All practice questions

Question 1741Question →

A parabola defined by the equation $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, has its vertex in the second quadrant of the coordinate plane. If the parabola passes through the point $(0,0)$ , which of the following inequalities must be true?

ab > 0

ab < 0

a > 0

b > 0

Show answer & explanation

Answer: $ab > 0$

Answer

The inequality

ab > 0

must be true.

Since the parabola passes through the point

(0,0)

and its vertex

(h, k)

lies in the second quadrant where

k > 0

, the vertex must represent a maximum value. Therefore, the parabola opens downward, so the leading coefficient

a

is negative. The x-coordinate of the vertex,

h

, is also negative because it lies in the second quadrant. The vertex x-coordinate is defined by

h = -\frac{b}{2a}

, which means

b = -2ah

. Because both

a

and

h

are negative, their product

ah

is positive, which makes

b

negative when multiplied by

-2

. Finally, since both

a

and

b

are negative, their product

ab

must be positive.

Step-by-Step Solution

1

Determine the signs of the vertex coordinates from its quadrant.

The vertex

(h, k)

lies in the second quadrant, which means

h < 0

and

k > 0

.

Points in the second quadrant have negative x-coordinates and positive y-coordinates.

2

Determine the value of the constant

c

.

c = 0

Since the parabola passes through

(0,0)

, substituting

x = 0

must result in

y = 0

.

3

Determine the sign of the leading coefficient

a

.

a < 0

The vertex is at

(h, k)

with

k > 0

, and the graph passes through

(0, 0)

where the y-value is

0

. Since the maximum value must be at least the value at any other point, the vertex

(h,k)

is a maximum. A parabola with a maximum opens downward, so

a < 0

.

4

Determine the sign of the coefficient

b

using the vertex formula.

b < 0

The x-coordinate of the vertex is

h = -\frac{b}{2a}

, which can be rewritten as

b = -2ah

. Since

a < 0

and

h < 0

, their product

ah

is positive. Multiplying this positive product by

-2

yields a negative value for

b

.

5

Find the sign of the product

ab

.

ab > 0

Since both

a

and

b

are negative, their product

ab

must be positive.

Key Concept

Analyzing quadratic coefficients and vertex properties in the coordinate plane.

Question 1742Question →

A chemist cools a liquid compound in a laboratory refrigerator. The temperature of the liquid decreases at a constant rate. After $4$ minutes of cooling, the temperature of the liquid is $22^\circ\text{C}$ . After $10$ minutes of cooling, the temperature of the liquid is $13^\circ\text{C}$ . Which of the following equations represents the relationship between the temperature, $T$ , in degrees Celsius, of the liquid and the number of minutes, $m$ , it has been cooling?

2m + 3T = 74

T = 4m + 22

3m + 2T = 56

2T - 3m = 56

Show answer & explanation

Answer: $3m + 2T = 56$

Answer

The equation

3m + 2T = 56

correctly represents the relationship between the temperature

T

and the cooling time

m

.

The relationship between the temperature

T

and the cooling time

m

is linear, which can be represented by the equation

T = -1.5m + 28

. Rearranging this equation by adding

1.5m

to both sides yields

1.5m + T = 28

. Multiplying the entire equation by

2

to clear the decimal gives

3m + 2T = 56

, which is the correct linear relationship.

Step-by-Step Solution

1

Calculate the rate of temperature change (slope,

k

) per minute of cooling.

k = \frac{13 - 22}{10 - 4} = \frac{-9}{6} = -1.5^\circ\text{C}

per minute.

A linear relationship has a constant rate of change, which is found by dividing the change in temperature by the change in time.

2

Use the point-slope form to write the equation of the line.

T - 22 = -1.5(m - 4)

, which simplifies to

T = -1.5m + 28

.

This models the relationship between

T

and

m

by using one of the given data points and the calculated slope.

3

Rearrange the equation into standard form with integer coefficients.

1.5m + T = 28 \implies 3m + 2T = 56

.

The options are written in standard form, so adding

1.5m

to both sides and multiplying the entire equation by

2

matches the target format.

Key Concept

Linear Equations in Two Variables

Question 1743Question →

The graph of the function $f$ in the $xy$ -plane is translated $4$ units to the right and $3$ units down to create the graph of the function $g$ . Which of the following equations defines $g(x)$ in terms of $f(x)$ ?

g(x) = f(x + 4) - 3

g(x) = f(x - 4) + 3

g(x) = f(x - 4) - 3

g(x) = f(x + 4) + 3

Show answer & explanation

Answer: $g(x) = f(x - 4) - 3$

Answer

The equation that defines

g(x)

in terms of

f(x)

is

g(x) = f(x - 4) - 3

.

The equation

g(x) = f(x - 4) - 3

correctly represents the translated function. A translation of

4

units to the right is represented by replacing

x

with

x - 4

within the function's input, and a translation of

3

units down is represented by subtracting

3

from the outer function.

Step-by-Step Solution

1

Apply the horizontal translation to the function input.

f(x - 4)

Translating a function

f(x)

horizontally to the right by

h

units is represented by substituting

x

with

x - h

. For a shift of

4

units to the right, we replace

x

with

x - 4

.

2

Apply the vertical translation to the function output.

f(x - 4) - 3

Translating a function vertically down by

k

units is represented by subtracting

k

from the entire function. For a shift of

3

units down, we subtract

3

.

3

Write the final equation for

g(x)

.

g(x) = f(x - 4) - 3

Combining the horizontal and vertical transformations yields the function

g(x)

.

Key Concept

Graph transformations of functions

Question 1744Question →

A linear function $f$ is defined such that its graph passes through the points $(2, 5)$ and $(4, 11)$ . The graph of another linear function, $g$ , is perpendicular to the graph of $f$ . If the graph of $g$ passes through the point $(3, 10)$ , what is the $x$ -coordinate of the $x$ -intercept of the graph of $g$ ?

Show answer & explanation

Answer: 33

Answer

The correct answer is

33

.

The slope of the linear function

f

is found using the two given points:

\frac{11 - 5}{4 - 2} = 3

. Because the graph of the linear function

g

is perpendicular to the graph of

f

, its slope is the negative reciprocal of

3

, which is

-\frac{1}{3}

. Using the point-slope form with the point

(3, 10)

, the equation of

g

is

y - 10 = -\frac{1}{3}(x - 3)

, which simplifies to

y = -\frac{1}{3}x + 11

. The

x

-intercept is the value of

x

when

y = 0

. Setting

y

to

0

gives

0 = -\frac{1}{3}x + 11

, which yields

x = 33

.

Step-by-Step Solution

1

Calculate the slope of the linear function

f

The slope of

f

is

3

.

The slope 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, 11)

yields

\frac{11 - 5}{4 - 2} = \frac{6}{2} = 3

.

2

Determine the slope of the perpendicular linear function

g

The slope of

g

is

-\frac{1}{3}

.

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other. The negative reciprocal of

3

is

-\frac{1}{3}

.

3

Write the equation representing function

g

y = -\frac{1}{3}x + 11

Using the point-slope equation

y - y_1 = m(x - x_1)

with the slope

m = -\frac{1}{3}

and the point

(3, 10)

, we get

y - 10 = -\frac{1}{3}(x - 3)

. Distributing and simplifying yields

y - 10 = -\frac{1}{3}x + 1 \implies y = -\frac{1}{3}x + 11

.

4

Find the

x

-intercept of the graph of

g

33

The

x

-intercept is the point on the graph where

y = 0

. Substituting

0

for

y

in the equation of

g

gives

0 = -\frac{1}{3}x + 11 \implies \frac{1}{3}x = 11 \implies x = 33

.

Key Concept

The relationship between the slopes of perpendicular lines and finding intercepts of linear functions.

Estimated Time:1m 30s

Question 1745Question →

Deep-sea hydrothermal vents support unique ecosystems that thrive in the absence of sunlight. In these extreme environments, chemosynthetic bacteria convert chemicals into energy, sustaining a variety of specialized organisms. Among the most prominent are giant tube worms, whose survival depends on a symbiotic relationship with bacteria housed inside each ______ body. Which choice completes the text so that it conforms to the conventions of Standard English?

worm

worms

worm's

worms'

Show answer & explanation

Answer: worm's

Answer

The correct answer is the singular possessive noun 'worm's'.

The sentence requires a singular possessive noun to modify 'body' and show ownership. The word 'each' is a singular determiner, indicating that the singular form 'worm' is required, rather than the plural 'worms'. To show possession, an apostrophe followed by 's' must be added, resulting in the singular possessive form 'worm's'.

Step-by-Step Solution

1

Analyze the grammatical structure of the sentence around the blank.

The blank is followed by the noun 'body' and preceded by the determiner 'each' within the prepositional phrase 'inside each ______ body'.

Identifying the surrounding words helps determine the required part of speech, number, and case.

2

Determine the number of the noun required by the context clues.

The word 'each' is singular, indicating that the noun completing the blank must be singular ('worm' rather than 'worms').

Singular determiners like 'each' must modify singular nouns.

3

Determine the case of the noun required by the context clues.

The noun must modify 'body' to show possession (whose body), which requires the possessive case ('worm's').

A noun modifying another noun to indicate ownership requires the possessive form.

4

Combine the singular and possessive requirements to select the correct form.

The correct form is the singular possessive 'worm's'.

This satisfies both the singular requirement of 'each' and the possessive requirement of 'body'.

Key Concept

Plural and Possessive Nouns and Pronouns

Estimated Time:1m 0s

Question 1746Question →

A polynomial $p(x)$ has a remainder of $3$ when divided by $x - 5$ . Which of the following equations must be true?

p(-5) = 3

p(5) = 0

p(5) = 3

p(3) = 0

Show answer & explanation

Answer: $p(5) = 3$

Answer

The equation

p(5) = 3

must be true.

According to the Remainder Theorem, when a polynomial

p(x)

is divided by a linear expression of the form

x - c

, the remainder is equal to

p(c)

. Here, the divisor is

x - 5

, which gives

c = 5

. The remainder is given as

3

. Therefore, evaluating the polynomial at

x = 5

must yield

3

, which is written as

p(5) = 3

.

Step-by-Step Solution

1

Identify the divisor and the remainder from the given problem statement.

The divisor is

x - 5

and the remainder is

3

.

Applying the Remainder Theorem requires identifying the value of

c

in the divisor form

x - c

and the remainder value.

2

Apply the Remainder Theorem to relate the divisor and the remainder to the polynomial function.

Since the divisor is

x - 5

, we have

c = 5

. The Remainder Theorem states that the remainder when

p(x)

is divided by

x - c

is

p(c)

. Therefore,

p(5) = 3

.

This establishes the mathematical relationship directly showing which equation must be true.

Key Concept

The Remainder Theorem

Estimated Time:45s

Question 1747Question →

In the polynomial function $p(x) = x^3 - 5x^2 + 2x + k$ , the constant $k$ is chosen such that $p(x)$ is divisible by $x - 4$ . What is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

The value of

k

is 8.

According to the Factor Theorem, a polynomial

p(x)

is divisible by

x - c

if and only if

p(c) = 0

. Since

p(x)

is divisible by

x - 4

, we must have

p(4) = 0

. Substituting

4

for

x

in the equation for

p(x)

yields

4^3 - 5(4)^2 + 2(4) + k = 0

. Simplifying the numerical terms gives

64 - 80 + 8 + k = 0

, which simplifies further to

-8 + k = 0

. Therefore, the value of the constant

k

is

8

.

Step-by-Step Solution

1

Apply the Factor Theorem to relate the factor

x - 4

to the value of the polynomial.

Since

x - 4

is a factor of

p(x)

,

p(4) = 0

.

By the Factor Theorem, if

x - c

is a factor of a polynomial

p(x)

, then

p(c) = 0

.

2

Substitute

x = 4

into the polynomial expression.

4^3 - 5(4)^2 + 2(4) + k = 0

We substitute

x = 4

into

p(x) = x^3 - 5x^2 + 2x + k

and set the expression to

0

.

3

Simplify the numerical expression to solve for

k

.

64 - 80 + 8 + k = 0 \implies -8 + k = 0 \implies k = 8

Evaluate powers, multiply, and solve the resulting linear equation for

k

.

Key Concept

Factor Theorem

Estimated Time:1m 0s

Question 1748Question →

The table below shows the number of employees in Division A and Division B of a company in 2024.

Division	Number of employees (2024)
Division A	160
Division B	240

In 2025, the number of employees in Division A increased by 35%, and the total number of employees in the company increased by 2%. What was the percent decrease in the number of employees in Division B from 2024 to 2025?

12%

20%

25%

80%

Show answer & explanation

Answer: 20%

Answer

The correct answer is 20%.

The correct answer is 20%. To find this, first calculate the total number of employees in 2024, which is

160 + 240 = 400

. In 2025, Division A's employees increase by 35% to

160 \times 1.35 = 216

, and the total number of employees increases by 2% to

400 \times 1.02 = 408

. The number of employees in Division B in 2025 is the difference:

408 - 216 = 192

. The decrease in Division B is

240 - 192 = 48

employees. Finally, the percent decrease is calculated relative to Division B's original count of 240:

\frac{48}{240} \times 100\% = 20\%

.

Step-by-Step Solution

1

Find the total number of employees in the company in 2024.

The total number of employees in 2024 is

160 + 240 = 400

.

To find changes in the total company size, we need to establish the baseline total.

2

Calculate the number of employees in Division A in 2025.

The number of employees in Division A in 2025 is

160 \times 1.35 = 216

.

Division A's workforce increased by 35% from its 2024 value of 160.

3

Calculate the total number of employees in the company in 2025.

The total number of employees in 2025 is

400 \times 1.02 = 408

.

The total company workforce increased by 2% from its 2024 value of 400.

4

Determine the number of employees in Division B in 2025.

The number of employees in Division B in 2025 is

408 - 216 = 192

.

The employees in Division B are the remaining employees after subtracting Division A's employees from the company total.

5

Calculate the percent decrease in Division B's employees from 2024 to 2025.

The decrease in Division B is

240 - 192 = 48

employees. The percent decrease is

\frac{48}{240} \times 100\% = 20\%

.

The percent decrease is the change divided by the initial value of Division B (240), multiplied by 100.

Key Concept

Solving multi-step percent problems with varying base values.

Alternative Method

Instead of calculating the absolute number of employees, you can use the weighted average of the percent changes. Since Division A represents

\frac{160}{400} = 40\%

of the company and Division B represents

\frac{240}{400} = 60\%

, the weighted percent change equation is:

0.40(35\%) + 0.60(x) = 2\%

, where

x

is the percent change for Division B. Solving for

x

gives:

14\% + 0.60x = 2\% \Rightarrow 0.60x = -12\% \Rightarrow x = -20\%

, representing a

20\%

decrease.

Estimated Time:2m 0s

Question 1749Question →

The price of a ticket to a museum is $25. If a$ 6\%$ sales tax is added to the price of the ticket, what is the total cost of the ticket, in dollars?

Show answer & explanation

Answer: 26.5

Answer

The correct answer is 26.5.

To find the total cost of the museum ticket, the

6\%

sales tax must be added to the original price. The tax amount is calculated by converting the percent to a decimal and multiplying it by the original price:

25 \times 0.06 = 1.5

dollars. Adding this tax to the original price of

25

gives the total cost:

25 + 1.5 = 26.5

dollars. Alternatively, this can be solved by multiplying the original price by

1.06

:

25 \times 1.06 = 26.5

dollars.

Step-by-Step Solution

1

Calculate the amount of sales tax.

1.5

dollars

To find

6\%

of

25

, convert the percentage to a decimal by dividing by 100 (

6\% = 0.06

) and multiply by the ticket price:

25 \times 0.06 = 1.5

.

2

Add the sales tax to the original ticket price to get the total cost.

26.5

dollars

The total cost is the sum of the original price and the tax:

25 + 1.5 = 26.5

.

Key Concept

Calculating a final value after a percentage increase

Alternative Method

The total cost of the ticket including a

6\%

tax can be found in one step by multiplying the original price by

1.06

:

25 \times 1.06 = 26.5

.

Estimated Time:45s

Question 1750Question →

For all $x \neq 2$ , the expression $\frac{3x^2 + kx - 8}{x - 2}$ is equivalent to $3x + 10 + \frac{12}{x - 2}$ , 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

.

The correct answer is

4

. By rewriting the right-hand side of the equation with a common denominator of

x - 2

, the expression becomes

\frac{(3x + 10)(x - 2) + 12}{x - 2}

. Expanding and simplifying the numerator yields

3x^2 + 4x - 8

. Comparing this to the numerator of the left-hand side,

3x^2 + kx - 8

, shows that the coefficient of the

x

term,

k

, must equal

4

.

Step-by-Step Solution

1

Set up the equation representing the equivalence of the two expressions.

\frac{3x^2 + kx - 8}{x - 2} = 3x + 10 + \frac{12}{x - 2}

This establishes the relationship that must hold for all

x \neq 2

.

2

Find a common denominator for the terms on the right-hand side.

3x + 10 + \frac{12}{x - 2} = \frac{(3x + 10)(x - 2)}{x - 2} + \frac{12}{x - 2}

To combine the terms into a single rational expression, they must have the same denominator.

3

Expand and simplify the numerator on the right-hand side.

(3x + 10)(x - 2) + 12 = 3x^2 + 4x - 8

Expanding the product of binomials and combining like terms yields the simplified numerator.

4

Equate the numerators of the simplified expressions to solve for

k

.

3x^2 + kx - 8 = 3x^2 + 4x - 8

, which means

k = 4

.

Since the denominators are equal, the numerators must be identical for all values of

x \neq 2

, meaning their corresponding coefficients must match.

Key Concept

Equivalent Algebraic Expressions

Alternative Method

An alternative method is to substitute a convenient value for

x

that is not equal to

2

. Substituting

x = 1

into both sides of the equivalence gives:

\frac{3(1)^2 + k(1) - 8}{1 - 2} = 3(1) + 10 + \frac{12}{1 - 2}

. Simplifying this yields

\frac{k - 5}{-1} = 13 - 12

, which simplifies to

-k + 5 = 1

. Solving for

k

results in

k = 4

.

Estimated Time:1m 30s

Question 1751Question →

For which of the following values of $k$ does the equation $\frac{x^2 - kx + 9}{x - 3} = 0$ have exactly one real solution?

6

3

-6

-3

Show answer & explanation

Answer: $-6$

Answer

The correct value is

-6

.

For the equation

\frac{x^2 - kx + 9}{x - 3} = 0

to have a solution, we must have

x \neq 3

. Multiplying both sides by the denominator yields the quadratic equation

x^2 - kx + 9 = 0

. For a quadratic equation to have exactly one real solution, its discriminant must equal zero:

(-k)^2 - 4(1)(9) = k^2 - 36 = 0

, which gives

k = 6

or

k = -6

. If

k = 6

, the equation is

(x-3)^2 = 0

, which has the single solution

x = 3

. Since

x = 3

is excluded by the domain restriction, the equation has zero real solutions. If

k = -6

, the equation is

(x+3)^2 = 0

, which has the single solution

x = -3

. Since

-3

is a valid value, this value of

k

yields exactly one real solution.

Step-by-Step Solution

1

Identify the domain restriction of the rational equation.

The denominator

x - 3

cannot be zero, which means

x \neq 3

.

Division by zero is undefined, so

x = 3

cannot be a solution to the equation.

2

Simplify the equation by multiplying both sides by the denominator.

x^2 - kx + 9 = 0

for

x \neq 3

.

A rational expression equals zero when its numerator is zero and its denominator is non-zero.

3

Set the discriminant of the quadratic equation to zero to find the values of

k

that yield exactly one real root.

\Delta = (-k)^2 - 4(1)(9) = k^2 - 36 = 0 \implies k = 6

or

k = -6

.

A quadratic equation has exactly one real solution (a double root) when its discriminant is zero.

4

Verify if the roots for each value of

k

are valid under the domain restriction

x \neq 3

.

If

k = 6

,

x^2 - 6x + 9 = (x - 3)^2 = 0 \implies x = 3

(invalid/extraneous). If

k = -6

,

x^2 + 6x + 9 = (x + 3)^2 = 0 \implies x = -3

(valid).

The value

k = 6

makes

x = 3

the only solution, which is excluded. Therefore, only

k = -6

yields exactly one valid real solution (

x = -3

).

Key Concept

Solving rational equations that simplify to quadratic equations, using the discriminant to find the number of solutions, and identifying extraneous solutions due to domain restrictions.

Estimated Time:2m 0s

Question 1752Question →

In the nineteenth century, agricultural chemist Justus von Liebig developed the "Law of the Minimum," which states that plant growth is controlled not by the total amount of resources available, but by the scarcest resource_______ a plant's development will be limited by whichever essential nutrient is in shortest supply.

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

resource, a

resource; although a

resource: a

resource a

Show answer & explanation

Answer: resource: a

Answer

resource: a

The correct answer is the option that reads "resource: a". The text preceding the blank is an independent clause, and the text following the blank is also an independent clause that explains the concept in the first clause (specifically, how the Law of the Minimum works). A colon is standard and correct for joining two independent clauses when the second explains or illustrates the first.

Step-by-Step Solution

1

Analyze the grammatical structure of the clauses before and after the blank.

The clause before the blank ("In the nineteenth century... scarcest resource") is independent, meaning it can stand alone as a complete sentence. The clause after the blank ("a plant's development... in shortest supply") is also an independent clause.

Identifying the clause boundaries and types is necessary to determine the correct punctuation.

2

Determine the relationship between the two independent clauses.

The second clause directly explains and clarifies the concept of the "Law of the Minimum" introduced in the first clause.

Understanding the semantic relationship helps select the appropriate punctuation mark, as colons can be used when the second clause explains or illustrates the first.

3

Evaluate the choices based on standard punctuation rules for independent clauses.

The option with a colon correctly joins the two independent clauses and signals that the second clause explains the first. The option with a comma creates a comma splice. The option with a semicolon followed by "although" is incorrect because "although" turns the second clause into a dependent clause. The option with no punctuation creates a run-on sentence.

Eliminating grammatically incorrect choices leaves the one correct option.

Key Concept

Using a colon to link two independent clauses when the second explains, illustrates, or expands upon the first.

Question 1753Question →

If

x

is a real solution to the equation

\frac{2x - 5}{x - 3} + \frac{x - 5}{x^2 - 4x + 3} = -\frac{1}{x - 1}

what is the value of

1 - 2x

?

-5

2

-2

0

Show answer & explanation

Answer: 2

Answer

2

The correct answer is 2. Multiplying the entire equation by the least common denominator

(x - 3)(x - 1)

simplifies the rational equation to the quadratic equation

2x^2 - 5x - 3 = 0

. This quadratic factors into

(2x + 1)(x - 3) = 0

, yielding two potential solutions:

x = -1/2

and

x = 3

. However,

x = 3

is an extraneous solution because it results in a denominator of zero in the original equation. Thus, the only valid real solution is

x = -1/2

. Substituting this into the expression

1 - 2x

yields

1 - 2(-1/2) = 2

.

Step-by-Step Solution

1

Factor the quadratic expression in the denominator to identify the least common denominator.

The quadratic in the denominator of the second term factors as

x^2 - 4x + 3 = (x - 3)(x - 1)

. Thus, the least common denominator for all terms in the equation is

(x - 3)(x - 1)

, with the restrictions that

x \neq 3

and

x \neq 1

.

Finding a common denominator allows us to multiply both sides of the equation to clear all rational expressions.

2

Multiply both sides of the equation by the least common denominator

(x - 3)(x - 1)

.

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

This clears the fractions and converts the rational equation into a polynomial equation.

3

Expand and simplify both sides of the equation.

(2x^2 - 7x + 5) + (x - 5) = -x + 3

2x^2 - 6x = -x + 3

Expanding the terms allows us to combine like terms and set up a quadratic equation.

4

Rearrange the terms to set the quadratic equation equal to zero.

2x^2 - 5x - 3 = 0

Quadratic equations must be set to zero to be solved by factoring or using the quadratic formula.

5

Solve the quadratic equation by factoring.

(2x + 1)(x - 3) = 0

This gives two potential solutions:

x = -1/2

and

x = 3

.

Factoring is the most direct algebraic method to find the roots of the quadratic equation.

6

Check the potential solutions against the original domain restrictions to identify any extraneous solutions.

For

x = 3

, the denominators

x - 3

and

x^2 - 4x + 3

equal zero, so

x = 3

is an extraneous solution. For

x = -1/2

, the denominators are non-zero, so

x = -1/2

is the only valid solution.

Squaring or multiplying by variable expressions can introduce solutions that are undefined in the original equation.

7

Substitute the valid solution

x = -1/2

into the expression

1 - 2x

.

1 - 2\left(-\frac{1}{2}\right) = 1 + 1 = 2

The question asks for the value of the expression

1 - 2x

, not the value of

x

itself.

Key Concept

Solving rational equations by clearing denominators and checking for extraneous solutions.

Question 1754Question →

Two different cultures of bacteria, culture $A$ and culture $B$ , begin growing at the same time. The population of culture $A$ increases by $300\%$ every $8$ hours, and the population of culture $B$ increases by $700\%$ every $9$ hours. If both cultures start with the same initial population, after how many hours will the population of culture $B$ be $4$ times the population of culture $A$ ?

9.6

24

2

48

Show answer & explanation

Answer: 24

Answer

The population of culture B will be 4 times the population of culture A after 24 hours.

The correct answer is the option stating 24 hours. The growth model for culture A is

P_A(t) = P_0 \cdot 4^{t/8}

and for culture B is

P_B(t) = P_0 \cdot 8^{t/9}

. Setting

P_B(t) = 4 P_A(t)

and converting all terms to base 2 yields

2^{t/3} = 2^{2 + t/4}

. Equating exponents gives

t/3 = 2 + t/4

, which simplifies to

t/12 = 2

and solves to

t = 24

hours.

Step-by-Step Solution

1

Determine the growth factors and set up the population equations for both cultures.

For culture

A

, a

300\%

increase means the population becomes

1 + 3.00 = 4

times its previous value every

8

hours, so

P_A(t) = P_0 \cdot 4^{t/8}

. For culture

B

, a

700\%

increase means the population becomes

1 + 7.00 = 8

times its previous value every

9

hours, so

P_B(t) = P_0 \cdot 8^{t/9}

.

Establishing the correct exponential growth models with the appropriate growth factors and time constants is necessary to relate the populations.

2

Set up the equation to find when the population of culture

B

is

4

times that of culture

A

.

P_B(t) = 4 P_A(t) \implies P_0 \cdot 8^{t/9} = 4 \cdot P_0 \cdot 4^{t/8} \implies 8^{t/9} = 4 \cdot 4^{t/8}

This sets up the equation that must be solved for

t

by dividing both sides by the non-zero initial population

P_0

.

3

Express all terms with a common base of

2

.

(2^3)^{t/9} = 2^2 \cdot (2^2)^{t/8} \implies 2^{t/3} = 2^2 \cdot 2^{t/4}

Expressing bases as powers of

2

allows the use of exponent rules to simplify the equation.

4

Apply the product rule of exponents to combine the terms on the right side.

2^{t/3} = 2^{2 + t/4}

The rule

2^a \cdot 2^b = 2^{a+b}

simplifies the multiplication of exponential terms with the same base.

5

Equate the exponents and solve the resulting linear equation for

t

.

t/3 = 2 + t/4 \implies t/3 - t/4 = 2 \implies t/12 = 2 \implies t = 24

Since the bases are equal, their exponents must be equal. Solving the linear equation yields the time in hours.

Key Concept

Solving exponential equations by converting to a common base and applying exponent laws.

Estimated Time:3m 0s

Question 1755Question →

Urban areas often experience significantly higher temperatures than their rural surroundings, a phenomenon known as the urban heat island effect. Dark materials like asphalt absorb and retain solar heat _______ areas cool the local environment through evapotranspiration. Consequently, city planners are increasingly incorporating green roofs and parks into urban designs to mitigate this warming.

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

vegetated

, vegetated

; vegetated

; while vegetated

Show answer & explanation

Answer: ; vegetated

Answer

; vegetated

The text contains two independent clauses: 'Dark materials like asphalt absorb and retain solar heat' and 'vegetated areas cool the local environment through evapotranspiration.' The option with a semicolon followed by 'vegetated' correctly joins these two independent clauses without creating a grammatical error.

Step-by-Step Solution

1

Identify the clause boundaries in the sentence.

The sentence consists of two independent clauses: 'Dark materials like asphalt absorb and retain solar heat' and 'vegetated areas cool the local environment through evapotranspiration.'

To determine how to link them, we must first know if each can stand alone as a complete sentence.

2

Evaluate the options to see which punctuation or conjunction correctly joins two independent clauses.

A semicolon alone can link two independent clauses.

A semicolon is grammatically sufficient to connect two closely related independent clauses without any additional coordinating conjunction.

Key Concept

Clause Boundaries and Linking

Estimated Time:1m 0s

Question 1756Question →

The following passage is adapted from an essay about earth science. Read the passage and determine which punctuation mark best completes the sentence to conform to the conventions of Standard English. Write the correct punctuation mark in the blank.

Fill in the blanks below

In 1912, German geophysicist Alfred Wegener proposed the theory of continental drift, suggesting that all landmasses were once joined in a supercontinent called Pangaeamany of his contemporaries, however, rejected this hypothesis because Wegener could not explain the physical mechanism that drove the movement of the continents.

Show answer & explanation

Answer

A semicolon (;)

A semicolon is the correct punctuation mark because it properly joins two independent clauses without a coordinating conjunction. In this sentence, the transition word 'however' is functioning parenthetically within the second clause and does not act as a coordinating conjunction.

Step-by-Step Solution

1

Identify the structure of the clauses before and after the blank space.

The clause before the blank ('In 1912... called Pangaea') and the clause after the blank ('many of his contemporaries... of the continents') are both independent clauses.

Determining clause boundaries helps identify what kind of punctuation is grammatically required to link the parts of the sentence.

2

Analyze the logical relationship between the two independent clauses to choose the correct punctuation mark.

The two independent clauses are closely related but present contrasting ideas (signaled by 'however'), and they are not joined by a coordinating conjunction.

Two independent clauses not joined by a coordinating conjunction must be linked by a semicolon or a colon, depending on their logical relationship.

3

Select the correct punctuation mark based on grammar conventions.

A semicolon is the correct mark to separate these two independent clauses. A colon is incorrect because the second clause does not explain or expand on the first, and a comma would result in a comma splice.

Semicolons are the standard tool for connecting independent clauses without coordinating conjunctions when a colon's explanatory relationship is absent.

Key Concept

Semicolons are used to link two independent clauses that are closely related in thought but are not joined by a coordinating conjunction.

Question 1757Question →

A construction company rents a concrete mixer. The total cost, in dollars, of renting the concrete mixer is a linear function of the rental time, in hours. The table below shows some values of the rental time and the corresponding total cost.

Time (hours)	Total cost (dollars)
$3$	$54$
$5$	$78$
$8$	$114$

Which of the following is the total cost, in dollars, of renting the concrete mixer for $12$ hours?

55

144

162

168

Show answer & explanation

Answer: $162$

Answer

162

dollars

The cost function is linear, represented by

C(t) = mt + b

. The rate of change (slope) is calculated as

m = \frac{78 - 54}{5 - 3} = 12

dollars per hour. Using the point

(3, 54)

, we solve for the initial fee (y-intercept)

b

:

54 = 12(3) + b

, which gives

b = 18

. Thus, the linear model is

C(t) = 12t + 18

. For a rental time of

12

hours, the total cost is

C(12) = 12(12) + 18 = 162

dollars.

Step-by-Step Solution

1

Calculate the rate of change (slope) of the linear cost function.

m = 12

Since the relationship is linear, the rate of change is constant. Using the coordinate points

(3, 54)

and

(5, 78)

from the table, the slope is calculated as

m = \frac{78 - 54}{5 - 3} = 12

.

2

Determine the y-intercept (initial fee) of the function.

b = 18

Substitute one of the points, such as

(3, 54)

, and the slope

m = 12

into the slope-intercept form equation

C(t) = mt + b

. This gives

54 = 12(3) + b \Rightarrow 54 = 36 + b \Rightarrow b = 18

.

3

Evaluate the cost function for a rental time of

12

hours.

C(12) = 162

Using the complete linear equation

C(t) = 12t + 18

, substitute

t = 12

to find the total cost:

C(12) = 12(12) + 18 = 144 + 18 = 162

.

Key Concept

Finding and evaluating a linear function from tabular data

Estimated Time:1m 30s

Question 1758Question →

A manufacturing plant uses two types of machines, Machine X and Machine Y, to produce a certain component. Machine X produces $15$ components per hour, and Machine Y produces $25$ components per hour. On a certain day, both machines were operated for a combined total of $14$ hours, producing a total of $270$ components. For how many hours was Machine X operated?

Show answer & explanation

Answer: 8

Answer

Machine X was operated for 8 hours.

By setting up the system of linear equations where

x

represents the hours Machine X operated and

y

represents the hours Machine Y operated, we get

x + y = 14

and

15x + 25y = 270

. Substituting

y = 14 - x

into the second equation yields

15x + 25(14 - x) = 270

. Expanding and simplifying results in

-10x + 350 = 270

, which solves to

x = 8

. Thus, Machine X operated for 8 hours.

Step-by-Step Solution

1

Define variables for the hours each machine operated and set up the system of linear equations.

Let

x

be the hours Machine X was operated and

y

be the hours Machine Y was operated. The system is:

x + y = 14

15x + 25y = 270

This translates the verbal descriptions in the problem into algebraic expressions and equations.

2

Express

y

in terms of

x

from the first equation and substitute it into the second equation.

y = 14 - x

15x + 25(14 - x) = 270

Substitution reduces the system from two variables to a single equation in terms of

x

.

3

Solve the linear equation for

x

by distributing the constant and combining like terms.

15x + 350 - 25x = 270

-10x + 350 = 270

-10x = -80

x = 8

Isolating

x

gives the specific number of hours Machine X was operated.

Key Concept

Solving systems of linear equations in two variables using substitution or elimination.

Alternative Method

We can solve the system by multiplying the first equation by 25 to get

25x + 25y = 350

. Subtracting the second equation

15x + 25y = 270

from this yields

10x = 80

, which directly gives

x = 8

.

Estimated Time:1m 30s

Question 1759Question →

A digital printing press uses paper from a large roll. The total weight $W$ , in pounds, of the paper remaining on the roll after $m$ minutes of continuous printing is given by the equation $W = 120 - 1.5m$ . According to the equation, after how many minutes of continuous printing will the weight of the remaining paper on the roll be exactly $45$ pounds?

Show answer & explanation

Answer: 50

Answer

The correct answer is 50.

To find the number of minutes,

m

, when the remaining weight is

45

pounds, substitute

W = 45

into the given equation

W = 120 - 1.5m

. This yields the equation

45 = 120 - 1.5m

. Subtracting

120

from both sides gives

-75 = -1.5m

. Dividing both sides by

-1.5

results in

m = 50

. Therefore, after

50

minutes, the remaining weight of the paper on the roll is

45

pounds.

Step-by-Step Solution

1

Substitute the target weight into the linear equation.

45 = 120 - 1.5m

We are given that the weight of the remaining paper,

W

, is

45

pounds, and we need to solve for the time in minutes,

m

.

2

Isolate the variable term by subtracting

120

from both sides of the equation.

-75 = -1.5m

To solve for

m

, we first need to isolate the term containing the variable on one side of the equation.

3

Divide both sides by

-1.5

to find the value of

m

.

m = 50

Dividing by the coefficient of the variable completes the isolation and gives the final value.

Key Concept

Interpreting values and solving linear equations in context

Alternative Method

Alternatively, you can determine the total weight of paper consumed:

120 - 45 = 75

pounds. Since the paper is used at a rate of

1.5

pounds per minute, the time required is

\frac{75}{1.5} = 50

minutes.

Estimated Time:1m 30s

Question 1760Question →

An online learning platform has $500$ active subscribers. If the number of active subscribers increases by $12\%$ next month, what is the total number of active subscribers the platform will have next month?

440

512

560

600

Show answer & explanation

Answer: $560$

Answer

The total number of active subscribers the platform will have next month is

560

.

To find the total number of active subscribers after a

12\%

increase, first calculate

12\%

of

500

. This is done by converting

12\%

to its decimal equivalent,

0.12

, and multiplying by the base:

500 \times 0.12 = 60

. Then, add the increase to the initial value:

500 + 60 = 560

. Alternatively, multiply the initial value by

1.12

to directly get the new total:

500 \times 1.12 = 560

. Therefore, the total number of active subscribers next month is

560

.

Step-by-Step Solution

1

Calculate the amount of increase by finding

12\%

of the initial

500

subscribers.

500 \times 0.12 = 60

To find the number of new subscribers, the percentage of increase must be multiplied by the initial amount.

2

Add the increase to the initial number of subscribers to find the new total.

500 + 60 = 560

An increase means adding the calculated amount of change to the baseline value.

Key Concept

Calculating a percentage increase involves finding the specified percent of the original value and adding it to that original value.

Answer

Step-by-Step Solution

Key Concept

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