All practice questions

Question 1261Question →

The passage below discusses prehistoric cave paintings in western Europe. Which word or phrase completes the passage so that it conforms to the conventions of Standard English?

Fill in the blanks below

In his study of prehistoric cave paintings in western Europe, archaeologist Dr. Liam Torres argues that these ancient artworks served a dual purpose in early human societies. According to Torres, the detailed illustrations of animals were designed not only to document successful hunting strategies to convey complex spiritual narratives to subsequent generations.

Show answer & explanation

Answer

but also

The phrase 'but also' correctly completes the correlative conjunction 'not only... but also', which connects the two parallel infinitive phrases 'to document' and 'to convey'.

Step-by-Step Solution

1

Identify the correlative conjunction pattern in the sentence.

The sentence uses the correlative conjunction 'not only' to introduce the first parallel element ('to document successful hunting strategies').

Correlative conjunctions work in pairs to link parallel elements.

2

Determine the coordinating partner for 'not only' and verify parallelism.

The partner for 'not only' is 'but also'. Placing 'but also' in the blank links 'to document...' with 'to convey...', maintaining parallel infinitive phrases.

Using 'but also' completes the coordinate pair and ensures that both parts of the sentence have the same grammatical form.

Key Concept

Parallel structure with correlative conjunctions requires matching grammatical elements and correct conjunction pairs.

Question 1262Question →

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

\begin{aligned} y &\geq 3x + 1 \\ y &< -2x + 5 \end{aligned}

(0, 2)

(1, 2)

(0, 6)

(3, 10)

Show answer & explanation

Answer: $(0, 2)$

Answer

(0, 2)

The ordered pair

(0, 2)

is correct because substituting these values into both inequalities results in true statements. Specifically,

2 \geq 3(0) + 1

becomes

2 \geq 1

, which is true, and

2 < -2(0) + 5

becomes

2 < 5

, which is also true.

Step-by-Step Solution

1

Substitute the coordinates of the correct ordered pair into the first inequality,

y \geq 3x + 1

.

2 \geq 3(0) + 1 \Rightarrow 2 \geq 1

, which is true.

To be a solution to the system, an ordered pair must satisfy both inequalities in the system.

2

Substitute the coordinates of the same ordered pair into the second inequality,

y < -2x + 5

.

2 < -2(0) + 5 \Rightarrow 2 < 5

, which is true.

To verify the solution, we check if it satisfies the second constraint.

Key Concept

Testing coordinate solutions in a system of linear inequalities

Question 1263Question →

An artist sells small prints for $4$ dollars each and large prints for $9$ dollars each. The artist wants to sell at least $15$ prints in total and earn at least $80$ dollars from these sales. If the artist sells $5$ small prints, what is the minimum number of large prints the artist must sell to meet both conditions?

Show answer & explanation

Answer: 10

Answer

The minimum number of large prints the artist must sell is 10.

Substituting the value of small prints (

x = 5

) into the total count inequality

x + y \geq 15

gives

5 + y \geq 15

, which simplifies to

y \geq 10

. Substituting it into the earnings inequality

4x + 9y \geq 80

yields

20 + 9y \geq 80

, which simplifies to

y \geq 6.67

. To satisfy both conditions, the value of

y

must be at least 10.

Step-by-Step Solution

1

Set up the inequalities for the system.

Let

x

represent the number of small prints and

y

represent the number of large prints. The constraint for the minimum number of prints is

x + y \geq 15

. The constraint for the minimum earnings is

4x + 9y \geq 80

.

To represent the given conditions as a system of linear inequalities in two variables.

2

Substitute the known value of small prints (

x = 5

) into the inequalities.

Substituting

x = 5

into the first inequality gives

5 + y \geq 15

, which simplifies to

y \geq 10

. Substituting

x = 5

into the second inequality gives

4(5) + 9y \geq 80

, which simplifies to

20 + 9y \geq 80

, then

9y \geq 60

, resulting in

y \geq \frac{20}{3} \approx 6.67

.

To determine the range of values for the number of large prints (

y

) under both constraints.

3

Determine the minimum integer value for

y

that satisfies both inequalities.

The first constraint requires

y \geq 10

, and the second constraint requires

y \geq 6.67

. Since the number of prints must be an integer and both conditions must be satisfied, the minimum value is

10

.

To satisfy both inequalities simultaneously with the smallest possible integer value.

Key Concept

Systems of Linear Inequalities in Two Variables

Estimated Time:1m 0s

Question 1264Question →

The system of linear equations shown below contains constants

a

and

b

.

\begin{aligned} \frac{1}{2}(ax - 4y) &= x - \frac{1}{2}y + 6 \\ 2x - y &= b \end{aligned}

If the system has infinitely many solutions, what is the value of

a + b

?

Show answer & explanation

Answer: 12

Answer

The correct answer is 12.

To find the value of

a + b

that yields infinitely many solutions, we rewrite both equations in the standard form

Ax + By = C

. Simplifying the first equation gives

(a - 2)x - 3y = 12

. Multiplying the second equation

2x - y = b

by

3

gives

6x - 3y = 3b

. For the system to have infinitely many solutions, the two equations must be equivalent, meaning

a - 2 = 6

(which gives

a = 8

) and

3b = 12

(which gives

b = 4

). The sum of these values is

8 + 4 = 12

.

Step-by-Step Solution

1

Distribute the fraction

\frac{1}{2}

on the left side of the first equation.

\frac{1}{2}ax - 2y = x - \frac{1}{2}y + 6

To expand the expression and prepare it for simplification.

2

Group the

x

and

y

terms on the left side of the equation and the constants on the right side.

\left(\frac{1}{2}a - 1\right)x - \frac{3}{2}y = 6

To write the equation in standard linear form.

3

Multiply the entire equation by

2

to eliminate the fractional coefficients.

(a - 2)x - 3y = 12

To simplify comparison with the second equation by working with integer coefficients.

4

Multiply the second equation,

2x - y = b

, by

3

to align the

y

-coefficients with the first equation.

6x - 3y = 3b

Two linear equations have infinitely many solutions if they represent the same line, which requires matching coefficients and constants.

5

Equate the corresponding

x

-coefficients and constant terms from

(a - 2)x - 3y = 12

and

6x - 3y = 3b

.

a - 2 = 6 \implies a = 8

and

3b = 12 \implies b = 4

To solve for the values of the constants

a

and

b

.

6

Calculate the sum of

a

and

b

.

8 + 4 = 12

To find the final requested value of

a + b

.

Key Concept

Determining parameters for infinitely many solutions in a system of linear equations

Question 1265Question →

A company plans to install standard-charging ports and fast-charging ports at its office building. The company must install at least $15$ charging ports in total. Each fast-charging port requires $12$ kilowatts of power, and each standard-charging port requires $4$ kilowatts of power. The electrical grid can supply a maximum of $120$ kilowatts of power for these ports. Additionally, the number of standard-charging ports must be at least twice the number of fast-charging ports. What is the maximum number of fast-charging ports the company can install?

Show answer & explanation

Answer: 6

Answer

The maximum number of fast-charging ports the company can install is 6.

Representing the number of fast-charging ports by

x

and standard-charging ports by

y

yields the system of inequalities:

x + y \geq 15

,

12x + 4y \leq 120

, and

y \geq 2x

. The power constraint simplifies to

3x + y \leq 30

, or

y \leq 30 - 3x

. Combining this with

y \geq 2x

gives

2x \leq 30 - 3x

, which simplifies to

5x \leq 30

, or

x \leq 6

. Substituting

x = 6

into the constraints gives

y = 12

, which satisfies the total port constraint because

6 + 12 = 18 \geq 15

. Therefore, the maximum number of fast-charging ports is 6.

Step-by-Step Solution

1

Define variables and write the system of inequalities representing the constraints.

Let

x

be the number of fast-charging ports and

y

be the number of standard-charging ports. The constraints are:
1)

x + y \geq 15

2)

12x + 4y \leq 120

3)

y \geq 2x

4)

x \geq 0, y \geq 0

This translates the word problem constraints into a system of linear inequalities.

2

Simplify the power capacity inequality.

3x + y \leq 30

, which can be rewritten as

y \leq 30 - 3x

.

Dividing the terms by 4 simplifies the coefficients, making algebraic manipulation easier.

3

Combine the simplified inequality with the charging port ratio constraint to find the upper limit for

x

.

Since

2x \leq y

and

y \leq 30 - 3x

, we have

2x \leq 30 - 3x

.
Adding

3x

to both sides yields

5x \leq 30

.
Dividing by 5 gives

x \leq 6

.

This determines the maximum possible value for the number of fast-charging ports.

4

Verify that the upper limit

x = 6

satisfies all system constraints with integer values.

If

x = 6

, then

y \geq 2(6) = 12

and

y \leq 30 - 3(6) = 12

, which means

y = 12

.
Checking the total port constraint:

x + y = 6 + 12 = 18

. Since

18 \geq 15

, the point

(6, 12)

satisfies all constraints.

Since the number of ports must be integers, we must confirm that

x = 6

yields a valid integer coordinate

(6, 12)

that lies within the feasible region.

Key Concept

Maximizing a variable under a system of linear inequalities in two variables.

Question 1266Question →

In a study of forest ecosystems, ecologist Dr. Suzanne Simard observed that Douglas fir trees utilize underground mycorrhizal networks to interact with one another. According to Simard, these fungal networks allow the trees not only to share essential carbon with neighboring seedlings but also ________ during periods of drought.

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

the transportation of water to shallower roots

transporting water to shallower roots

water is transported to shallower roots

to transport water to shallower roots

Show answer & explanation

Answer: to transport water to shallower roots

Answer

to transport water to shallower roots

The sentence uses the correlative conjunction 'not only... but also...' to join two parallel elements describing what the fungal networks allow the trees to do. The first element, following 'not only', is the infinitive phrase 'to share essential carbon with neighboring seedlings'. To maintain parallel structure, the second element, following 'but also', must also be an infinitive phrase. The phrase 'to transport water to shallower roots' completes the parallel structure perfectly.

Step-by-Step Solution

1

Identify the grammatical construction used to connect ideas in the sentence.

The sentence uses the correlative conjunction structure 'not only [X] but also [Y]'.

Recognizing the structure helps determine that the elements [X] and [Y] must share the same grammatical form.

2

Analyze the grammatical form of the first element in the correlative pair.

The phrase following 'not only' is 'to share essential carbon with neighboring seedlings', which is an infinitive phrase starting with 'to'.

This establishes the parallel structure that the second element must match.

3

Evaluate the choices to find the one that matches this infinitive phrase structure.

The phrase 'to transport water to shallower roots' is also an infinitive phrase starting with 'to', maintaining perfect parallelism.

Selecting this option ensures the sentence conforms to Standard English parallel structure conventions.

Key Concept

Parallel structure with correlative conjunctions (not only... but also...)

Question 1267Question →

Which of the following expressions is equivalent to $(8x^6)^{\frac{2}{3}} \cdot (4x^{-2})^{-\frac{1}{2}}$ for all positive values of $x$ ?

2x^3

2x^4

2x^5

8x^5

Show answer & explanation

Answer: $2x^5$

Answer

2x^5

To find the equivalent expression, simplify each part of the product. The first part,

(8x^6)^{\frac{2}{3}}

, simplifies to

8^{\frac{2}{3}} \cdot x^{6 \cdot \frac{2}{3}} = 4x^4

. The second part,

(4x^{-2})^{-\frac{1}{2}}

, simplifies to

4^{-\frac{1}{2}} \cdot x^{-2 \cdot -\frac{1}{2}} = \frac{1}{2}x

. Multiplying these two simplified expressions gives

4x^4 \cdot \frac{1}{2}x = 2x^5

. This matches the expression

2x^5

.

Step-by-Step Solution

1

Simplify the first term,

(8x^6)^{\frac{2}{3}}

, by distributing the exponent to both the coefficient and the variable.

(8x^6)^{\frac{2}{3}} = 8^{\frac{2}{3}} \cdot (x^6)^{\frac{2}{3}} = (\sqrt[3]{8})^2 \cdot x^{6 \cdot \frac{2}{3}} = 2^2 \cdot x^4 = 4x^4

The power of a product rule states that

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

, and the power of a power rule states that

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

.

2

Simplify the second term,

(4x^{-2})^{-\frac{1}{2}}

, by distributing the exponent to both the coefficient and the variable.

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

Applying the same power rules, negative exponent properties, and multiplying negative exponents yields a positive exponent:

-2 \cdot -\frac{1}{2} = 1

.

3

Multiply the simplified expressions obtained in Step 1 and Step 2.

4x^4 \cdot \frac{1}{2}x = (4 \cdot \frac{1}{2}) \cdot (x^4 \cdot x^1) = 2x^{4+1} = 2x^5

Multiply the coefficients and add the exponents of the same base variable according to the product rule

a^m \cdot a^n = a^{m+n}

.

Key Concept

Simplifying algebraic expressions containing rational and negative exponents by using the rules of exponents.

Question 1268Question →

In the late nineteenth century, chemists struggled to isolate elemental fluorine due to its extreme reactivity, which presented unprecedented laboratory challenges. Attempts to extract the element repeatedly ended in failure ______ the highly reactive gas immediately bonded with any surrounding materials, including the glass apparatus used to contain it, often causing dangerous explosions.

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

failure; the

failure, the

failure the

failure, although

Show answer & explanation

Answer: failure; the

Answer

The option containing the semicolon correctly links the two independent clauses without introducing grammatical errors or incorrect logical transitions.

The correct option uses a semicolon to link the two independent clauses. A semicolon is a standard grammatical method for separating two independent clauses without a coordinating conjunction.

Step-by-Step Solution

1

Identify the structure of the clauses on both sides of the blank.

The clause before the blank ('Attempts to extract the element repeatedly ended in failure') and the clause after the blank ('the highly reactive gas immediately bonded with any surrounding materials...') are both independent clauses.

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

2

Evaluate standard punctuation rules for linking independent clauses.

Two independent clauses can be joined by a period, a semicolon, or a coordinating conjunction (such as 'and', 'but', 'for') preceded by a comma.

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

3

Analyze the logical relationship between the two clauses to evaluate conjunctions.

The second clause explains why the attempts ended in failure, indicating a causal relationship. A contrast subordinator is logically incorrect.

This ensures the sentence is both grammatically correct and semantically logical.

Key Concept

Clause Boundaries and Linking

Estimated Time:1m 30s

Question 1269Question →

In the $xy$ -plane, the graph of the linear function $f$ passes through the points $(-2, 5)$ and $(4, 1)$ . If the linear function $g$ is defined by $g(x) = f(x) + 3$ , what is the $x$ -intercept of the graph of $g$ ?

6

\frac{11}{2}

10

\frac{20}{3}

Show answer & explanation

Answer: $10$

Answer

10

The correct answer is

10

. First, find the slope of

f

using the formula

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

, which gives

m = \frac{1 - 5}{4 - (-2)} = -\frac{2}{3}

. Using the point

(4, 1)

in the point-slope formula, the equation of

f

is

f(x) = -\frac{2}{3}x + \frac{11}{3}

. Since

g(x) = f(x) + 3

, the equation of

g

is

g(x) = -\frac{2}{3}x + \frac{20}{3}

. Setting

g(x) = 0

to find the

x

-intercept gives

x = 10

.

Step-by-Step Solution

1

Calculate the slope of the linear function

f

m = -\frac{2}{3}

The slope

m

of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

. Substituting

(-2, 5)

and

(4, 1)

yields

m = \frac{1 - 5}{4 - (-2)} = \frac{-4}{6} = -\frac{2}{3}

.

2

Find the equation of function

f

f(x) = -\frac{2}{3}x + \frac{11}{3}

Using the point-slope form

y - y_1 = m(x - x_1)

with the point

(4, 1)

, we get

y - 1 = -\frac{2}{3}(x - 4) \Rightarrow y = -\frac{2}{3}x + \frac{8}{3} + 1 = -\frac{2}{3}x + \frac{11}{3}

.

3

Determine the equation of function

g

g(x) = -\frac{2}{3}x + \frac{20}{3}

Since

g(x) = f(x) + 3

, we add

3

to the expression for

f(x)

:

g(x) = -\frac{2}{3}x + \frac{11}{3} + 3 = -\frac{2}{3}x + \frac{20}{3}

.

4

Find the

x

-intercept of the graph of

g

x = 10

The

x

-intercept is the value of

x

when

g(x) = 0

. Setting

g(x) = 0

gives

0 = -\frac{2}{3}x + \frac{20}{3} \Rightarrow \frac{2}{3}x = \frac{20}{3} \Rightarrow x = 10

.

Key Concept

Finding the equation of a linear function from two points and applying vertical translations to find key features such as intercepts.

Estimated Time:1m 30s

Question 1270Question →

For $x > 0$ , which of the following is equivalent to the expression $\frac{\sqrt{x^3 \cdot \sqrt{x}}}{\sqrt[4]{x^3}}$ ?

x^4

x\sqrt[4]{x}

x

x^2\sqrt{x}

Show answer & explanation

Answer: $x$

Answer

The expression is equivalent to

x

.

To simplify the expression, convert each radical into an expression with a rational exponent. The inner term of the numerator contains

\sqrt{x} = x^{1/2}

. Multiplying this by

x^3

yields

x^3 \cdot x^{1/2} = x^{3 + 1/2} = x^{7/2}

. Applying the outer square root to this product yields

\sqrt{x^{7/2}} = (x^{7/2})^{1/2} = x^{7/4}

. The denominator is

\sqrt[4]{x^3} = x^{3/4}

. Dividing the simplified numerator by the denominator is performed by subtracting the exponent of the denominator from that of the numerator:

x^{7/4} / x^{3/4} = x^{7/4 - 3/4} = x^{4/4} = x

. Therefore, the expression simplifies to

x

.

Step-by-Step Solution

1

Convert the inner radical in the numerator into a fractional exponent.

The term

\sqrt{x}

becomes

x^{1/2}

, so the product inside the outer square root becomes

x^3 \cdot x^{1/2}

.

Converting radicals to rational exponents allows the application of standard exponent properties.

2

Combine the terms inside the outer square root using exponent addition rules.

Since

3 + 1/2 = 7/2

, the product is

x^{7/2}

.

When multiplying exponential terms with the same base, their exponents are added.

3

Simplify the entire numerator by applying the outer square root.

The numerator becomes

\sqrt{x^{7/2}} = (x^{7/2})^{1/2} = x^{7/4}

.

Applying a square root to a term is equivalent to raising that term to the power of

1/2

.

4

Convert the denominator into a fractional exponent.

The denominator

\sqrt[4]{x^3}

becomes

x^{3/4}

.

Converting the denominator's radical to a rational exponent allows division with the numerator.

5

Divide the numerator by the denominator using exponent subtraction rules.

The final expression is

x^{7/4} / x^{3/4} = x^{7/4 - 3/4} = x^{4/4} = x

.

When dividing exponential terms with the same base, the exponent of the denominator is subtracted from the exponent of the numerator.

Key Concept

Simplification of equivalent algebraic expressions involving radical and rational exponents.

Alternative Method

An alternative approach is to substitute a convenient value for

x

that is a perfect power of 4, such as

x = 16

. The numerator becomes

\sqrt{16^3 \cdot \sqrt{16}} = \sqrt{4096 \cdot 4} = \sqrt{16384} = 128

. The denominator becomes

\sqrt[4]{16^3} = \sqrt[4]{4096} = 8

. Evaluating the fraction yields

128 / 8 = 16

. Since substituting

x = 16

yields a value of

16

, the expression must be equivalent to

x

.

Estimated Time:1m 15s

Question 1271Question →

In the $xy$ -plane, the graph of the equation $ax + by = 24$ , where $a$ and $b$ are constants, is a line. The line has an $x$ -intercept of $(d, 0)$ and a $y$ -intercept of $(0, d - 6)$ , where $d > 6$ is a constant. If the line passes through the point $(2, 5)$ , what is the value of $a + b$ ?

Show answer & explanation

Answer: 6

Answer

6

Substituting the intercepts

(d, 0)

and

(0, d-6)

into

ax + by = 24

gives

a = 24/d

and

b = 24/(d-6)

. Using the point

(2, 5)

, we get

2(24/d) + 5(24/(d-6)) = 24

, which simplifies to

2/d + 5/(d-6) = 1

. Solving for

d

yields

d^2 - 13d + 12 = 0

, giving

d = 12

or

d = 1

. Given

d > 6

, we have

d = 12

. Substituting

d = 12

back gives

a = 2

and

b = 4

, so

a + b = 6

.

Step-by-Step Solution

1

Find expressions for

a

and

b

in terms of

d

by substituting the given intercepts.

a = \frac{24}{d}

and

b = \frac{24}{d-6}

The

x

-intercept

(d, 0)

and

y

-intercept

(0, d-6)

lie on the line

ax + by = 24

.

2

Substitute the point

(2, 5)

into the line's equation.

2a + 5b = 24

The point

(2, 5)

lies on the line.

3

Substitute the expressions for

a

and

b

into

2a + 5b = 24

and simplify.

\frac{2}{d} + \frac{5}{d-6} = 1

To create a single equation in terms of the variable

d

.

4

Solve the rational equation for

d

.

d = 12

Multiplying by the common denominator

d(d-6)

leads to the quadratic equation

d^2 - 13d + 12 = 0

, which factors as

(d-12)(d-1) = 0

. Since the problem states

d > 6

, the only valid solution is

d = 12

.

5

Calculate

a

and

b

using

d = 12

, and find their sum.

a = 2

,

b = 4

, and

a + b = 6

Substituting

d = 12

into the expressions for

a

and

b

yields the constant coefficients, and summing them provides the final requested value.

Key Concept

Using intercepts and a given point on a line in the coordinate plane to solve for the parameters of its linear equation.

Question 1272Question →

\begin{aligned} x - 2y &= 5 \\ 3x - y &= 10 \end{aligned}

If $(x, y)$ is the solution to the system of equations above, what is the value of $x + y$ ?

2

3

6

8

Show answer & explanation

Answer: 2

Answer

2

The correct answer is 2. Isolating

x

in the first equation gives

x = 2y + 5

. Substituting this expression into the second equation yields

3(2y + 5) - y = 10

. Distributing the 3 gives

6y + 15 - y = 10

, which simplifies to

5y + 15 = 10

. Subtracting 15 from both sides gives

5y = -5

, so

y = -1

. Substituting

y = -1

back into the equation for

x

gives

x = 2(-1) + 5 = 3

. The sum of the solutions is

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

.

Step-by-Step Solution

1

Isolate

x

in the first equation.

x = 2y + 5

Isolating one variable makes it straightforward to substitute it into the other equation.

2

Substitute the expression for

x

into the second equation and solve for

y

.

y = -1

Substituting

x = 2y + 5

gives

3(2y + 5) - y = 10

. Distributing the 3 yields

6y + 15 - y = 10

. Combining like terms gives

5y + 15 = 10

. Subtracting 15 from both sides gives

5y = -5

, so

y = -1

.

3

Substitute

y = -1

back into the expression for

x

to find its value.

x = 3

Using the isolated expression

x = 2y + 5

, substituting

y = -1

gives

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

.

4

Calculate the sum

x + y

.

2

The question asks for the value of

x + y

, which is

3 + (-1) = 2

.

Key Concept

Solving systems of linear equations in two variables using substitution or elimination, and evaluating algebraic combinations of the variables.

Alternative Method

The system can also be solved by elimination. Multiplying the second equation by 2 gives

6x - 2y = 20

. Subtracting the first equation (

x - 2y = 5

) from this new equation eliminates

y

, resulting in

5x = 15

, or

x = 3

. Substituting

x = 3

into either original equation yields

y = -1

. Finally, the sum is

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

.

Estimated Time:1m 30s

Question 1273Question →

A recent investigation into the subglacial lakes of Antarctica revealed that a unique community of extremophilic bacteria thrives in complete darkness, obtaining ______ energy from chemical compounds dissolved in the water.

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

their

its

it's

they're

Show answer & explanation

Answer: its

Answer

its

The singular possessive pronoun 'its' correctly refers to the singular collective noun antecedent 'community'. The intervening plural prepositional phrase 'of extremophilic bacteria' does not change the singular nature of the main antecedent.

Step-by-Step Solution

1

Identify the antecedent of the pronoun needed in the blank.

The noun phrase serving as the antecedent is 'a unique community of extremophilic bacteria'. The head of this phrase is the singular collective noun 'community'.

Determining the grammatical number of the antecedent is necessary to choose a pronoun that agrees with it.

2

Determine the correct case and grammatical category for the pronoun.

Since the pronoun modifies the noun 'energy', it must be possessive. Since the antecedent 'community' is singular, the pronoun must be singular possessive ('its').

Pronouns must agree in number (singular vs. plural) and case (possessive vs. subject/object) with their antecedents.

3

Evaluate the choices to find the singular possessive pronoun.

The word 'its' is the singular possessive pronoun. The word 'their' is plural, and the options 'it's' and 'they're' are contractions rather than possessives.

Eliminating grammatically incorrect options leaves the correct answer.

Key Concept

Pronoun-Antecedent Agreement and Case

Question 1274Question →

Deep within Earth's lower mantle, temperatures and pressures are high enough to synthesize unique crystalline structures. The mineral bridgmanite ______ remains unstable and rapidly decomposes when brought to the surface. Consequently, geophysicists must study this abundant substance indirectly by simulating the extreme conditions of the deep Earth in high-pressure laboratory chambers.

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

, which constitutes approximately thirty-eight percent of the planet's total volume—

—which constitutes approximately thirty-eight percent of the planet's total volume—

—which constitutes approximately thirty-eight percent of the planet's total volume,

—which constitutes approximately thirty-eight percent of the planet's total volume

Show answer & explanation

Answer: —which constitutes approximately thirty-eight percent of the planet's total volume—

Answer

The correct choice is the option that sets off the nonessential clause 'which constitutes approximately thirty-eight percent of the planet's total volume' with matching em dashes on both sides.

The text setting off the nonessential relative clause with matching em dashes is correct because nonessential elements must be set off by matching pairs of punctuation (such as two em dashes, two commas, or two parentheses). In this sentence, the relative clause 'which constitutes approximately thirty-eight percent of the planet's total volume' is nonessential and must be separated from the main clause using matching em dashes on both sides.

Step-by-Step Solution

1

Identify the nonessential element in the sentence.

The relative clause 'which constitutes approximately thirty-eight percent of the planet's total volume' is nonessential because removing it does not alter the core meaning of the independent clause.

Determining whether a clause is essential or nonessential dictates how it must be punctuated.

2

Check the options for matching pairs of punctuation.

The correct option must use matching punctuation marks on both sides of the nonessential clause, such as two commas or two em dashes.

Standard English conventions require nonessential elements to be set off symmetrically.

3

Eliminate options with asymmetrical punctuation.

Options that mix a comma with an em dash, or fail to provide a closing punctuation mark, are incorrect.

Asymmetry violates punctuation rules for parenthetical elements.

Key Concept

Punctuation of Nonessential Elements

Estimated Time:45s

Question 1275Question →

Although marine biologists have long studied the ecological dynamics of giant kelp forests, they continue to be amazed by the plants' rapid growth under optimal conditions. A single kelp frond can grow up to two feet per day in ideal ______ this rapid expansion provides vital shelter and nutrition for hundreds of diverse marine species.

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

conditions;

conditions,

conditions

conditions although

Show answer & explanation

Answer: conditions;

Answer

conditions;

The correct answer correctly uses a semicolon to separate two independent clauses. Since both clauses can stand alone as complete sentences, a semicolon is a grammatically valid method of joining them without a coordinating conjunction.

Step-by-Step Solution

1

Identify the clause boundaries in the sentence.

The sentence contains two independent clauses: 'A single kelp frond can grow up to two feet per day in ideal conditions' and 'this rapid expansion provides vital shelter and nutrition for hundreds of diverse marine species.'

Determining whether the clauses are independent or dependent guides the punctuation required to join them.

2

Evaluate the grammatical rules for linking two independent clauses.

Two independent clauses must be joined by a period, a semicolon, a colon, or a comma paired with a coordinating conjunction (such as 'and', 'but', or 'so').

This narrows down the correct punctuation options and eliminates grammatically incorrect structures.

3

Select the choice that correctly links the clauses without creating grammatical or logical errors.

The choice containing a semicolon successfully separates the two independent clauses, whereas the comma alone creates a comma splice, the lack of punctuation creates a run-on, and the transition word 'although' creates an illogical relationship.

Ensures adherence to the conventions of Standard English.

Key Concept

Clause Boundaries and Linking

Estimated Time:1m 0s

Question 1276Question →

A researcher is preparing a growth medium for a bacterial culture by mixing Nutrient A and Nutrient B. Let $x$ represent the number of grams of Nutrient A and $y$ represent the number of grams of Nutrient B in the mixture. The mixture must satisfy the following conditions:

* The total mass of the nutrients in the mixture is at most $15$ grams.
* The mass of Nutrient B is at most twice the difference of the mass of Nutrient A and $2$ grams.
* The mass of Nutrient B is at least half the mass of Nutrient A.

What is the maximum possible integer number of grams of Nutrient B that can be used in the mixture?

5

6

8

9

Show answer & explanation

Answer: 8

Answer

The maximum possible integer number of grams of Nutrient B that can be used is 8.

The system of inequalities bounding the feasible region is

x + y \leq 15

,

y \leq 2x - 4

, and

y \geq \frac{1}{2}x

. The maximum value of

y

(representing Nutrient B) occurs at the intersection of the boundary lines

x + y = 15

and

y = 2x - 4

. Solving this system gives

x = 19/3

and

y = 26/3 \approx 8.67

. Since the researcher needs an integer number of grams, the largest integer value of

y

that falls within the region is

8

. Setting

y = 8

yields the valid range

6 \leq x \leq 7

, confirming that

y = 8

is a possible value.

Step-by-Step Solution

1

Translate the verbal constraints into a system of linear inequalities.

The system of inequalities is:
1)

x + y \leq 15

2)

y \leq 2(x - 2) \implies y \leq 2x - 4

3)

y \geq \frac{1}{2}x

Establishing the mathematical model is necessary to define the feasible region in the coordinate plane.

2

Find the intersection points of the boundary lines to locate the vertices of the feasible region.

Vertex 1 (Intersection of lines 1 and 2):

x + (2x - 4) = 15 \implies 3x = 19 \implies x = 19/3

, so

y = 26/3

.

Vertex 2 (Intersection of lines 1 and 3):

x + \frac{1}{2}x = 15 \implies \frac{3}{2}x = 15 \implies x = 10

, so

y = 5

.

Vertex 3 (Intersection of lines 2 and 3):

2x - 4 = \frac{1}{2}x \implies \frac{3}{2}x = 4 \implies x = 8/3

, so

y = 4/3

.

The maximum value of a coordinate in a bounded linear inequality system occurs at one of the vertices of the feasible region.

3

Identify the vertex that maximizes the

y

-coordinate.

Comparing the

y

-coordinates of the vertices:
-

y = 4/3 \approx 1.33

at

(8/3, 4/3)

-

y = 5

at

(10, 5)

-

y = 26/3 \approx 8.67

at

(19/3, 26/3)

The maximum value of

y

in the feasible region is

26/3 \approx 8.67

.

Comparing the coordinates determines the upper limit of the variable

y

in the shaded region.

4

Find the maximum possible integer value for

y

within the feasible region.

The maximum value is

8.67

. The largest integer less than or equal to

8.67

is

8

. Verify that

y = 8

is achievable by substituting it into the inequalities:

x + 8 \leq 15 \implies x \leq 7

8 \leq 2x - 4 \implies 2x \geq 12 \implies x \geq 6

8 \geq \frac{1}{2}x \implies x \leq 16

This yields the interval

6 \leq x \leq 7

, which contains valid solutions (such as the integer points

(6, 8)

and

(7, 8)

).

Ensures that the integer value chosen actually lies within the set of valid coordinates for the system.

Key Concept

Maximizing a variable under a system of linear constraints

Estimated Time:3m 0s

Question 1277Question →

In temperate forests, mycorrhizal networks formed by underground fungi connect the root systems of multiple distinct plants, facilitating the shared distribution of essential carbon and nitrogen. Research by ecologist Pamela Cabrera suggests that these subterranean pathways do not merely benefit isolated specimens; rather, they can significantly enhance the neighboring _______ survival rates during prolonged periods of severe environmental stress.

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

seedling

seedlings

seedling's

seedlings'

Show answer & explanation

Answer: seedlings'

Answer

seedlings'

The sentence requires a modifier for the noun phrase 'survival rates' to show whose survival rates are being enhanced. The context mentions 'multiple distinct plants' and 'isolated specimens,' which establishes that the reference is plural. Therefore, the plural possessive noun 'seedlings'' is correct.

Step-by-Step Solution

1

Analyze the grammatical function of the blank in the sentence.

The blank modifies 'survival rates,' a noun phrase, indicating that the target word must be in the possessive case to show association.

Determining the correct case ensures the sentence is structurally sound and conveys the intended relationship of ownership or association.

2

Determine the number (singular or plural) of the target noun from context clues.

The context references 'multiple distinct plants' and 'isolated specimens' in the plural, establishing that the reference must be plural.

Matching the number to existing plural nouns in the context maintains logical agreement across the passage.

3

Combine the possession and plurality requirements to identify the correct form.

The correct form is the plural possessive noun 'seedlings'.

This is the only form that satisfies both the plural context and the possessive modifier requirement.

Key Concept

Plural and Possessive Nouns

Question 1278Question →

In the $xy$ -plane, the system of linear equations below has infinitely many solutions, where $a$ , $b$ , and $c$ are constants and $c > 0$ :

\begin{aligned} 3x - 4y &= 10 \\ ax + by &= c \end{aligned}

If the graph of the second equation in the system passes through the point $(a, b)$ , what is the value of $c$ ?

Show answer & explanation

Answer: 4

Answer

The correct answer is 4.

Since the system has infinitely many solutions, the two equations are equivalent. This means the coefficients are proportional, so we can write

a = 3k

,

b = -4k

, and

c = 10k

for some constant

k

. Because the line passes through

(a, b)

, substituting these coordinates into the second equation gives

a^2 + b^2 = c

. Substituting the expressions in terms of

k

results in

(3k)^2 + (-4k)^2 = 10k

, which simplifies to

25k^2 = 10k

. Since

c > 0

, we have

k > 0

, and dividing by

k

gives

25k = 10

, so

k = 0.4

. Finally,

c = 10(0.4) = 4

.

Step-by-Step Solution

1

Set up a proportionality constant to relate the coefficients of the two equations.

a = 3k

,

b = -4k

, and

c = 10k

for a constant

k

.

Since the system has infinitely many solutions, the equations represent the same line, meaning their coefficients and constants must be proportional.

2

Substitute the point

(a, b)

into the second equation

ax + by = c

.

a^2 + b^2 = c

The graph of the equation passes through the point

(a, b)

, so the coordinates must satisfy the equation.

3

Substitute the parametric expressions of

a

,

b

, and

c

into the equation

a^2 + b^2 = c

.

25k^2 = 10k

This allows us to solve for the parameter

k

using a single variable quadratic equation.

4

Solve the equation

25k^2 = 10k

for

k

, given that

k > 0

.

k = 0.4

Since

c > 0

and

c = 10k

,

k

must be strictly positive, allowing us to divide both sides by

k

.

5

Compute the final value of

c

using the value of

k

.

c = 4

Substituting

k = 0.4

back into the expression

c = 10k

gives the value of

c

.

Key Concept

Systems of linear equations with infinitely many solutions and coordinate geometry constraints

Question 1279Question →

Which of the following is equivalent to the expression $\frac{(x^2)^3}{x^{-2}}$ for all positive values of $x$ ?

x^3

x^4

x^8

x^7

Show answer & explanation

Answer: $x^8$

Answer

The expression

x^8

is equivalent to the given expression.

The correct expression is

x^8

. First, simplify the numerator using the power of a power rule:

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

. Next, divide by the denominator using the quotient rule of exponents, which requires subtracting the exponent in the denominator from the exponent in the numerator:

x^6 / x^{-2} = x^{6 - (-2)} = x^{6 + 2} = x^8

.

Step-by-Step Solution

1

Apply the power of a power rule to the numerator:

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

.

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

To raise an exponential term to a power, multiply the exponents.

2

Apply the quotient rule for exponents:

\frac{x^a}{x^b} = x^{a - b}

.

\frac{x^6}{x^{-2}} = x^{6 - (-2)} = x^{6 + 2} = x^8

To divide powers with the same base, subtract the exponent in the denominator from the exponent in the numerator.

Key Concept

Simplifying algebraic expressions containing negative and positive exponents using the rules of exponents.

Question 1280Question →

If $\frac{2}{3}(6x - 9) - \frac{3}{4}(4x - 8) = \frac{1}{2}(x + 10)$ , what is the value of $x$ ?

Show answer & explanation

Answer: 10

Answer

The correct answer is 10.

Distributing the fractions on the left-hand side of the equation simplifies the expression to

x

. Setting this equal to the right-hand side yields the equation

x = \frac{1}{2}(x + 10)

. Multiplying both sides by 2 gives

2x = x + 10

, and subtracting

x

from both sides gives the correct value of 10.

Step-by-Step Solution

1

Distribute the fraction

\frac{2}{3}

to the terms inside the first parentheses,

(6x - 9)

, and distribute the fraction

-\frac{3}{4}

to the terms inside the second parentheses,

(4x - 8)

.

4x - 6 - 3x + 6 = \frac{1}{2}(x + 10)

Applying the distributive property removes the parentheses so that like terms can be combined.

2

Combine the variable terms and constant terms on the left side of the equation.

x = \frac{1}{2}(x + 10)

Combining

4x

and

-3x

yields

x

, while

-6

and

6

cancel each other out.

3

Multiply both sides of the equation by 2 to eliminate the fraction.

2x = x + 10

Multiplying by the denominator simplifies the equation by removing the fraction.

4

Subtract

x

from both sides of the equation to isolate the variable.

x = 10

Subtracting

x

isolates the variable

x

on the left side of the equation.

Key Concept

Solving linear equations in one variable by applying the distributive property and combining like terms.

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

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer