All practice questions

Question 1381Question →

For all $x > 1$ , which of the following is equivalent to the expression $\frac{x^2 - \sqrt{x}}{x - \sqrt{x}} - \sqrt{x}$ ?

x - 2\sqrt{x} + 1

x - \sqrt{x} - 1

x + 1

x - 1

Show answer & explanation

Answer: $x + 1$

Answer

The expression is equivalent to

x + 1

.

The expression can be simplified by substituting

u = \sqrt{x}

, which gives

x = u^2

and

x^2 = u^4

. Substituting these into the original expression yields

\frac{u^4 - u}{u^2 - u} - u

. Factoring out

u

from the numerator and denominator gives

\frac{u(u^3 - 1)}{u(u - 1)} - u = \frac{u^3 - 1}{u - 1} - u

. Factoring the difference of cubes in the numerator as

(u - 1)(u^2 + u + 1)

and canceling the common factor of

u - 1

leaves

u^2 + u + 1 - u = u^2 + 1

. Substituting back

x = u^2

yields the equivalent expression

x + 1

.

Step-by-Step Solution

1

Substitute

u = \sqrt{x}

into the expression, which implies

x = u^2

and

x^2 = u^4

.

The expression becomes

\frac{u^4 - u}{u^2 - u} - u

.

Using a substitution simplifies the fractional exponents and makes the polynomial structure easier to recognize.

2

Factor out

u

from both the numerator and the denominator of the fraction.

\frac{u(u^3 - 1)}{u(u - 1)} - u = \frac{u^3 - 1}{u - 1} - u

.

Since

x > 1

, we have

u > 1

, which means

u \neq 0

. Therefore, we can cancel the common factor

u

from the numerator and denominator.

3

Factor the difference of cubes in the numerator:

u^3 - 1 = (u - 1)(u^2 + u + 1)

.

\frac{(u - 1)(u^2 + u + 1)}{u - 1} - u

.

Factoring the numerator allows us to simplify the rational expression by canceling the common binomial factor in the denominator.

4

Cancel the common factor

u - 1

and simplify the remaining terms.

(u^2 + u + 1) - u = u^2 + 1

.

Since

u > 1

, we have

u - 1 \neq 0

, allowing us to divide out

u - 1

. Subtracting

u

from

u^2 + u + 1

leaves

u^2 + 1

.

5

Substitute

x

back in place of

u^2

.

x + 1

.

Converting the simplified expression back to the original variable gives the final equivalent algebraic expression.

Key Concept

Simplifying rational expressions with fractional exponents by substitution and factoring.

Question 1382Question →

Ancient Roman concrete structures have survived for millennia in harsh marine environments, baffling modern materials scientists who study infrastructure durability. Researchers recently discovered that this extraordinary longevity is due to a rare mineral called aluminous tobermorite ________ mineral crystallizes when seawater reacts with the volcanic ash in the concrete mixture, reinforcing the matrix over time.

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

, this

; this

this

whereas this

Show answer & explanation

Answer: ; this

Answer

The semicolon followed by the word 'this' is the correct choice because it properly joins the two independent clauses without creating a grammatical error.

The correct option is the one that contains a semicolon followed by 'this'. The clause preceding the blank and the clause succeeding the blank are both independent, meaning they could each stand alone as complete sentences. A semicolon is a grammatically standard way to link two independent clauses that are closely related in meaning.

Step-by-Step Solution

1

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

The clause before the blank ('Researchers recently discovered that this extraordinary longevity is due to a rare mineral called aluminous tobermorite') and the clause after the blank ('this mineral crystallizes when seawater reacts with the volcanic ash in the concrete mixture, reinforcing the matrix over time') are both independent clauses.

Identifying the grammatical nature of the clauses is necessary to determine how they can be linked.

2

Evaluate the grammar rules for linking two independent clauses.

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

This establishes the valid punctuation marks and linking words that conform to Standard English conventions.

3

Assess the options to find the grammatically correct link.

The option with a semicolon followed by 'this' correctly uses a semicolon to link two independent clauses. The option with a comma creates an invalid comma splice. The option with no punctuation creates an invalid run-on sentence. The option with 'whereas' creates an illogical relationship because the second clause supports the first clause rather than contrasting with it.

This determines which specific choice complies with the grammatical and logical requirements of the sentence.

Key Concept

Clause Boundaries and Linking

Question 1383Question →

For a constant $k$ , the inequality $\frac{k - 3x}{2} > \frac{5x + 3}{4}$ has exactly 4 positive integer solutions for $x$ . If $k$ is an integer, how many possible values of $k$ are there?

Show answer & explanation

Answer: 6

Answer

The number of possible integer values for

k

is 6.

Simplifying the inequality gives

x < \frac{2k - 3}{11}

. For this inequality to have exactly 4 positive integer solutions, the solution set must contain only the integers 1, 2, 3, and 4. This requires the boundary

\frac{2k - 3}{11}

to satisfy

4 < \frac{2k - 3}{11} \le 5

. Solving for

k

gives

23.5 < k \le 29

. The integers in this interval are 24, 25, 26, 27, 28, and 29, which total 6 values.

Step-by-Step Solution

1

Multiply both sides of the inequality by 4.

2(k - 3x) > 5x + 3

Clearing the denominators makes the inequality easier to solve.

2

Distribute the 2 on the left side.

2k - 6x > 5x + 3

Expanding terms allows us to group variables.

3

Add

6x

to both sides and subtract 3 from both sides.

11x < 2k - 3

Grouping

x

on one side and parameter terms on the other side.

4

Divide by 11.

x < \frac{2k - 3}{11}

Isolating

x

gives the boundary for the solution set.

5

Establish the range for the boundary term

\frac{2k - 3}{11}

.

4 < \frac{2k - 3}{11} \le 5

For the inequality to contain exactly the positive integers 1, 2, 3, and 4, the boundary must lie strictly above 4 and less than or equal to 5.

6

Solve the compound inequality for

k

.

23.5 < k \le 29

Performing algebraic operations on all parts of the inequality to isolate

k

.

7

Identify and count all integer solutions for

k

.

6 integers (24, 25, 26, 27, 28, 29)

Counting the integers in the range

(23.5, 29]

yields the final answer.

Key Concept

Solving linear inequalities in one variable with parameter constraints and identifying integer solution sets.

Estimated Time:3m 0s

Question 1384Question →

For all $x > 4$ , the expression $\frac{x^2 - 4x}{x - 2\sqrt{x}} + \frac{x\sqrt{x} + 8}{\sqrt{x} + 2}$ can be written in the form $ax + b$ , where $a$ and $b$ are constants. What is the value of $a + b$ ?

Show answer & explanation

Answer: 6

Answer

6

Factoring the numerator of the first term yields

x(\sqrt{x}-2)(\sqrt{x}+2)

and its denominator yields

\sqrt{x}(\sqrt{x}-2)

. Simplifying this term gives

x + 2\sqrt{x}

. Factoring the numerator of the second term as a sum of cubes gives

(\sqrt{x}+2)(x - 2\sqrt{x} + 4)

, which simplifies to

x - 2\sqrt{x} + 4

. Summing both simplified terms results in

2x + 4

. Matching this to the form

ax+b

gives

a=2

and

b=4

, so

a+b=6

.

Step-by-Step Solution

1

Simplify the first term of the expression.

\frac{x^2 - 4x}{x - 2\sqrt{x}} = x + 2\sqrt{x}

Factor

x

from the numerator to get

x(x-4)

and

\sqrt{x}

from the denominator to get

\sqrt{x}(\sqrt{x}-2)

. Rewrite

x-4

as the difference of squares

(\sqrt{x}-2)(\sqrt{x}+2)

, then cancel the common factor

\sqrt{x}-2

and simplify

\frac{x}{\sqrt{x}}

to

\sqrt{x}

.

2

Simplify the second term of the expression.

\frac{x\sqrt{x} + 8}{\sqrt{x} + 2} = x - 2\sqrt{x} + 4

Recognize

x\sqrt{x} + 8

as a sum of cubes,

(\sqrt{x})^3 + 2^3

. Factor it as

(\sqrt{x}+2)(x - 2\sqrt{x} + 4)

and cancel the common factor of

\sqrt{x}+2

in the denominator.

3

Add the simplified terms together.

2x + 4

Combine

(x + 2\sqrt{x})

and

(x - 2\sqrt{x} + 4)

by grouping like terms: the

2\sqrt{x}

and

-2\sqrt{x}

cancel out, leaving

2x + 4

.

4

Identify the values of

a

and

b

and find

a+b

.

6

Comparing

2x + 4

to

ax + b

gives

a = 2

and

b = 4

. Therefore,

a + b = 2 + 4 = 6

.

Key Concept

Simplifying rational expressions involving radicals by factoring (difference of squares and sum of cubes).

Question 1385Question →

The passage below discusses the tool-use behavior of New Caledonian crows. Which word completes the passage so that it conforms to the conventions of Standard English parallel structure?

Fill in the blanks below

Researchers studying the New Caledonian crow (*Corvus moneduloides*) have observed that these intelligent birds are highly skilled at crafting tools to retrieve food from hard-to-reach crevices in their environment. In laboratory settings, the crows demonstrate remarkable cognitive flexibility: they shave the barbed edges of pandanus leaves, shape the twigs of local trees, and metal wires into functional hooks.

Show answer & explanation

Answer

bend (or another grammatically parallel base verb like 'twist' or 'mold')

To maintain parallel structure in a list, all items must share the same grammatical form. The list consists of three actions performed by the crows: 'they shave...', 'shape...', and 'and [verb]...'. Since 'shave' and 'shape' are base/present-tense plural verbs, the blank must be completed by a parallel present-tense base verb, such as 'bend', 'twist', or 'mold'.

Step-by-Step Solution

1

Identify the structure of the sentence containing the blank. The sentence features a list of actions performed by the crows: 'they shave...', 'shape...', and 'and [blank]...'.

A parallel list structure is identified.

Understanding the sentence structure helps locate the elements that need to be parallel.

2

Determine the grammatical form of the existing list items ('shave' and 'shape').

Both 'shave' and 'shape' are base-form, present-tense plural verbs that agree with the subject 'they'.

The correct answer must match this grammatical form to maintain parallelism.

3

Choose a verb that completes the list in the same grammatical form.

A base verb such as 'bend', 'twist', or 'mold' is selected.

This completes the parallel series grammatically.

Key Concept

Parallel Structure

Question 1386Question →

A custom t-shirt printing company charges a one-time design setup fee plus a fixed price for each t-shirt printed. The total cost, $C$ , in dollars, for an order of $n$ t-shirts is given by the equation $C = 15n + 50$ . What does the number $50$ represent in this equation?

The one-time design setup fee, in dollars

The price, in dollars, for each printed t-shirt

The total cost, in dollars, for an order of 1 t-shirt

The number of t-shirts that are printed in an order that has no setup fee

Show answer & explanation

Answer: The one-time design setup fee, in dollars

Answer

The one-time design setup fee, in dollars

The constant term

50

in the linear equation

C = 15n + 50

represents the initial value of the cost function when no t-shirts are ordered (

n = 0

). In this context, this represents the one-time design setup fee.

Step-by-Step Solution

1

Identify the structure of the linear equation

C = 15n + 50

.

The equation is in the slope-intercept form

y = mx + b

, where

C

is the total cost,

n

is the number of t-shirts,

15

is the slope, and

50

is the y-intercept.

Analyzing the linear equation structure helps map its parts to the contextual scenario.

2

Interpret the constant term

50

within the context.

The constant term represents the value of

C

when

n = 0

, which is

C = 15(0) + 50 = 50

dollars.

When no t-shirts are printed, the only cost incurred is the initial flat setup fee.

Key Concept

Interpreting the y-intercept of a linear relationship in context

Question 1387Question →

For all $x > 1$ , which of the following is equivalent to the expression $\frac{4x^2 - 1}{2x - 1} - \frac{6x^2 + x - 2}{3x + 2}$ ?

0

2

-2

4x

Show answer & explanation

Answer: 2

Answer

2

Factoring the numerators allows the expression to be simplified. The first term becomes

2x + 1

, and the second term becomes

2x - 1

. Subtracting the second term from the first and distributing the negative sign results in

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

.

Step-by-Step Solution

1

Factor the numerator of the first rational term,

4x^2 - 1

, using the difference of squares identity.

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

To identify and divide out common factors between the numerator and denominator.

2

Simplify the first term by dividing the factored numerator by its denominator,

2x - 1

.

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

Since

x > 1

,

2x - 1 \neq 0

, allowing the division.

3

Factor the numerator of the second rational term,

6x^2 + x - 2

.

6x^2 + x - 2 = (2x - 1)(3x + 2)

To find common factors that can be simplified with the denominator.

4

Simplify the second term by dividing the factored numerator by its denominator,

3x + 2

.

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

Since

x > 1

,

3x + 2 \neq 0

, allowing the division.

5

Subtract the second simplified expression from the first, ensuring that the negative sign is correctly distributed to all terms.

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

To combine the terms and find the final equivalent value of the entire expression.

Key Concept

Equivalent Algebraic Expressions

Question 1388Question →

In the $xy$ -plane, a point with coordinates $(a, b)$ lies in the solution set of the system of inequalities below:

\begin{aligned} y &\ge -2x + 8 \\ y &< \frac{1}{2}x + 3 \end{aligned}

If $a$ and $b$ are integers, what is the minimum possible value of $a$ ?

1

2

3

4

Show answer & explanation

Answer: 3

Answer

3

The correct value is 3. Substituting

x = 3

into the system gives

y \ge 2

and

y < 4.5

. Since the coordinate

b

must be an integer,

b

can be 2, 3, or 4, showing that a valid integer coordinate pair exists. For any integer

x < 3

, there are no integer solutions for

y

.

Step-by-Step Solution

1

Find the intersection point of the boundary lines of the system of inequalities by setting their corresponding equations equal.

-2x + 8 = \frac{1}{2}x + 3 \implies 5 = 2.5x \implies x = 2

This determines the x-coordinate where the boundaries of the two inequality regions intersect.

2

Evaluate the system of inequalities at the intersection point

x = 2

.

y \ge -2(2) + 8 \implies y \ge 4

and

y < \frac{1}{2}(2) + 3 \implies y < 4

. This yields

4 \le y < 4

, which has no solution.

This checks if any valid solution exists at the boundary intersection point.

3

Test consecutive integer values of

x

greater than 2 to find the first value that allows at least one integer

y

.

Substituting

x = 3

gives

y \ge -2(3) + 8 \implies y \ge 2

and

y < \frac{1}{2}(3) + 3 \implies y < 4.5

. The integers that satisfy

2 \le y < 4.5

are

y = 2, 3,

and

4

.

Since

a = 3

is the smallest integer greater than 2 that yields integer solutions for

y

, it is the minimum possible value.

Key Concept

Identifying integer coordinate solutions in systems of linear inequalities

Estimated Time:2m 0s

Question 1389Question →

An oceanographic research vessel measures hydrostatic pressure during deep-sea dives. During a dive in temperate waters, the pressure $P$ , in atmospheres (atm), at a depth of $d$ meters below the surface is modeled by the linear equation $P = \rho d + 1.0$ , where $\rho$ is a constant representing the rate of change of pressure with respect to depth. At a depth of $150$ meters, the pressure is $16.0$ atm. During a second dive in a high-salinity region, the rate of change of pressure with respect to depth is $25\%$ greater than the rate in the temperate waters. According to the model for the second dive, at what depth, in meters, is the pressure $26.0$ atm?

Show answer & explanation

Answer: 200

Answer

200

To find the depth on the second dive where the pressure is

26.0

atm, we first find the slope of the linear model for the first dive. Given

P = \rho d + 1.0

, substituting the point

(150, 16.0)

gives

16.0 = 150\rho + 1.0

, which yields

\rho = 0.1

atm/m. The slope for the second dive is

25\%

greater, so the new slope is

0.1 \times 1.25 = 0.125

atm/m. The equation for the second dive is

P = 0.125d + 1.0

. Substituting

P = 26.0

yields

26.0 = 0.125d + 1.0

. Solving for

d

gives

25.0 = 0.125d

, which results in a depth of

200

meters.

Step-by-Step Solution

1

Find the rate of change of pressure with respect to depth in temperate waters.

\rho = 0.1

atm/m

Substitute the depth

d = 150

and pressure

P = 16.0

into the linear equation

P = \rho d + 1.0

to get

16.0 = 150\rho + 1.0

. Solving for

\rho

yields

15.0 = 150\rho

, which means

\rho = 0.1

.

2

Calculate the rate of change of pressure with respect to depth for the second dive.

\rho_{\text{new}} = 0.125

atm/m

The rate of change for the second dive is

25\%

greater than the rate of

0.1

atm/m from the first dive. Thus,

\rho_{\text{new}} = 0.1 \times 1.25 = 0.125

.

3

Calculate the depth where the pressure is

26.0

atm using the new rate of change.

d = 200

meters

Set up the equation for the second dive:

P = 0.125d + 1.0

. Substitute

P = 26.0

to get

26.0 = 0.125d + 1.0

. Subtracting

1.0

from both sides gives

25.0 = 0.125d

. Solving for

d

gives

d = \frac{25.0}{0.125} = 200

.

Key Concept

Interpreting and manipulating slope in a linear relationship context

Question 1390Question →

A scientist is monitoring the volume of liquid nitrogen in a storage tank. The volume $V$ , in liters, of liquid nitrogen remaining in the tank $t$ hours after a cooling system malfunction is modeled by the equation $V = 240 - 1.5t$ . After how many hours of malfunction will there be exactly $180$ liters of liquid nitrogen remaining in the tank?

Show answer & explanation

Answer: 40

Answer

The cooling system malfunction must last 40 hours for there to be exactly 180 liters of liquid nitrogen remaining in the tank.

To find the number of hours of malfunction until

180

liters of liquid nitrogen remain in the tank, substitute

180

for

V

in the given model equation, yielding

180 = 240 - 1.5t

. Subtracting

240

from both sides of the equation results in

-60 = -1.5t

. Dividing both sides by

-1.5

gives the final value of

t = 40

hours.

Step-by-Step Solution

1

Substitute the target volume of liquid nitrogen into the linear equation.

180 = 240 - 1.5t

The question asks for the time

t

when the remaining volume

V

is exactly

180

liters.

2

Isolate the term containing the variable by subtracting the initial constant volume from both sides.

-60 = -1.5t

Subtracting

240

from both sides of the equation begins the process of isolating the variable

t

.

3

Divide both sides of the equation by the rate coefficient to solve for time.

t = 40

Dividing by

-1.5

isolates the variable

t

and gives the solution in hours.

Key Concept

Interpreting values and solving equations in linear contexts

Estimated Time:45s

Question 1391Question →

Established in the late nineteenth century, the museum, which houses one of the largest collections of avian specimens in North America, recently completed the restoration of _______ historical library. This newly renovated space now offers visiting scholars unprecedented access to rare field guides, hand-colored illustrations, and original correspondence from early ornithologists.

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

its

their

it's

its'

Show answer & explanation

Answer: its

Answer

The singular possessive pronoun 'its' correctly completes the sentence by agreeing with the singular antecedent 'the museum'.

The singular possessive pronoun 'its' correctly agrees in number with the singular antecedent 'the museum'. The intervening nonessential relative clause contains plural nouns like 'collections' and 'specimens', but the pronoun must agree with the main subject of the clause, which is singular.

Step-by-Step Solution

1

Identify the antecedent of the blank pronoun by analyzing the sentence structure.

The main subject and antecedent is the singular noun phrase 'the museum'.

The relative clause 'which houses one of the largest collections of avian specimens in North America' is nonessential and contains plural nouns that do not act as the true antecedent.

2

Determine the grammatical role and case required for the blank.

The blank precedes the noun phrase 'historical library', requiring a possessive pronoun.

The sentence describes the library belonging to the museum, so a possessive form is necessary.

3

Select the singular possessive pronoun that correctly matches the antecedent.

The correct choice is 'its'.

The pronoun 'its' is the correct singular possessive form, while other options are either plural, contractions, or spelling errors.

Key Concept

Pronoun-Antecedent Agreement in Number

Alternative Method

To easily identify the correct pronoun, temporarily remove the intervening relative clause: 'the museum... recently completed the restoration of [its/their] historical library.' This highlights that the singular pronoun 'its' is required to refer back to the singular noun 'museum'.

Estimated Time:1m 0s

Question 1392Question →

In the early twentieth century, Swedish meteorologist Vilhelm Bjerknes proposed a revolutionary method ______ that shifted meteorology from a descriptive, observational science to a rigorous, predictive discipline. Bjerknes argued that because the atmosphere obeys the laws of classical physics, its future states could be calculated using thermodynamic and hydrodynamic equations.

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

—numerical weather prediction—

,numerical weather prediction—

—numerical weather prediction,

;numerical weather prediction;

Show answer & explanation

Answer: —numerical weather prediction—

Answer

—numerical weather prediction—

The correct choice uses a pair of em-dashes to set off the nonessential explanatory phrase 'numerical weather prediction'. In English grammar, nonessential elements that provide additional information can be enclosed by a matching pair of em-dashes.

Step-by-Step Solution

1

Identify the grammatical role of the phrase 'numerical weather prediction' within the sentence.

The phrase is an appositive that provides explanatory details about the 'revolutionary method'. Removing this phrase leaves the rest of the sentence grammatically complete and coherent.

This establishes that the phrase is a nonessential element.

2

Apply the punctuation rules for setting off nonessential elements.

Nonessential elements must be set off from the rest of the sentence using matching punctuation marks, such as a pair of commas or a pair of em-dashes.

Using asymmetrical punctuation (like a comma and a dash) or incorrect punctuation (like semicolons) violates standard English conventions.

3

Select the option that uses a matching pair of punctuation marks to enclose the nonessential phrase.

The option with a pair of em-dashes correctly sets off the phrase.

This conforms to standard conventions.

Key Concept

Punctuation of Nonessential Elements

Estimated Time:1m 0s

Question 1393Question →

A line has the equation $y = mx + k$ in the $xy$ -plane, where $m$ and $k$ are constants. If the line passes through the points $(2, p)$ and $(p, -4)$ , and has a $y$ -intercept of $(0, 3p)$ for some constant $p > 0$ , what is the slope, $m$ , of the line?

-4

4

-1

1

Show answer & explanation

Answer: $-4$

Answer

-4

The correct answer is

-4

. Substituting the

y

-intercept

(0, 3p)

into the slope-intercept form equation

y = mx + k

gives

k = 3p

. Substituting the point

(2, p)

into

y = mx + 3p

gives

p = 2m + 3p

, which simplifies to

m = -p

. Then, substituting the point

(p, -4)

into

y = mx + 3p

gives

-4 = mp + 3p

. Replacing

m

with

-p

results in

-4 = -p^2 + 3p

, which can be rearranged to the quadratic equation

p^2 - 3p - 4 = 0

. Factoring this equation yields

(p - 4)(p + 1) = 0

. Since

p > 0

, the only valid solution is

p = 4

. Substituting

p = 4

back into the expression for

m

gives the slope

m = -4

.

Step-by-Step Solution

1

Determine the y-intercept constant

k

in terms of

p

.

k = 3p

The equation of the line is

y = mx + k

. Since the

y

-intercept is given as

(0, 3p)

, substituting

x = 0

and

y = 3p

gives

k = 3p

.

2

Find the relationship between the slope

m

and the parameter

p

.

m = -p

Since the line passes through

(2, p)

, we substitute these coordinates into the equation

y = mx + 3p

to get

p = 2m + 3p

. Solving for

m

yields

2m = -2p

, or

m = -p

.

3

Substitute the point

(p, -4)

into the linear equation and solve the resulting quadratic equation for

p

.

p = 4

Substituting the coordinates of

(p, -4)

into

y = mx + 3p

gives

-4 = mp + 3p

. Substituting

m = -p

gives

-4 = -p^2 + 3p

, which simplifies to the quadratic equation

p^2 - 3p - 4 = 0

. Factoring this equation gives

(p - 4)(p + 1) = 0

. Since

p > 0

, we must have

p = 4

.

4

Calculate the slope

m

.

m = -4

Since

m = -p

and we found

p = 4

, the slope is

m = -4

.

Key Concept

Determining the slope of a line from given points and a parameter in a linear equation.

Alternative Method

Instead of finding the equation of the line first, the slope

m

can be expressed directly using the slope formula between the points

(2, p)

and

(p, -4)

:

m = \frac{-4 - p}{p - 2}

. Since the slope of the line connecting

(0, 3p)

and

(2, p)

is also

m = \frac{p - 3p}{2 - 0} = -p

, we can set these two expressions for

m

equal to each other:

-p = \frac{-4 - p}{p - 2}

. Multiplying both sides by

p - 2

yields

-p(p - 2) = -4 - p \implies -p^2 + 2p = -4 - p

. Rearranging the terms gives the quadratic equation

p^2 - 3p - 4 = 0

. Factoring gives

(p - 4)(p + 1) = 0

. Since

p > 0

, we find

p = 4

, and the slope is

m = -p = -4

.

Estimated Time:2m 0s

Question 1394Question →

A manufacturing company uses the equation below to model the total daily cost, $K$ , in thousands of dollars, of operating a factory when $x$ units are produced:

K = 0.08(x - 150) + 22

where $150 \le x \le 800$ . The company plans to transition to a new production setup that will increase the daily fixed operating cost by $\$3,000$ but will decrease the cost to produce each unit by $\$20$ . Which of the following equations best models the new total daily cost, $K_{\text{new}}$ , in thousands of dollars, to produce $x$ units under the new setup?

K_{\text{new}} = 0.06x + 13

K_{\text{new}} = 0.06x + 25

K_{\text{new}} = 0.08x + 25

K_{\text{new}} = 0.06x + 10

Show answer & explanation

Answer: $K_{\text{new}} = 0.06x + 13$

Answer

The equation that best models the new total daily cost is

K_{\text{new}} = 0.06x + 13

.

The correct equation is determined by first expressing the original cost equation in slope-intercept form:

K = 0.08x + 10

. This reveals that the original variable cost is

0.08

thousand dollars (

\$80

) per unit and the original fixed cost is

10

thousand dollars (

\$10,000

). Decreasing the variable cost by

\$20

(

0.02

thousand dollars) results in a new slope of

0.06

. Increasing the fixed cost by

\$3,000

(

3

thousand dollars) yields a new y-intercept of

13

. Thus, the new cost model is

K_{\text{new}} = 0.06x + 13

.

Step-by-Step Solution

1

Expand the original equation to identify the baseline fixed and variable costs.

K = 0.08x - 12 + 22 \implies K = 0.08x + 10

Converting the equation to slope-intercept form,

y = mx + b

, allows direct identification of the variable cost per unit (slope,

m

) and the fixed cost (y-intercept,

b

).

2

Convert the baseline parameters to their corresponding contextual values in dollars.

Variable cost:

0.08

thousand dollars =

\$80

per unit. Fixed cost:

10

thousand dollars =

\$10,000

daily.

Since the cost

K

is in thousands of dollars, a coefficient of

0.08

represents

0.08 \times 1,000 = \$80

per unit, and a constant of

10

represents

10 \times 1,000 = \$10,000

.

3

Apply the described operational changes to determine the new variable and fixed costs.

New variable cost:

\$80 - \$20 = \$60

per unit, or

0.06

thousand dollars. New fixed cost:

\$10,000 + \$3,000 = \$13,000

daily, or

13

thousand dollars.

The unit cost decreases by

\$20

, lowering it from

\$80

to

\$60

. The fixed cost increases by

\$3,000

, raising it from

\$10,000

to

\$13,000

.

4

Assemble the new linear equation using the updated parameters.

K_{\text{new}} = 0.06x + 13

Combining the new variable cost per unit (

0.06

) and the new fixed cost (

13

) yields the new daily cost equation.

Key Concept

Converting and modifying linear equations in a real-world context by identifying and altering slopes and y-intercepts.

Estimated Time:3m 0s

Question 1395Question →

For all $x \neq -2.5$ , the expression $\frac{6x^2 + 17x + 8}{2x + 5}$ is equivalent to $3x + 1 + \frac{a}{2x + 5}$ , where $a$ is a constant. What is the value of $a$ ?

Show answer & explanation

Answer: 3

Answer

The value of

a

is

3

.

Multiplying both sides of the equivalent relation by

2x + 5

yields

6x^2 + 17x + 8 = (3x + 1)(2x + 5) + a

. Expanding the right side gives

6x^2 + 17x + 5 + a

. Since the expressions are equivalent, the constant terms must be equal:

8 = 5 + a

, which simplifies to

a = 3

.

Step-by-Step Solution

1

Multiply both sides of the equation by

2x + 5

to clear the denominators.

6x^2 + 17x + 8 = (3x + 1)(2x + 5) + a

This clears the rational expressions so we can work with polynomials directly.

2

Expand the expression

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

using the distributive property.

6x^2 + 17x + 8 = 6x^2 + 15x + 2x + 5 + a

To write the right side as a polynomial in standard form.

3

Combine like terms on the right side of the equation.

6x^2 + 17x + 8 = 6x^2 + 17x + (5 + a)

To group coefficients of like powers of

x

for easy comparison.

4

Equate the constant terms from both sides of the equation to solve for

a

.

8 = 5 + a

, which gives

a = 3

.

For the two polynomial expressions to be equivalent for all values of

x

, their corresponding coefficients and constants must be equal.

Key Concept

Rewriting rational expressions by clearing denominators or polynomial long division to find equivalent expressions.

Question 1396Question →

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

y = 3x^2 - 5x + 4

y = 7x + k

If the system has exactly one real solution $(x, y)$ , what is the value of $k$ ?

8

-8

16

40

Show answer & explanation

Answer: $-8$

Answer

-8

To find the number of solutions for the system of equations, we set the equations equal to each other:

3x^2 - 5x + 4 = 7x + k

. Subtracting

7x

and

k

from both sides gives the standard quadratic equation

3x^2 - 12x + (4 - k) = 0

. For a quadratic equation to have exactly one real solution, its discriminant,

b^2 - 4ac

, must equal

0

. Substituting

a = 3

,

b = -12

, and

c = 4 - k

into the discriminant formula gives

(-12)^2 - 4(3)(4 - k) = 0

. Simplifying this equation yields

144 - 12(4 - k) = 0

, which simplifies to

144 - 48 + 12k = 0

, or

96 + 12k = 0

. Solving for

k

gives

k = -8

.

Step-by-Step Solution

1

Equate the two equations to eliminate

y

and form a single quadratic equation.

3x^2 - 5x + 4 = 7x + k \Rightarrow 3x^2 - 12x + (4 - k) = 0

To find the points of intersection of the system, we set the expressions for

y

equal to each other.

2

Identify the coefficients of the quadratic equation in standard form

ax^2 + bx + c = 0

.

a = 3

,

b = -12

, and

c = 4 - k

These coefficients are needed to calculate the discriminant of the quadratic equation.

3

Set the discriminant

\Delta = b^2 - 4ac

equal to zero and solve for

k

.

(-12)^2 - 4(3)(4 - k) = 0 \Rightarrow 144 - 12(4 - k) = 0 \Rightarrow 12(4 - k) = 144 \Rightarrow 4 - k = 12 \Rightarrow k = -8

A quadratic equation has exactly one real solution if and only if its discriminant is equal to zero.

Key Concept

Solving nonlinear systems of equations by setting them equal to each other and using the discriminant of the resulting quadratic equation to determine the number of solutions.

Estimated Time:1m 30s

Question 1397Question →

What are the solutions to the quadratic equation $x^2 - 4x - 12 = 0$ ?

x = -6

and

x = 2

x = 6

and

x = -2

x = 6

and

x = 2

x = -6

and

x = -2

Show answer & explanation

Answer: $x = 6$ and $x = -2$

Answer

The solutions are

x = 6

and

x = -2

.

The correct answer is

x = 6

and

x = -2

. Factoring the quadratic equation

x^2 - 4x - 12 = 0

involves finding two numbers whose product is

-12

and whose sum is

-4

. These numbers are

-6

and

2

. Rewriting the equation as

(x - 6)(x + 2) = 0

and setting each factor to zero yields the solutions

x = 6

and

x = -2

.

Step-by-Step Solution

1

Identify the quadratic equation to factor.

The equation is

x^2 - 4x - 12 = 0

.

To find the solutions, we need to rewrite the quadratic expression in factored form.

2

Find two numbers that multiply to

-12

and add up to

-4

.

The numbers are

-6

and

2

.

These numbers satisfy both conditions:

(-6) \times 2 = -12

and

-6 + 2 = -4

.

3

Write the quadratic equation in factored form and solve for

x

.

(x - 6)(x + 2) = 0

, which means

x - 6 = 0

or

x + 2 = 0

. Thus,

x = 6

or

x = -2

.

Applying the zero product property allows us to solve for the individual values of

x

.

Key Concept

Solving quadratic equations by factoring

Question 1398Question →

In the $xy$ -plane, the graph of the linear equation $ax + by = c$ , where $a$ , $b$ , and $c$ are constants, has the property that the value of $y$ decreases by $3$ for every increase of $5$ in the value of $x$ . If the graph of this equation passes through the point $(4, -2)$ and $a = 6$ , what is the value of $c$ ?

Show answer & explanation

Answer: 4

Answer

The correct answer is

4

.

The correct answer is

4

. The slope of the line described is

m = \frac{\Delta y}{\Delta x} = \frac{-3}{5} = -0.6

. Rearranging

ax + by = c

into slope-intercept form

y = -\frac{a}{b}x + \frac{c}{b}

shows that the slope is

-\frac{a}{b}

. Setting these equal gives

-\frac{a}{b} = -\frac{3}{5} \implies \frac{a}{b} = \frac{3}{5}

. Since

a = 6

, we find

b = 10

. Substituting

a = 6

,

b = 10

, and the coordinates of the point

(4, -2)

into the original equation yields

6(4) + 10(-2) = 4

.

Step-by-Step Solution

1

Determine the slope of the line from the described relationship between

x

and

y

.

The slope

m

of the line is

-\frac{3}{5}

.

The slope represents the change in

y

divided by the change in

x

. Since

y

decreases by

3

(

\Delta y = -3

) for every increase of

5

in

x

(

\Delta x = 5

), the slope is

m = \frac{\Delta y}{\Delta x} = -\frac{3}{5}

.

2

Express the slope of the line in terms of the coefficients from the standard form equation

ax + by = c

.

The slope of the line is

-\frac{a}{b}

.

Rewriting the equation

ax + by = c

in slope-intercept form gives

by = -ax + c \implies y = -\frac{a}{b}x + \frac{c}{b}

. The coefficient of

x

is the slope, so

m = -\frac{a}{b}

.

3

Equate the two expressions for the slope and solve for

b

using the given value of

a = 6

.

b = 10

Setting

-\frac{a}{b} = -\frac{3}{5}

gives

\frac{a}{b} = \frac{3}{5}

. Substituting

a = 6

yields

\frac{6}{b} = \frac{3}{5}

, which simplifies to

3b = 30

, so

b = 10

.

4

Substitute the point

(4, -2)

and the values of

a

and

b

into the equation

ax + by = c

to solve for

c

.

c = 4

Substituting

a = 6

,

b = 10

,

x = 4

, and

y = -2

into

ax + by = c

gives

6(4) + 10(-2) = c \implies 24 - 20 = c \implies c = 4

.

Key Concept

Linear Equations in Two Variables

Question 1399Question →

A bakery sells blueberry muffins and bran muffins. The total cost of $6$ blueberry muffins and $4$ bran muffins is $\$22.00$ . The total cost of $5$ blueberry muffins and $8$ bran muffins is $\$26.50$ . What is the cost, in dollars, of $1$ blueberry muffin?

Show answer & explanation

Answer: 2.5

Answer

The cost of one blueberry muffin is 2.5 dollars.

The correct answer is 2.5. By setting up the system of linear equations

6x + 4y = 22

and

5x + 8y = 26.5

, where

x

represents the cost of one blueberry muffin and

y

represents the cost of one bran muffin, we can solve for

x

. Multiplying the first equation by 2 gives

12x + 8y = 44

. Subtracting the second equation,

5x + 8y = 26.5

, from the modified first equation yields

7x = 17.5

. Dividing by 7 results in

x = 2.5

.

Step-by-Step Solution

1

Set up a system of equations.

6x + 4y = 22

and

5x + 8y = 26.5

Translate the context into mathematical statements where

x

is the price of a blueberry muffin and

y

is the price of a bran muffin.

2

Multiply the first equation by 2.

12x + 8y = 44

Align the coefficients of

y

to facilitate elimination.

3

Subtract the second equation from the modified first equation.

7x = 17.5

Eliminate the variable

y

to solve for

x

.

4

Divide by 7.

x = 2.5

Isolate the variable

x

to find the cost of a blueberry muffin.

Key Concept

Solving systems of two linear equations using elimination.

Alternative Method

Alternatively, solve for

y

in terms of

x

using the first equation:

y = 5.5 - 1.5x

. Substitute this expression into the second equation:

5x + 8(5.5 - 1.5x) = 26.5

, which simplifies to

5x + 44 - 12x = 26.5 \\Rightarrow -7x = -17.5 \\Rightarrow x = 2.5

.

Estimated Time:1m 30s

Question 1400Question →

A linear equation and a quadratic equation form a system, as shown.

y = x^2 + 5

y = 3x + 5

If the ordered pair $(x, y)$ is a solution to the system where $x$ is positive, what is the value of $x$ ?

Show answer & explanation

Answer: 3

Answer

The value of

x

is 3.

Equating the two expressions for

y

gives the equation

x^2 + 5 = 3x + 5

. Subtracting 5 from both sides results in

x^2 = 3x

. Subtracting

3x

from both sides gives the quadratic equation

x^2 - 3x = 0

, which can be factored as

x(x - 3) = 0

. This yields two solutions for

x

:

0

and

3

. Since the question specifies that

x

is positive, the correct value is 3.

Step-by-Step Solution

1

Equate the equations

x^2 + 5 = 3x + 5

Since both equations are solved for

y

, we can substitute the quadratic expression into the linear equation.

2

Simplify the equation

x^2 - 3x = 0

Subtracting 5 from both sides and then subtracting

3x

from both sides collects all terms on one side.

3

Factor and solve

x(x - 3) = 0

, so

x = 0

or

x = 3

Factoring out the greatest common factor,

x

, allows us to apply the zero product property to find the individual roots.

4

Apply the constraint

x = 3

The problem states that

x

must be positive, which excludes the solution

x = 0

.

Key Concept

Solving a system consisting of a linear equation and a quadratic equation by substitution.

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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