony of leafcutter ants can consist of millions of individual workers, _______ functions as a superorganism, coordinating complex foraging routes to harvest vegetation.

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

they

their

it

its

Show answer & explanation

Answer: it

Answer

The singular subject pronoun 'it' correctly completes the sentence by agreeing in number with the singular antecedent 'colony' and serving as the subject of the clause.

The correct option is the singular subject pronoun 'it'. The pronoun refers back to the singular collective noun 'colony', not the plural nouns 'ants' or 'workers' within the prepositional phrases. Therefore, a singular pronoun is required. Additionally, because the pronoun serves as the subject of the verb 'functions', it must be in the subject case rather than the possessive case.

Step-by-Step Solution

1

Identify the antecedent of the blank space.

The antecedent is the singular noun phrase 'a colony of leafcutter ants,' where the main noun is the singular collective noun 'colony.'

Finding the noun being replaced by the pronoun determines whether the pronoun must be singular or plural.

2

Determine the grammatical function of the pronoun in the clause.

The pronoun precedes the verb 'functions' and acts as the subject of the independent clause.

Knowing the function of the pronoun helps identify the correct case (subject vs. object or possessive).

3

Select the pronoun that matches both the singular number of the antecedent and the subject case.

The singular subject pronoun 'it' fits both requirements, while plural options like 'they' and 'their' or possessive options like 'its' do not.

Standard English conventions require agreement in both number and grammatical case.

Key Concept

Pronoun-Antecedent Agreement and Case

Answer

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