All practice questions

Question 1581Question →

Data collected by the Cassini spacecraft revealed that Enceladus, a moon of Saturn, harbors a subsurface liquid water ocean. Hydrothermal activity on the ocean floor spews silica nanoparticles and organic compounds into the ________ materials then erupt through cracks in the icy crust, forming a towering plume of vapor.

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

water, these

water these

water; these

water; since these

Show answer & explanation

Answer: water; these

Answer

water; these

The sentence contains two independent clauses: the first ends with the word 'water,' and the second begins with the word 'these.' To join two independent clauses without a coordinating conjunction (like 'and' or 'but'), a semicolon is required. The option 'water; these' correctly uses a semicolon to separate the two independent clauses.

Step-by-Step Solution

1

Identify the clause structure of the sentence containing the blank.

The sentence consists of two independent clauses: 'Hydrothermal activity on the ocean floor spews silica nanoparticles and organic compounds into the water' and 'these materials then erupt through cracks in the icy crust, forming a towering plume of vapor.'

Understanding the boundary between independent clauses is necessary to determine the correct punctuation and conjunctions needed to link them.

2

Evaluate the choices to find the correct method of linking two independent clauses.

Two independent clauses can be separated by a period, a semicolon, or a comma paired with a coordinating conjunction (FANBOYS). The option 'water; these' correctly uses a semicolon.

A semicolon is a grammatically valid way to link two closely related independent clauses without any conjunctions.

3

Verify why the other options are grammatically incorrect.

The option 'water, these' is a comma splice. The option 'water these' is a run-on. The option 'water; since these' uses a semicolon followed by a subordinating conjunction, which incorrectly tries to link an independent clause to a dependent clause.

Eliminating grammatically incorrect choices confirms the correct option.

Key Concept

Clause Boundaries and Linking

Estimated Time:1m 0s

Question 1582Question →

Which of the following is a solution to the equation $x^2 - 2x - 8 = 0$ ?

-4

2

4

-8

Show answer & explanation

Answer: 4

Answer

The correct solution is 4.

To find the solutions to the equation

x^2 - 2x - 8 = 0

, we factor the quadratic expression on the left side of the equation. We need two numbers that multiply to

-8

and add to

-2

. These numbers are

-4

and

2

. Thus, the equation can be rewritten in factored form as

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

. Setting each factor to zero yields the solutions

x = 4

and

x = -2

. Among the given options,

4

is the only correct solution.

Step-by-Step Solution

1

Factor the quadratic expression

x^2 - 2x - 8

.

The expression factors into

(x - 4)(x + 2)

.

We look for two numbers that multiply to the constant term

-8

and add to the linear coefficient

-2

. These numbers are

-4

and

2

.

2

Set each factor equal to zero to solve for

x

.

x - 4 = 0

or

x + 2 = 0

, which gives

x = 4

or

x = -2

.

By the zero product property, if the product of two factors is zero, then at least one of the factors must be zero.

3

Identify the solution that appears in the options.

x = 4

is a solution.

Checking the given options,

4

is listed while

-2

is not.

Key Concept

Solving quadratic equations by factoring

Question 1583Question →

Which punctuation mark should be placed in the blank to ensure the sentence conforms to the conventions of Standard English?

Fill in the blanks below

During the excavation of the ancient Roman port, archaeologists uncovered artifacts from various Mediterranean regions: transport amphorae from Carthage, modern-day Tunisia glass vessels from Alexandria, Egypt; and decorative mosaics from Antioch, Turkey.

Show answer & explanation

Answer

A semicolon (;) must be used to separate items in a list when those items already contain internal commas.

A semicolon (;) is the correct punctuation mark because the sentence presents a complex list where the items already contain internal commas. To prevent confusion and maintain parallel structure with the semicolon used later in the sentence, a semicolon must be used to separate the items.

Step-by-Step Solution

1

Identify the structure of the sentence and the elements being listed.

The sentence lists three types of archaeological artifacts: amphorae from Carthage, glass vessels from Alexandria, and mosaics from Antioch.

Determining the parts of the list helps establish the appropriate level of punctuation required.

2

Examine the internal punctuation of each list item.

Each individual item in the list contains a comma separating the city of origin from the modern-day country (e.g., 'Carthage, modern-day Tunisia').

Since the list items contain internal punctuation, standard commas cannot be used to separate the items themselves without creating confusion.

3

Apply the rule for complex lists with internal commas.

A semicolon must be inserted in the blank to separate the first item from the second item, matching the semicolon that separates the second item from the third.

Semicolons function as 'super-commas' to clearly divide complex items in a list.

Key Concept

Semicolons in Complex Lists

Question 1584Question →

If $27^{2x - 1} \cdot 9^{y + 2} = 3^{13}$ and $8^{x - y} = 2^{y - 9}$ , what is the value of $x + y$ ?

2

4

6

9

Show answer & explanation

Answer: 4

Answer

4

To solve the system, we convert all exponential terms to their prime bases. The first equation becomes

3^{6x - 3} \cdot 3^{2y + 4} = 3^{13}

, which simplifies to

3^{6x + 2y + 1} = 3^{13}

. Equating the exponents yields

6x + 2y + 1 = 13

, which simplifies to

3x + y = 6

. The second equation becomes

2^{3x - 3y} = 2^{y - 9}

. Equating these exponents yields

3x - 4y = -9

. Solving this system of equations gives

x = 1

and

y = 3

. The sum of these values is

1 + 3 = 4

.

Step-by-Step Solution

1

Convert the bases of the first equation to a common base of 3.

Since

27 = 3^3

and

9 = 3^2

, we can rewrite the equation

27^{2x - 1} \cdot 9^{y + 2} = 3^{13}

as

(3^3)^{2x - 1} \cdot (3^2)^{y + 2} = 3^{13}

. Applying the power of a power rule, this simplifies to

3^{6x - 3} \cdot 3^{2y + 4} = 3^{13}

. Using the product rule of exponents, we add the exponents to get

3^{6x + 2y + 1} = 3^{13}

.

This allows us to equate the exponents and form a linear equation.

2

Set the exponents equal to each other to obtain the first linear equation.

6x + 2y + 1 = 13 \implies 6x + 2y = 12 \implies 3x + y = 6

.

Since the bases are identical on both sides, their exponents must be equal.

3

Convert the bases of the second equation to a common base of 2 and equate their exponents.

Since

8 = 2^3

, rewrite

8^{x - y} = 2^{y - 9}

as

(2^3)^{x - y} = 2^{y - 9} \implies 2^{3x - 3y} = 2^{y - 9}

. Equating the exponents gives

3x - 3y = y - 9 \implies 3x - 4y = -9

.

This gives us a second linear equation to form a system of equations.

4

Solve the system of equations for the variables.

From the first equation, we have

3x = 6 - y

. Substituting this into the second equation gives

(6 - y) - 4y = -9 \implies 6 - 5y = -9 \implies -5y = -15 \implies y = 3

. Substituting

y = 3

back into the first equation yields

3x + 3 = 6 \implies 3x = 3 \implies x = 1

.

Solving the system of linear equations provides the individual values of the variables.

5

Calculate the sum of the variables.

x + y = 1 + 3 = 4

.

This answers the question asking for the sum of the variables.

Key Concept

Expressing exponential terms with a common base to form and solve a system of linear equations.

Question 1585Question →

A certain substance decays radioactively such that the mass of the substance, in grams, remaining after $t$ days is modeled by the function $M(t) = 80 \cdot 2^{-\frac{t}{5}}$ . After how many days will the mass of the substance be $10$ grams?

Show answer & explanation

Answer: 15

Answer

15

To find the number of days after which the mass of the substance is

10

grams, we substitute

10

for

M(t)

in the given function, yielding

10 = 80 \cdot 2^{-\frac{t}{5}}

. Dividing both sides by

80

gives

\frac{10}{80} = 2^{-\frac{t}{5}}

, which simplifies to

\frac{1}{8} = 2^{-\frac{t}{5}}

. Since

\frac{1}{8}

can be written as

2^{-3}

, the equation becomes

2^{-3} = 2^{-\frac{t}{5}}

. Because the bases are the same, we equate the exponents:

-3 = -\frac{t}{5}

. Multiplying both sides by

-5

gives

t = 15

.

Step-by-Step Solution

1

Set the mass

M(t)

equal to

10

in the given function.

10 = 80 \cdot 2^{-\frac{t}{5}}

We want to find the value of

t

when the remaining mass of the substance is

10

grams.

2

Divide both sides of the equation by

80

.

\frac{1}{8} = 2^{-\frac{t}{5}}

To isolate the exponential expression.

3

Express both sides of the equation with a common base of

2

.

2^{-3} = 2^{-\frac{t}{5}}

Since

8 = 2^3

, the fraction

\frac{1}{8}

can be written as

2^{-3}

. Having the same base on both sides allows us to equate the exponents.

4

Set the exponents equal to each other and solve for

t

.

t = 15

Because the bases are equal, the exponents must be equal:

-3 = -\frac{t}{5}

.

Key Concept

Solving exponential equations by finding a common base.

Question 1586Question →

An IT administrator uses the equation $T = 0.06r + 40$ to model the total time $T$ , in milliseconds, required to retrieve $r$ records from a cloud database. The constant term in the model represents the total setup time, which consists of network latency and database connection overhead. If the database connection overhead is 20% of the total setup time, which of the following is the best interpretation of the value 8 in this model?

The increase in database connection overhead, in milliseconds, for each additional record retrieved.

The network latency, in milliseconds, for retrieving any number of records.

The database connection overhead, in milliseconds, for retrieving any number of records.

The rate at which the total time increases, in milliseconds per record, for the first 8 records retrieved.

Show answer & explanation

Answer: The database connection overhead, in milliseconds, for retrieving any number of records.

Answer

The database connection overhead, in milliseconds, for retrieving any number of records.

In the linear model

T = 0.06r + 40

, the constant term of

40

represents the total setup time in milliseconds, which is independent of the number of records

r

retrieved. The database connection overhead is

20\%

of this total setup time, which equals

0.20 \times 40 = 8

milliseconds. Since it is part of the constant term, it represents a fixed overhead of

8

milliseconds for retrieving any number of records.

Step-by-Step Solution

1

Identify the constant term in the linear model

T = 0.06r + 40

.

The constant term (y-intercept) is

40

, which represents the total setup time of

40

milliseconds when

r = 0

records are retrieved.

In a linear relationship of the form

y = mx + b

, the constant term

b

represents the initial value or y-intercept.

2

Calculate the database connection overhead.

The database connection overhead is

20\%

of the

40

milliseconds total setup time, which is

0.20 \times 40 = 8

milliseconds.

The problem states that the database connection overhead is 20% of the total setup time.

3

Interpret the calculated value of 8 in the context of the model.

Since the

40

milliseconds is a constant term that does not depend on the number of records

r

, the

8

milliseconds represents a fixed database connection overhead for any number of records retrieved.

The constant term in a linear model represents a fixed value that does not scale with the independent variable.

Key Concept

Interpreting the constant term (y-intercept) and its components in a linear relationship context.

Question 1587Question →

A community theater sold adult tickets and student tickets for its opening weekend performances. On Friday, the theater sold $20$ adult tickets and $40$ student tickets for a total of $\$760$ . On Saturday, the theater sold $35$ adult tickets and $30$ student tickets for a total of $\$930$ . What is the price, in dollars, of one adult ticket?

10

16

18

30

Show answer & explanation

Answer: 18

Answer

The price of one adult ticket is 18 dollars.

The price of one adult ticket is

18

. By setting up the system of linear equations representing the total sales on Friday (

20x + 40y = 760

) and Saturday (

35x + 30y = 930

), where

x

is the adult ticket price and

y

is the student ticket price, we can simplify them to

x + 2y = 38

and

7x + 6y = 186

. Multiplying the first simplified equation by

3

gives

3x + 6y = 114

. Subtracting this equation from

7x + 6y = 186

eliminates

y

and yields

4x = 72

, which simplifies to

x = 18

.

Step-by-Step Solution

1

Define variables and write the system of linear equations based on the given information.

Let

x

represent the price of an adult ticket and

y

represent the price of a student ticket. The system of equations is:

20x + 40y = 760

35x + 30y = 930

Variables represent the unknown quantities, and the equations relate these quantities to the total sales on Friday and Saturday.

2

Simplify both equations by dividing by their greatest common divisors.

Divide the first equation by

20

:

x + 2y = 38

Divide the second equation by

5

:

7x + 6y = 186

Simplifying equations reduces the coefficient values and minimizes the chance of arithmetic errors in later steps.

3

Use the elimination method to solve for

x

. Multiply the first simplified equation by

3

to align the

y

-coefficients.

3(x + 2y) = 3(38) \Rightarrow 3x + 6y = 114

Matching the coefficients of

y

allows us to eliminate the variable

y

by subtracting the equations.

4

Subtract the equation obtained in Step 3 from the second simplified equation.

(7x + 6y) - (3x + 6y) = 186 - 114 \Rightarrow 4x = 72

Subtracting the equations eliminates

y

, leaving a single linear equation in terms of

x

.

5

Solve for

x

by dividing both sides of the equation by

4

.

x = 18

Isolating

x

provides the price of one adult ticket.

Key Concept

Solving a system of linear equations using substitution or elimination.

Alternative Method

Instead of using elimination on the simplified equations, you can use substitution. Express

x

in terms of

y

using the first simplified equation:

x = 38 - 2y

. Substitute this expression into the second equation:

7(38 - 2y) + 6y = 186

. Expanding this gives

266 - 14y + 6y = 186

, which simplifies to

266 - 8y = 186

. Subtracting

266

from both sides gives

-8y = -80

, which solves to

y = 10

. Finally, substitute

y = 10

back into

x = 38 - 2y

to find

x = 38 - 2(10) = 18

.

Estimated Time:1m 30s

Question 1588Question →

If $x > 0$ and $x^2 - 3x - 18 = 0$ , what is the value of $x + 2$ ?

5

6

8

-1

Show answer & explanation

Answer: 8

Answer

8

Factoring the quadratic equation

x^2 - 3x - 18 = 0

yields

(x - 6)(x + 3) = 0

. This gives two solutions:

x = 6

and

x = -3

. Since the condition specifies

x > 0

, we choose

x = 6

. The question asks for the value of

x + 2

, which is

6 + 2 = 8

.

Step-by-Step Solution

1

Factor the quadratic equation

x^2 - 3x - 18 = 0

by finding two numbers that multiply to

-18

and add to

-3

.

(x - 6)(x + 3) = 0

Factoring allows us to find the individual roots of the quadratic equation.

2

Solve for the possible values of

x

by setting each factor equal to zero.

x = 6

or

x = -3

If the product of two factors is zero, then at least one of the factors must equal zero.

3

Apply the given constraint that

x > 0

.

x = 6

The problem specifies that

x

must be a positive number, so the negative root

x = -3

must be discarded.

4

Substitute the value of

x

into the expression

x + 2

.

8

The question asks for the value of

x + 2

, so we evaluate

6 + 2

.

Key Concept

Solving quadratic equations by factoring and applying constraints on roots

Estimated Time:1m 0s

Question 1589Question →

For all $x > 1$ , the expression $\frac{2x^2 + 7x - 4}{x^2 - 1} \cdot \frac{x - 1}{2x - 1}$ is equivalent to $\frac{x+k}{x+1}$ , where $k$ is a constant. What is the value of $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of the constant

k

is 4.

Factoring the numerator

2x^2 + 7x - 4

yields

(2x - 1)(x + 4)

and factoring the denominator

x^2 - 1

yields

(x - 1)(x + 1)

. Substituting these factored forms into the given product gives

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

. Canceling the common factors

(2x - 1)

and

(x - 1)

simplifies the expression to

\frac{x + 4}{x + 1}

. Comparing this to

\frac{x + k}{x + 1}

shows that

k = 4

.

Step-by-Step Solution

1

Factor the quadratic expression in the numerator:

2x^2 + 7x - 4

.

(2x - 1)(x + 4)

Factoring the numerator helps identify common factors that can be simplified.

2

Factor the difference of squares in the denominator:

x^2 - 1

.

(x - 1)(x + 1)

Factoring the denominator helps identify common factors that can be simplified.

3

Multiply the rational expressions and cancel out the common factors.

\frac{x + 4}{x + 1}

Since

x > 1

, the terms

(2x - 1)

and

(x - 1)

are not equal to zero and can be canceled.

4

Compare the resulting expression with

\frac{x + k}{x + 1}

to find the value of

k

.

k = 4

By matching the numerators of the equivalent expressions,

x + 4 = x + k

, which gives

k = 4

.

Key Concept

Factoring and simplifying products of rational expressions

Question 1590Question →

\begin{aligned} y &= 3x^2 - 4x - 5 \\ y - 2x &= 4 \end{aligned}

If $(x_1, y_1)$ and $(x_2, y_2)$ are the solutions to the system of equations above, and $y_1 > y_2$ , what is the value of $x_1 + y_2$ ?

-3

-1

5

1

Show answer & explanation

Answer: 5

Answer

5

To find the solutions to the system of equations, substitute the expression for

y

from the second equation into the first equation. First, rewrite the second equation as

y = 2x + 4

. Substituting this into the first equation yields

2x + 4 = 3x^2 - 4x - 5

. Rearranging terms to set the equation to zero gives

3x^2 - 6x - 9 = 0

. Dividing the entire equation by 3 simplifies it to

x^2 - 2x - 3 = 0

. Factoring this quadratic equation gives

(x - 3)(x + 1) = 0

, which yields the solutions

x_1 = 3

and

x_2 = -1

. Next, find the corresponding

y

-coordinates by substituting these

x

-values back into the linear equation

y = 2x + 4

. For

x = 3

,

y_1 = 2(3) + 4 = 10

. For

x = -1

,

y_2 = 2(-1) + 4 = 2

. We are given that

y_1 > y_2

, which confirms that

(x_1, y_1) = (3, 10)

and

(x_2, y_2) = (-1, 2)

. Finally, calculate

x_1 + y_2

, which is

3 + 2 = 5

.

Step-by-Step Solution

1

Rewrite the linear equation to express

y

in terms of

x

.

y = 2x + 4

This allows for substitution into the quadratic equation.

2

Substitute

y = 2x + 4

into the quadratic equation and set to zero.

3x^2 - 6x - 9 = 0

To find the

x

-coordinates of the intersection points.

3

Simplify and factor the quadratic equation.

(x - 3)(x + 1) = 0

, giving

x = 3

and

x = -1

.

To solve for the

x

-values of the intersection points.

4

Substitute the

x

-values back into the linear equation to find the corresponding

y

-values.

For

x = 3

,

y = 10

. For

x = -1

,

y = 2

. The intersection points are

(3, 10)

and

(-1, 2)

.

To find the complete coordinates of the intersection points.

5

Identify

(x_1, y_1)

and

(x_2, y_2)

using the condition

y_1 > y_2

, and compute

x_1 + y_2

.

Since

10 > 2

,

y_1 = 10

(with

x_1 = 3

) and

y_2 = 2

(with

x_2 = -1

). Then,

x_1 + y_2 = 3 + 2 = 5

.

To compute the required target expression.

Key Concept

Solving a nonlinear system of equations by substituting a linear expression into a quadratic equation and solving the resulting quadratic equation.

Question 1591Question →

If $3^{2x - 1} \cdot 9^{y + 1} = 27^{x + y}$ , which of the following equations expresses $x$ in terms of $y$ ?

x = 1 - y

x = y + 1

x = -y

x = -y - 1

Show answer & explanation

Answer: $x = 1 - y$

Answer

The equation expressing

x

in terms of

y

is

x = 1 - y

.

The correct option is

x = 1 - y

. To solve the equation, each term is rewritten with base 3:

3^{2x-1} \cdot (3^2)^{y+1} = (3^3)^{x+y}

. Using the exponent rule

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

, this becomes

3^{2x-1} \cdot 3^{2y+2} = 3^{3x+3y}

. Using the rule

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

, the left side simplifies to

3^{2x+2y+1}

. Equating the exponents gives

2x+2y+1 = 3x+3y

. Subtracting

2x

and

2y

from both sides results in

1 = x+y

, which isolated for

x

gives

x = 1-y

.

Step-by-Step Solution

1

Rewrite all bases in the equation as powers of 3.

Since

9 = 3^2

and

27 = 3^3

, the equation becomes

3^{2x - 1} \cdot (3^2)^{y + 1} = (3^3)^{x + y}

.

To apply exponent rules and combine terms, all bases must be identical.

2

Apply the power of a power rule

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

to simplify the exponents.

3^{2x - 1} \cdot 3^{2y + 2} = 3^{3x + 3y}

Multiplying the outer exponent by the inner exponent simplifies the terms on both sides.

3

Apply the product of powers rule

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

to combine the terms on the left side of the equation.

3^{(2x - 1) + (2y + 2)} = 3^{3x + 3y}

, which simplifies to

3^{2x + 2y + 1} = 3^{3x + 3y}

.

Adding exponents when multiplying expressions with the same base allows the left side to be represented as a single exponential term.

4

Set the exponents equal to each other and solve for

x

in terms of

y

.

2x + 2y + 1 = 3x + 3y \implies 1 = x + y \implies x = 1 - y

If two exponential expressions with the same positive base (other than 1) are equal, their exponents must be equal.

Key Concept

Solving exponential equations by expressing terms with a common base and applying exponent laws.

Estimated Time:1m 30s

Question 1592Question →

In cognitive linguistics, the concept of 'embodied cognition' suggests that human language is shaped by our physical interactions with the environment. Abstract concepts are often conceptualized through metaphoric extensions of physical ______ we regularly use spatial terms like 'up' and 'down' to describe emotional states, linking physical orientation directly to subjective feelings.

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

experience; for example,

experience, for example,

experience for example

experience, but

Show answer & explanation

Answer: experience; for example,

Answer

experience; for example,

The sentence contains two independent clauses: the first ends with 'experience' and the second begins with 'we regularly use'. To connect two independent clauses, standard English conventions require a period, a semicolon, or a coordinating conjunction with a comma. The punctuation 'experience; for example,' correctly uses a semicolon to separate the clauses and a comma after the transitional phrase 'for example'.

Step-by-Step Solution

1

Identify the boundary between the two clauses in the sentence.

The first clause is 'Abstract concepts are often conceptualized through metaphoric extensions of physical experience' and the second clause is 'we regularly use spatial terms like 'up' and 'down' to describe emotional states, linking physical orientation directly to subjective feelings.'

Both clauses contain a subject and a verb and can stand alone as complete sentences, meaning they are independent clauses.

2

Determine the grammatical rules for linking two independent clauses.

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

Joining them with only a comma creates a comma splice, and joining them with no punctuation creates a run-on sentence.

3

Evaluate the logical relationship between the two clauses to select the appropriate connector.

The second clause provides a specific instance of the phenomenon described in the first clause, so a transition meaning 'for example' is logical, whereas a contrastive coordinator like 'but' is inappropriate.

A semicolon followed by the transitional phrase 'for example' and a comma correctly and logically joins the clauses.

Key Concept

Clause Boundaries and Linking

Estimated Time:1m 15s

Question 1593Question →

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

y + 2x \leq 12

x - 2y \leq 6

x \geq 2

What is the maximum possible value of $y$ ?

Show answer & explanation

Answer: 8

Answer

8

The system of inequalities bounds the solution set. The upper boundary is given by

y \leq -2x + 12

. Because the slope is negative, the maximum value of

y

on this boundary occurs at the smallest possible value of

x

. The constraint

x \geq 2

dictates that the minimum value of

x

is 2. Substituting

x = 2

into the boundary equation yields

y = -2(2) + 12 = 8

. Checking this coordinate against the third inequality,

2 - 2(8) = -14 \leq 6

, verifies that

(2, 8)

is a valid solution.

Step-by-Step Solution

1

Express the first two inequalities in terms of y.

y \leq -2x + 12

and

y \geq \frac{1}{2}x - 3

This helps identify the upper and lower boundaries of the solution region.

2

Determine the boundary line that limits the maximum values of y.

The upper boundary is the line

y = -2x + 12

.

Since the inequality is

y \leq -2x + 12

, any solution must lie on or below this line.

3

Find the maximum value of y on this boundary line given the constraint

x \geq 2

.

y \leq -2(2) + 12 = 8

Since the slope of the boundary line is negative, y is maximized when x is at its minimum value, which is 2.

4

Verify that the point

(2, 8)

satisfies the inequality

x - 2y \leq 6

.

2 - 2(8) = -14 \leq 6

, which is true.

This confirms that the point

(2, 8)

is indeed in the solution set of the system.

Key Concept

Maximizing a coordinate value subject to a system of linear inequalities in two variables

Alternative Method

Instead of graphing or checking boundaries, we can algebraically solve for the boundary. Since

x \geq 2

, multiplying by

-2

and reversing the inequality gives

-2x \leq -4

. Adding 12 to both sides gives

-2x + 12 \leq 8

. Since

y \leq -2x + 12

, we get

y \leq 8

. Checking if

y = 8

and

x = 2

satisfies the second inequality

x - 2y \leq 6

confirms that

2 - 16 = -14 \leq 6

is true, meaning

y = 8

is indeed a valid solution and thus the maximum.

Estimated Time:1m 30s

Question 1594Question →

A shipping container has a maximum weight capacity of $24,150$ kilograms. The container is loaded with $12$ machinery units, each weighing $1,150$ kilograms. The remaining space will be filled with packing crates, each weighing $180$ kilograms. A safety regulation requires that a clearance weight of at least $15\%$ of the total loaded weight (the combined weight of the machinery units and the packing crates) must be left unused. What is the maximum number of packing crates that can be loaded into the container?

Show answer & explanation

Answer: 40

Answer

The maximum number of packing crates that can be loaded is 40.

By setting up the inequality representing the physical constraints, we find that the number of packing crates,

x

, must satisfy

x \le 40

. Since the question asks for the maximum number of packing crates, the maximum value is 40.

Step-by-Step Solution

1

Calculate the constant weight of the machinery units.

The total weight of the

12

machinery units is

12 \times 1,150 = 13,800

kilograms.

This establishes the base weight that is already loaded in the container.

2

Define the variable and write the expression for the total loaded weight.

Let

x

be the number of packing crates. The total loaded weight is

13,800 + 180x

kilograms.

This represents the combined weight of the machinery and the crates in terms of the variable

x

.

3

Set up the inequality representing the safety clearance requirement.

24,150 - (13,800 + 180x) \ge 0.15(13,800 + 180x)

The unused weight capacity (maximum capacity minus loaded weight) must be at least

15\%

of the loaded weight.

4

Solve the inequality for

x

.

24,150 \ge 1.15(13,800 + 180x) \implies 21,000 \ge 13,800 + 180x \implies 7,200 \ge 180x \implies x \le 40

.

Isolating

x

gives the range of allowable values for the number of packing crates.

Key Concept

Formulating and solving linear inequalities in one variable based on real-world constraints.

Question 1595Question →

If $(x - 4)^2 = 81$ and $x > 0$ , what is the value of $x$ ?

Show answer & explanation

Answer: 13

Answer

The value of

x

is 13.

Taking the square root of both sides of

(x - 4)^2 = 81

yields

x - 4 = 9

or

x - 4 = -9

. Solving these linear equations gives

x = 13

or

x = -5

. Since

x > 0

is specified, the correct value of

x

is 13.

Step-by-Step Solution

1

Take the square root of both sides of the equation.

x - 4 = 9

or

x - 4 = -9

Applying the square root property to solve the quadratic equation.

2

Solve each linear equation for

x

.

x = 13

or

x = -5

Adding 4 to both sides of each equation.

3

Apply the given constraint that

x > 0

.

x = 13

Since

-5

is not greater than 0, the only positive solution is 13.

Key Concept

Solving quadratic equations by taking square roots

Question 1596Question →

In the $xy$ -plane, the graph of the linear function $f$ passes through the points $(3, 11)$ and $(7, 23)$ . The function $g$ is defined by $g(x) = f(2x) - 5$ . What is the value of $g(4)$ ?

Show answer & explanation

Answer: 21

Answer

21

The correct answer is 21. First, find the slope of the linear function

f

using the given points

(3, 11)

and

(7, 23)

:

m = \frac{23 - 11}{7 - 3} = 3

. The equation of the line is

f(x) - 11 = 3(x - 3)

, which simplifies to

f(x) = 3x + 2

. To find

g(4)

, substitute

x = 4

into the definition of

g(x)

:

g(4) = f(2 \cdot 4) - 5 = f(8) - 5

. Evaluating

f(8)

gives

3(8) + 2 = 26

. Finally, subtracting

5

gives

g(4) = 26 - 5 = 21

.

Step-by-Step Solution

1

Determine the equation of the linear function

f(x)

f(x) = 3x + 2

First find the slope

m = \frac{23 - 11}{7 - 3} = 3

. Then, use the point-slope formula with

(3, 11)

to get

f(x) - 11 = 3(x - 3)

, which simplifies to

f(x) = 3x + 2

.

2

Express

g(4)

in terms of

f

g(4) = f(8) - 5

Substitute

x = 4

into the definition

g(x) = f(2x) - 5

to get

g(4) = f(2(4)) - 5

.

3

Calculate the value of

f(8)

and

g(4)

g(4) = 21

Evaluate

f(8) = 3(8) + 2 = 26

, then subtract

5

to obtain

g(4) = 26 - 5 = 21

.

Key Concept

Linear Functions and Graphs

Question 1597Question →

A scientist is monitoring the population of two species of bacteria, Species A and Species B, in a controlled environment. Initially, the combined population of the two species is $12,000$ . Over the next week, the population of Species A doubles, and the population of Species B triples. If the total combined population of both species is $29,000$ at the end of the week, what was the initial population of Species A?

Show answer & explanation

Answer: 7000

Answer

The initial population of Species A was 7000.

To find the initial population of Species A, we set up a system of linear equations representing the total initial population and the population after one week. Let

x

represent the initial population of Species A and

y

represent the initial population of Species B. The initial combined population is given by

x + y = 12,000

. After one week, the population of Species A doubles (

2x

) and Species B triples (

3y

), so the new combined population is

2x + 3y = 29,000

. We can express

y

in terms of

x

as

y = 12,000 - x

. Substituting this expression into the second equation gives

2x + 3(12,000 - x) = 29,000

. Distributing and simplifying yields

-x + 36,000 = 29,000

, which simplifies to

x = 7,000

. Therefore, the initial population of Species A was 7,000.

Step-by-Step Solution

1

Define the variables and set up the first equation.

x + y = 12,000

Let

x

be the initial population of Species A and

y

be the initial population of Species B. Their combined initial population is

12,000

.

2

Set up the second equation based on the growth after one week.

2x + 3y = 29,000

The population of Species A doubles to

2x

and Species B triples to

3y

, summing to a total of

29,000

.

3

Solve for

x

using substitution.

x = 7,000

From the first equation,

y = 12,000 - x

. Substituting this into the second equation gives

2x + 3(12,000 - x) = 29,000

. Distributing the

3

yields

2x + 36,000 - 3x = 29,000

. Combining like terms results in

-x + 36,000 = 29,000

. Subtracting

36,000

from both sides gives

-x = -7,000

, which simplifies to

x = 7,000

.

Key Concept

Solving systems of linear equations using substitution or elimination methods

Question 15980% of students answered this correctlyQuestion →

For all $x > 3$ , which of the following is equivalent to the expression $\frac{2x^2 - 5x - 3}{x^2 - 9} - \frac{x}{x+3}$ ?

\frac{x + 1}{x + 3}

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

\frac{3x + 1}{x + 3}

\frac{2x^2 - 6x - 3}{x^2 - 9}

Show answer & explanation

Answer: $\frac{x + 1}{x + 3}$

Answer

\frac{x + 1}{x + 3}

To simplify the expression, we factor the first term:

\frac{2x^2 - 5x - 3}{x^2 - 9} = \frac{(2x + 1)(x - 3)}{(x - 3)(x + 3)}

. Canceling the common factor of

x - 3

leaves

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

. Subtracting the second term gives

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

. This matches the correct expression.

Step-by-Step Solution

1

Factor the numerator and denominator of the first rational expression.

The numerator

2x^2 - 5x - 3

factors into

(2x + 1)(x - 3)

. The denominator

x^2 - 9

is a difference of squares and factors into

(x - 3)(x + 3)

.

Factoring allows for the simplification of the rational expression by identifying common factors in the numerator and denominator.

2

Simplify the first rational expression by canceling the common factor.

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

for all

x > 3

.

Since

x > 3

, the term

x - 3

is non-zero, allowing us to divide both the numerator and denominator by

x - 3

.

3

Subtract the second expression from the simplified first expression.

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

Since both fractions now have the same denominator,

x + 3

, we subtract their numerators.

Key Concept

Simplifying rational expressions by factoring polynomials and performing operations on fractions with common denominators.

Estimated Time:1m 30s

Question 1599Question →

A scientist models the population of a bacteria culture using the function $P(t) = P_0 \cdot b^t$ , where $P(t)$ is the population $t$ hours after the start of the experiment, $P_0$ is the initial population, and $b$ is a constant. The table below shows the population at two different times:

$t$ (hours)	$P(t)$
$2$	$1,800$
$5$	$48,600$

If the population of the bacteria culture is $1,312,200$ after $k$ hours, what is the value of $k$ ?

4

8

9

14

Show answer & explanation

Answer: 8

Answer

8

The correct value is 8. By setting up the ratio of the population at

t = 5

to

t = 2

, we find that

\frac{P(5)}{P(2)} = b^3 = \frac{48,600}{1,800} = 27

, which yields an hourly growth factor of

b = 3

. Using

P(2) = P_0 \cdot 3^2 = 1,800

, we determine the initial population

P_0

is 200. To find the hour

k

when the population is

1,312,200

, we solve

200 \cdot 3^k = 1,312,200

, which simplifies to

3^k = 6,561

. Since

3^8 = 6,561

, we find

k = 8

. Alternatively, we can calculate

\frac{P(k)}{P(5)} = \frac{1,312,200}{48,600} = 27 = 3^3

, meaning the population triples 3 more times after

t = 5

, giving

k = 5 + 3 = 8

.

Step-by-Step Solution

1

Set up the ratio of the population at

t = 5

to the population at

t = 2

to find the growth factor

b

.

\frac{P(5)}{P(2)} = \frac{P_0 \cdot b^5}{P_0 \cdot b^2} = b^3 = \frac{48,600}{1,800} = 27

Dividing the function values eliminates the initial population

P_0

and isolates the base

b

.

2

Solve for the growth factor

b

.

b = 3

Since

b^3 = 27

, taking the cube root of both sides gives

b = 3

.

3

Find the initial population

P_0

using

P(2) = 1,800

and

b = 3

.

1,800 = P_0 \cdot 3^2 \Rightarrow 1,800 = 9P_0 \Rightarrow P_0 = 200

Substituting the known values into the function formula allows us to solve for

P_0

.

4

Set up the equation for the population after

k

hours and solve for

k

.

1,312,200 = 200 \cdot 3^k \Rightarrow 3^k = \frac{1,312,200}{200} = 6,561

Dividing both sides by 200 isolates the exponential term

3^k

.

5

Find the exponent

k

by expressing 6,561 as a power of 3.

3^k = 3^8 \Rightarrow k = 8

Since

3^8 = 6,561

, the exponents must be equal, so

k = 8

.

Key Concept

Determining parameters of an exponential growth function from given data points and using the function to solve for an unknown time variable.

Alternative Method

Instead of solving for the initial population

P_0

, we can compare the target population of

1,312,200

to the population at

t = 5

. Since the population triples every hour (

b = 3

), the ratio of the population at

t = k

to the population at

t = 5

is

3^{k-5}

. We have

\frac{1,312,200}{48,600} = 27

, which is

3^3

. Therefore,

3^{k-5} = 3^3

, which directly gives

k - 5 = 3

, or

k = 8

.

Estimated Time:1m 15s

Question 1600Question →

An artist creates $x$ small sculptures and $y$ large sculptures. Each small sculpture requires 3 hours of crafting and 2 hours of painting. Each large sculpture requires 7 hours of crafting and 3 hours of painting. The artist can spend at most 120 hours on crafting and at most 50 hours on painting. Which of the following pairs of small and large sculptures can the artist create under these constraints?

5 small sculptures and 15 large sculptures

12 small sculptures and 9 large sculptures

10 small sculptures and 8 large sculptures

15 small sculptures and 12 large sculptures

Show answer & explanation

Answer: 10 small sculptures and 8 large sculptures

Answer

10 small sculptures and 8 large sculptures

The option featuring 10 small sculptures and 8 large sculptures is correct because substituting

x = 10

and

y = 8

satisfies both linear inequalities representing the constraints. Specifically, the crafting time of 86 hours is less than or equal to the maximum allowed 120 hours (

3(10) + 7(8) = 86 \leq 120

), and the painting time of 44 hours is less than or equal to the maximum allowed 50 hours (

2(10) + 3(8) = 44 \leq 50

).

Step-by-Step Solution

1

Set up the system of inequalities representing the constraints.

The crafting constraint is

3x + 7y \leq 120

. The painting constraint is

2x + 3y \leq 50

. The variables must be non-negative:

x \geq 0

and

y \geq 0

.

To model the limits on crafting hours and painting hours mathematically.

2

Substitute the values of each option into the system to verify which one satisfies both inequalities.

For the option with 10 small sculptures and 8 large sculptures (

x = 10, y = 8

):

3(10) + 7(8) = 30 + 56 = 86 \leq 120

(True)

2(10) + 3(8) = 20 + 24 = 44 \leq 50

(True)

Only a coordinate pair that makes both inequality statements true is a valid solution.

Key Concept

Systems of Linear Inequalities in Two Variables

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method