All practice questions

The table below shows several values of the function $f$ .

$x$	$f(x)$
$-3$	$8$
$-1$	$2$
$1$	$5$
$3$	$-1$

The function $g$ is defined by $g(x) = f(x + 2)$ . What is the value of $g(-1)$ ?

8

5

4

0

Show answer & explanation

Answer: 5

Answer

5

To find the value of

g(-1)

, substitute

x = -1

into the equation

g(x) = f(x + 2)

, which yields

g(-1) = f(-1 + 2) = f(1)

. Looking at the table, when the input is

1

, the output of the function

f

is

5

. Therefore, the value of

g(-1)

is

5

.

Step-by-Step Solution

1

Substitute the given input value of

-1

for

x

in the definition of

g(x)

.

g(-1) = f(-1 + 2)

To find

g(-1)

, we need to replace

x

with

-1

in the equation

g(x) = f(x + 2)

.

2

Simplify the expression inside the function notation.

g(-1) = f(1)

Performing the addition

-1 + 2

yields

1

.

3

Use the table to find the value of

f(1)

.

f(1) = 5

Looking at the row in the table where

x = 1

, the corresponding value of

f(x)

is

5

.

Key Concept

Function transformations and notation using tables

Question 1782Question →

A cubic polynomial function $p$ is defined by $p(x) = a(x + 3)(x - 1)(x - k)$ , where $a$ and $k$ are constants. In the $xy$ -plane, the $y$ -intercept of the graph of $y = p(x)$ is $(0, 6)$ . The graph of the shifted function $y = p(x - 2)$ passes through the point $(4, -30)$ . What is the third $x$ -intercept of the graph of $p$ ?

-1

9

\frac{1}{3}

-\frac{1}{2}

Show answer & explanation

Answer: $-1$

Answer

-1

The correct answer is

-1

because the

y

-intercept

(0, 6)

gives

3ak = 6

, which simplifies to

ak = 2

. The shift

y = p(x-2)

passing through

(4, -30)

implies

p(2) = -30

. Substituting

x=2

into the polynomial expression yields

5a(2-k) = -30

, which expands to

10a - 5ak = -30

. Substituting

ak = 2

gives

10a - 10 = -30

, solving to

a = -2

. Using

ak = 2

, we find

k = -1

. The third factor is

(x - k) = (x + 1)

, which corresponds to the third

x

-intercept at

x = -1

.

Step-by-Step Solution

1

Use the

y

-intercept of the graph of

y = p(x)

to establish a relationship between

a

and

k

.

p(0) = a(0 + 3)(0 - 1)(0 - k) = 3ak = 6 \Rightarrow ak = 2

.

The

y

-intercept is the point on the graph where

x = 0

.

2

Translate the point on the shifted graph back to the original function

p(x)

.

p(4 - 2) = p(2) = -30

.

Since the graph of

y = p(x - 2)

passes through

(4, -30)

, substituting

x = 4

yields

y = -30

.

3

Substitute

x = 2

into the expression for

p(x)

to set up the second equation.

p(2) = a(2 + 3)(2 - 1)(2 - k) = 5a(2 - k) = 10a - 5ak = -30

.

This establishes a system of equations with the relation from Step 1.

4

Solve the system of equations by substituting

ak = 2

into the second equation.

10a - 5(2) = -30 \Rightarrow 10a - 10 = -30 \Rightarrow 10a = -20 \Rightarrow a = -2

. Since

ak = 2

, we have

-2k = 2 \Rightarrow k = -1

.

Solving for the unknown constants

a

and

k

determines the specific polynomial expression.

5

Identify the third

x

-intercept from the factored form of the polynomial.

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

. The factors correspond to roots at

x = -3

,

x = 1

, and

x = -1

. The third

x

-intercept is

-1

.

The third factor

(x - k)

becomes

(x + 1)

when

k = -1

, yielding the root and intercept at

x = -1

.

Key Concept

Polynomial Factors and Graphs

Question 1783Question →

In a certain school, the number of students in the science club was $25\%$ of the number of students in the math club. During a membership drive, the number of students in the science club increased by $30\%$ , and the number of students in the math club increased by $10\%$ . After the drive, the total number of students in both clubs combined was $228$ . If no student is a member of both clubs, what was the total number of students in both clubs combined before the membership drive?

Show answer & explanation

Answer: 200

Answer

The total number of students in both clubs combined before the membership drive was 200.

To find the initial total number of students, we define the initial math club members as

M

. Since the science club has

25\%

of the size of the math club, its initial size is

0.25M

. After a

30\%

increase, the science club size is

1.30(0.25M) = 0.325M

. After a

10\%

increase, the math club size is

1.10M

. The new combined total is

0.325M + 1.10M = 228

. Simplifying gives

1.425M = 228

, which yields

M = 160

. The initial science club size is

0.25(160) = 40

. The total number of students before the drive was

160 + 40 = 200

.

Step-by-Step Solution

1

Define variables for the initial number of students in each club.

Let

M

be the initial number of students in the math club, and

S = 0.25M

be the initial number of students in the science club.

This establishes a relationship between the sizes of the two clubs based on the given percentage.

2

Express the new number of students in each club after the percent increases.

Science club:

1.30 \times S = 1.30(0.25M) = 0.325M

. Math club:

1.10M

.

An increase of

30\%

is represented by multiplying by

1.30

, and an increase of

10\%

is represented by multiplying by

1.10

.

3

Set up and solve an equation using the new combined total.

0.325M + 1.10M = 228 \implies 1.425M = 228 \implies M = 160

.

The sum of the new club sizes equals the new total of

228

students.

4

Calculate the initial science club members and the initial total combined members.

Initial science members:

S = 0.25 \times 160 = 40

. Total initial members:

M + S = 160 + 40 = 200

.

This answers the final question asking for the combined total before the membership drive.

Key Concept

Applying percent changes to individual components to determine initial values in a combined total system.

Alternative Method

Instead of setting up equations using

M

, you can choose an arbitrary starting value for the math club that is easily divisible, such as

100

. If the math club starts with

100

students, the science club starts with

25

students. After the increases, the math club has

110

students and the science club has

25 \times 1.3 = 32.5

students, for a total of

142.5

students. Since the actual final total is

228

, we scale our initial values by a factor of

\frac{228}{142.5} = 1.6

. The initial combined total is therefore

125 \times 1.6 = 200

.

Estimated Time:2m 0s

Question 1784Question →

The table below shows the total number of registered voters in two districts, District X and District Y, in 2020 and 2024.

District	2020 registered voters	2024 registered voters
District X	$10,000$	$15,000$
District Y	$15,000$	$18,000$

In 2020, $12\%$ of the registered voters in District X and $24\%$ of the registered voters in District Y were between the ages of 18 and 25. In 2024, $16\%$ of the registered voters in District X and $20\%$ of the registered voters in District Y were between the ages of 18 and 25. From 2020 to 2024, what was the percent increase in the total number of registered voters between the ages of 18 and 25 across both districts combined?

20%

25%

32%

50%

Show answer & explanation

Answer: 25%

Answer

The percent increase in the total number of registered voters between the ages of 18 and 25 across both districts combined is 25%.

To find the percent increase in the total number of registered voters between the ages of 18 and 25, we first determine the number of voters in this age group for both years. For 2020, District X had

10,000 \times 0.12 = 1,200

voters and District Y had

15,000 \times 0.24 = 3,600

voters, giving a combined total of

4,800

voters aged 18-25. For 2024, District X had

15,000 \times 0.16 = 2,400

voters and District Y had

18,000 \times 0.20 = 3,600

voters, giving a combined total of

6,000

voters aged 18-25. The percent increase is computed by dividing the change in this subpopulation by the initial 2020 value:

\frac{6,000 - 4,800}{4,800} \times 100\% = 25\%

. Therefore, the total number of registered voters in this age group increased by 25%.

Step-by-Step Solution

1

Calculate the number of registered voters aged 18-25 in both districts in 2020.

In District X:

10,000 \times 0.12 = 1,200

. In District Y:

15,000 \times 0.24 = 3,600

. Total 2020 voters aged 18-25 =

1,200 + 3,600 = 4,800

.

To find the overall percent change of the subpopulation, we need to know its initial size in 2020.

2

Calculate the number of registered voters aged 18-25 in both districts in 2024.

In District X:

15,000 \times 0.16 = 2,400

. In District Y:

18,000 \times 0.20 = 3,600

. Total 2024 voters aged 18-25 =

2,400 + 3,600 = 6,000

.

To find the overall percent change of the subpopulation, we need to know its final size in 2024.

3

Calculate the absolute increase in the registered voters aged 18-25.

6,000 - 4,800 = 1,200

voters.

The percent change formula requires the difference between the final and initial values.

4

Calculate the percent increase relative to the 2020 initial value.

\frac{1,200}{4,800} \times 100\% = 25\%

.

The percent increase is defined as the change divided by the initial value, multiplied by 100 to get a percentage.

Key Concept

Calculating percent change for a combined population by computing the absolute initial and final counts of the subpopulation first, rather than manipulating percent rates directly.

Estimated Time:3m 0s

Question 1785Question →

A municipal water utility charges customers a flat monthly fee plus a constant rate per thousand gallons of water used. In June, a household used $8$ thousand gallons of water and was charged a total of $\$46.00$ . In July, the same household used $12$ thousand gallons of water and was charged a total of $\$62.00$ . If $x$ represents the number of thousand gallons of water used in a month, and $y$ represents the total monthly charge, in dollars, which of the following equations represents the relationship between $x$ and $y$ ?

y = 0.25x + 44

y = 14x + 4

y = 4x + 14

y = 4x + 78

Show answer & explanation

Answer: $y = 4x + 14$

Answer

y = 4x + 14

The correct equation is

y = 4x + 14

. The constant rate of change (slope) is found by dividing the difference in monthly charges by the difference in water usage:

\frac{62 - 46}{12 - 8} = 4

. Using the point-slope form or slope-intercept form with the point

(8, 46)

yields

46 = 4(8) + b

, which simplifies to

b = 14

for the y-intercept (the flat monthly fee). Substituting these parameters back into the slope-intercept form yields the correct relationship.

Step-by-Step Solution

1

Calculate the constant rate of change (slope,

m

) using the two given data points,

(8, 46)

and

(12, 62)

.

m = \frac{62 - 46}{12 - 8} = \frac{16}{4} = 4

.

The constant rate of change represents the slope of the linear equation.

2

Use the slope-intercept form,

y = mx + b

, and substitute one of the data points, such as

(8, 46)

, to solve for the y-intercept (

b

).

46 = 4(8) + b \implies 46 = 32 + b \implies b = 14

.

Substituting a known point allows us to isolate and find the value of the flat monthly fee, which is the y-intercept.

3

Write the final equation by substituting the calculated slope (

m = 4

) and y-intercept (

b = 14

) back into the slope-intercept form.

y = 4x + 14

.

This combines the rate per thousand gallons and the flat fee into the final linear relationship.

Key Concept

Linear Equations in Two Variables

Estimated Time:1m 30s

Question 1786Question →

A small company had $150$ employees. If the number of employees decreased by $8\%$ , what is the current number of employees at the company?

12

138

142

162

Show answer & explanation

Answer: $138$

Answer

The correct answer is the option stating

138

.

The correct answer is the option stating

138

. To find the current number of employees, first find

8\%

of

150

, which is

0.08 \times 150 = 12

. Since the number of employees decreased, subtract

12

from the original count of

150

to get

150 - 12 = 138

.

Step-by-Step Solution

1

Calculate the number of employees represented by the

8\%

decrease.

0.08 \times 150 = 12

employees

To find the decrease in the number of employees, multiply the initial number of employees by the decimal equivalent of the percentage decrease.

2

Subtract the decrease from the initial number of employees to find the current count.

150 - 12 = 138

employees

Since the employee count decreased, subtracting the change from the original count yields the current number of employees.

Key Concept

Calculating a percentage decrease from an initial value.

Estimated Time:45s

Question 1787Question →

An equation is given as follows:

\frac{x}{x - 3} - \frac{4}{x + 2} = \frac{20}{x^2 - x - 6}

What is the only value of $x$ for which this equation is true?

Show answer & explanation

Answer: 4

Answer

4

To solve the equation, factor the denominator on the right side:

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

. The least common denominator is

(x - 3)(x + 2)

. Multiplying both sides by

(x - 3)(x + 2)

clears the fractions, resulting in

x(x + 2) - 4(x - 3) = 20

. Expanding the terms gives

x^2 + 2x - 4x + 12 = 20

. Simplifying and writing this in standard form yields

x^2 - 2x - 8 = 0

. Factoring the quadratic gives

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

, which gives the candidate solutions

x = 4

and

x = -2

. Substituting

x = -2

back into the original equation results in a denominator of zero, so

x = -2

is an extraneous solution. The only valid solution is

x = 4

.

Step-by-Step Solution

1

Factor the quadratic trinomial in the denominator of the right side of the equation.

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

Identifying the factors of the quadratic trinomial helps find the least common denominator of the rational equation.

2

Multiply the entire equation by the least common denominator

(x - 3)(x + 2)

to clear the fractions.

x(x + 2) - 4(x - 3) = 20

, with the constraints that

x \neq 3

and

x \neq -2

.

This simplifies the rational equation into a standard polynomial equation.

3

Distribute and combine like terms to write the equation in standard quadratic form.

x^2 - 2x - 8 = 0

Rewriting the equation in the form

ax^2 + bx + c = 0

is necessary to solve it by factoring.

4

Factor the quadratic equation.

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

Factoring allows us to apply the zero product property to find candidate solutions.

5

Check the candidate solutions

x = 4

and

x = -2

against the original equation to identify extraneous solutions.

x = 4

is the only valid solution because

x = -2

makes the denominators in the original equation equal to zero.

Any solution that makes a denominator in the original rational expression equal to zero is extraneous and must be excluded.

Key Concept

Solving rational equations by finding a common denominator, clearing fractions, and checking for extraneous solutions.

Question 1788Question →

If $81^{y-2} = 27^{y+1}$ , what is the value of $y$ ?

Show answer & explanation

Answer: 11

Answer

11

The correct answer is 11. By expressing both sides of the equation with the common base of 3, the equation simplifies from

81^{y-2} = 27^{y+1}

to

(3^4)^{y-2} = (3^3)^{y+1}

. Applying the power of a power rule gives

3^{4y-8} = 3^{3y+3}

. Equating the exponents yields

4y - 8 = 3y + 3

, which solves to

y = 11

.

Step-by-Step Solution

1

Express both bases as powers of 3

(3^4)^{y-2} = (3^3)^{y+1}

To solve an exponential equation algebraically, it is helpful to express both sides using a common base.

2

Apply the power of a power exponent rule

3^{4(y-2)} = 3^{3(y+1)}

The rule

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

allows us to simplify the exponent expressions by multiplying the exponents.

3

Equate the exponents

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

Since the bases are equal, the exponents must be equal for the equation to hold true.

4

Distribute the coefficients

4y - 8 = 3y + 3

Expanding the linear expressions prepares the equation for isolation of the variable.

5

Solve the linear equation for

y

y = 11

Subtract

3y

from both sides to get

y - 8 = 3

, then add 8 to both sides to isolate

y

.

Key Concept

Solving exponential equations by expressing bases in terms of a common base and equating exponents.

Alternative Method

Alternatively, substitute the value of 11 back into the original equation to verify that both sides are equal:

81^{11-2} = 81^9 = (3^4)^9 = 3^{36}

and

27^{11+1} = 27^{12} = (3^3)^{12} = 3^{36}

.

Estimated Time:1m 30s

Question 1789Question →

What is the sum of all positive real solutions to the equation

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

?

Show answer & explanation

Answer: 7

Answer

The sum of all positive real solutions is 7.

By substituting

u = x^2 - 3x

, the original rational equation simplifies to

\frac{u}{u - 2} + \frac{u - 2}{u} = \frac{5}{2}

. Multiplying both sides by the common denominator

2u(u-2)

and simplifying results in the quadratic equation

u^2 - 2u - 8 = 0

. Factoring gives

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

, so

u = 4

or

u = -2

. Substituting back

x^2 - 3x

for

u

leads to two quadratic equations:

x^2 - 3x = 4

(which has solutions

x = 4

and

x = -1

) and

x^2 - 3x = -2

(which has solutions

x = 2

and

x = 1

). Checking the denominators, none of these solutions make the original denominators zero, so they are all valid. The positive solutions are

1

,

2

, and

4

, and their sum is

1 + 2 + 4 = 7

.

Step-by-Step Solution

1

Introduce a substitution variable to simplify the rational equation.

Letting

u = x^2 - 3x

transforms the equation into

\frac{u}{u - 2} + \frac{u - 2}{u} = \frac{5}{2}

.

This reduces the degree of the rational expression and simplifies the algebraic manipulation required to solve it.

2

Eliminate the denominators by multiplying by the least common denominator.

Multiplying by

2u(u-2)

gives

2u^2 + 2(u-2)^2 = 5u(u-2)

, which simplifies to

u^2 - 2u - 8 = 0

.

This converts the rational equation into a standard quadratic equation in terms of

u

.

3

Solve the quadratic equation for

u

by factoring.

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

, which gives

u = 4

or

u = -2

.

Finding the values of

u

allows us to set up equations to solve for the original variable

x

.

4

Substitute back

x^2 - 3x

for

u

and solve the resulting quadratic equations for

x

.

From

x^2 - 3x = 4

, we get

(x-4)(x+1) = 0 \implies x = 4, -1

. From

x^2 - 3x = -2

, we get

(x-2)(x-1) = 0 \implies x = 2, 1

.

This yields all real values of

x

that satisfy the original algebraic structure.

5

Filter for positive real solutions and calculate their sum.

The positive solutions are

1

,

2

, and

4

. Their sum is

1 + 2 + 4 = 7

.

The question specifically asks for the sum of only the positive real solutions.

Key Concept

Solving rational equations using algebraic substitution and factoring quadratic equations.

Alternative Method

Instead of using substitution directly, the equation can be solved by multiplying by the common denominator

(x^2 - 3x - 2)(x^2 - 3x)

to get a fourth-degree polynomial:

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

. Letting

z = x^2 - 3x

at this stage simplifies this expression to

2z^2 + 2(z-2)^2 = 5z(z-2)

, which avoids full expansion into a fourth-degree polynomial and leads to the same quadratic in

z

.

Estimated Time:3m 0s

Question 1790Question →

The table below shows the population of a certain plant species in two conservation areas in 2024 and 2025.

Conservation Area	2024 Population	2025 Population
Area P	$120$	$150$
Area Q	$160$	$180$

The percent increase in the plant population of Area P from 2024 to 2025 is how many percentage points greater than the percent increase in the plant population of Area Q from 2024 to 2025?

7.5\%

10.0\%

12.5\%

8.9\%

Show answer & explanation

Answer: $12.5\%$

Answer

The percent increase in the plant population of Area P is

12.5\%

greater than that of Area Q.

To find the difference in the percent increases, we first calculate the percent increase for each conservation area. The percent increase is calculated by dividing the change in population by the original 2024 population, then multiplying by 100. For Area P, the percent increase is

\frac{150 - 120}{120} \times 100 = 25\%

. For Area Q, the percent increase is

\frac{180 - 160}{160} \times 100 = 12.5\%

. Subtracting these two percentages yields a difference of

25\% - 12.5\% = 12.5\%

.

Step-by-Step Solution

1

Calculate the percent increase in the plant population for Area P from 2024 to 2025.

25\%

The initial population in 2024 was

120

and the population in 2025 was

150

. The increase is

150 - 120 = 30

. The percent increase is calculated relative to the 2024 base:

\frac{30}{120} \times 100 = 25\%

.

2

Calculate the percent increase in the plant population for Area Q from 2024 to 2025.

12.5\%

The initial population in 2024 was

160

and the population in 2025 was

180

. The increase is

180 - 160 = 20

. The percent increase is calculated relative to the 2024 base:

\frac{20}{160} \times 100 = 12.5\%

.

3

Calculate the difference between the two percent increases in percentage points.

12.5

percentage points

Subtract the percent increase of Area Q from the percent increase of Area P:

25\% - 12.5\% = 12.5\%

.

Key Concept

Percent change calculations require identifying the correct baseline (initial) value.

Question 1791Question →

Due to rising global temperatures, the thickness $T$ , in meters, of a mountain glacier is decreasing at a constant rate. A researcher models the thickness of the glacier using the equation $T = 48.5 - 0.12w$ , where $w$ represents the number of weeks since the start of a monitoring study. According to the model, by how many meters does the thickness of the glacier decrease every $50$ weeks?

Show answer & explanation

Answer: 6

Answer

The glacier's thickness decreases by 6 meters.

In the linear model

T = 48.5 - 0.12w

, the thickness

T

decreases by

0.12

meters for each increase of

1

in

w

(each week). Therefore, over a period of

50

weeks, the total decrease in thickness is

0.12 \times 50 = 6

meters.

Step-by-Step Solution

1

Identify the weekly rate of decrease from the slope of the equation.

0.12 meters per week

In the linear model

T = 48.5 - 0.12w

, the coefficient of

w

is

-0.12

. This slope represents the rate of change, indicating a decrease of

0.12

meters in thickness per week.

2

Calculate the cumulative decrease over 50 weeks.

6

Multiply the weekly rate of decrease (

0.12

meters per week) by the duration (

50

weeks) to obtain the total decrease:

0.12 \times 50 = 6

.

Key Concept

Interpreting the slope of a linear relationship in context

Estimated Time:1m 30s

Question 1792Question →

If $9^{2x - 1} = \frac{27^{x + 2}}{3}$ , what is the value of $x$ ?

3

7

8

9

Show answer & explanation

Answer: 7

Answer

The correct value of

x

is 7.

To solve the equation

9^{2x - 1} = \frac{27^{x + 2}}{3}

, write all bases as powers of 3:

9 = 3^2

and

27 = 3^3

. Substituting these into the equation gives

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

. Applying the power rule of exponents, we get

3^{4x - 2} = \frac{3^{3x + 6}}{3^1}

. Using the quotient rule of exponents on the right side, we subtract the exponent in the denominator (which is 1) from the exponent in the numerator:

3^{3x + 6 - 1} = 3^{3x + 5}

. Now we have

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

. Since the bases are the same, we set the exponents equal to each other:

4x - 2 = 3x + 5

. Solving for

x

by subtracting

3x

and adding 2 to both sides gives the correct value of 7.

Step-by-Step Solution

1

Express all bases as powers of 3.

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

To solve exponential equations with different bases, we rewrite them using a common base to apply exponent laws.

2

Apply the power of a power rule to simplify exponents.

3^{4x - 2} = \frac{3^{3x + 6}}{3^1}

Multiply exponents when raising a power to another power:

(a^m)^n = a^{m \cdot n}

.

3

Apply the quotient rule to simplify the right side of the equation.

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

Subtract the exponent of the denominator from the numerator when dividing:

\frac{a^m}{a^n} = a^{m-n}

.

4

Set the exponents equal to each other and solve the linear equation.

x = 7

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

4x - 2 = 3x + 5

.

Key Concept

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

Question 1793Question →

Which singular possessive pronoun correctly completes the passage? Type the correct word in the blank.

Fill in the blanks below

Discovered in 1610, Jupiter's moon Europa has long fascinated astronomers because of the vast liquid water ocean that lies beneath icy crust. This subsurface ocean, kept warm by tidal heating, makes the moon one of the most promising locations in the solar system to search for extraterrestrial life.

Show answer & explanation

Answer

its

The blank requires a singular possessive pronoun to modify the noun phrase 'icy crust' and refer back to the singular noun 'Europa'. The correct spelling is 'its'. Possessive pronouns in English do not contain apostrophes. The form 'it's' is a contraction meaning 'it is' or 'it has', which is grammatically incorrect in this context.

Step-by-Step Solution

1

Identify the pronoun that needs to be completed in the sentence.

The blank requires a word that modifies the noun phrase 'icy crust' to show that the crust belongs to Jupiter's moon Europa (a singular, non-human noun).

This establishes that we need a singular possessive pronoun.

2

Select the correct grammatical form for the singular possessive pronoun.

The correct form is 'its' (without an apostrophe).

In English, possessive pronouns do not use apostrophes, while contractions (like 'it's' for 'it is') do.

Key Concept

Possessive Pronouns vs. Contractions

Question 1794Question →

Located in modern-day Jordan, the ancient city of Petra contains hundreds of monumental tombs carved directly into red sandstone cliffs by Nabataean builders. These extraordinary rock-cut structures have fascinated archaeologists and travelers for centuries. The _______ elaborate facades, which seamlessly blend Hellenistic architectural elements with traditional Arabian designs, demonstrate both a sophisticated understanding of engineering and a distinct artistic vision.

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

structures

structure's

structures'

structure

Show answer & explanation

Answer: structures'

Answer

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

The plural possessive noun 'structures'' is correct because the passage refers to multiple rock-cut structures ('These extraordinary rock-cut structures') and indicates that the 'facades' belong to them. For regular plural nouns ending in '-s', the possessive is formed by adding only an apostrophe at the end.

Step-by-Step Solution

1

Identify the noun being modified and determine if possession is required.

The blank is followed by the noun phrase 'elaborate facades', which belong to the structures mentioned in the preceding sentence. Therefore, a possessive noun is required.

Establishing possession is necessary to determine whether the noun requires an apostrophe.

2

Determine the number (singular or plural) of the noun.

The preceding sentence refers to 'These extraordinary rock-cut structures', which establishes that there are multiple structures. Therefore, a plural noun is required.

Determining the number helps identify whether to use a singular or plural possessive form.

3

Form the plural possessive of the noun 'structure'.

To make the plural noun 'structures' possessive, add an apostrophe to the end of the word, resulting in 'structures''.

Adding an apostrophe after the plural '-s' is the standard rule for forming plural possessive nouns.

Key Concept

Plural and Possessive Nouns

Question 1795Question →

Last year, a gardener grew $80$ tomato plants. This year, the gardener grew $92$ tomato plants. What is the percent increase in the number of tomato plants from last year to this year?

Show answer & explanation

Answer: 15

Answer

The percent increase in the number of tomato plants is

15

.

To find the percent increase, subtract the original value from the new value to get the absolute increase:

92 - 80 = 12

. Next, divide this increase by the original value:

\frac{12}{80} = 0.15

. Finally, multiply by

100

to convert the decimal to a percentage, resulting in

15

.

Step-by-Step Solution

1

Subtract the original number of tomato plants from the new number of plants

12

plants

To find the absolute change in the quantity of plants.

2

Divide the change by the original number of plants

0.15

To find the ratio of the increase relative to the starting amount.

3

Multiply the ratio by

100

to convert it to a percentage

15

To express the ratio as a percentage value.

Key Concept

Calculating percent change from an initial value to a final value

Question 1796Question →

\begin{aligned} 4x - 3y &= 25 \\ -2x + 5y &= -9 \end{aligned}

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

Show answer & explanation

Answer: 6

Answer

6

To solve the system of equations, we can use the elimination method. First, multiply the second equation by 2 to align the coefficients of

x

:

2(-2x + 5y) = 2(-9) \implies -4x + 10y = -18

Next, add this new equation to the first equation to eliminate

x

:

(4x - 3y) + (-4x + 10y) = 25 + (-18)

7y = 7

y = 1

Substitute

y = 1

back into the second equation to solve for

x

:

-2x + 5(1) = -9

-2x + 5 = -9

-2x = -14

x = 7

Finally, calculate the value of the requested expression

x - y

:

x - y = 7 - 1 = 6

Thus, the correct response is 6.

Step-by-Step Solution

1

Multiply the second equation by 2 to prepare for the elimination of

x

.

-4x + 10y = -18

This creates coefficients for

x

in both equations that are additive opposites, allowing

x

to be eliminated when the equations are added.

2

Add the first equation and the modified second equation together to solve for

y

.

7y = 7

, which simplifies to

y = 1

Adding the equations eliminates the

x

terms and leaves a single-variable equation in terms of

y

.

3

Substitute

y = 1

back into one of the original equations to solve for

x

.

-2x + 5(1) = -9

, which simplifies to

-2x = -14

, yielding

x = 7

Now that the value of

y

is known, it can be substituted into either equation to find the corresponding value of

x

.

4

Calculate the value of the expression

x - y

.

7 - 1 = 6

The question asks specifically for the value of the difference

x - y

, so we subtract the value of

y

from the value of

x

.

Key Concept

Solving systems of linear equations using the elimination method and evaluating linear combinations of the solutions.

Estimated Time:1m 30s

Question 1797Question →

The function $f$ is defined by $f(x) = (x - 4)(x - 2)(x + k)$ , where $k$ is a constant. If the $y$ -intercept of the graph of $y = f(x)$ in the $xy$ -plane is $(0, 24)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 3

Answer

3

The

y

-intercept of the graph of

y = f(x)

is the point where

x = 0

. Since the

y

-intercept is

(0, 24)

, it follows that

f(0) = 24

. Substituting

0

for

x

in the equation for

f(x)

gives

f(0) = (0 - 4)(0 - 2)(0 + k) = (-4)(-2)(k) = 8k

. Setting this equal to the

y

-value of the intercept yields

8k = 24

. Dividing both sides of the equation by

8

gives

k = 3

.

Step-by-Step Solution

1

Use the definition of the

y

-intercept to find the value of

f(0)

f(0) = 24

The

y

-intercept of a graph is the point where the graph crosses the

y

-axis, corresponding to

x = 0

. Given the point

(0, 24)

,

f(0)

must equal

24

.

2

Evaluate the polynomial at

x = 0

in terms of

k

f(0) = 8k

Substituting

0

for

x

in

f(x) = (x - 4)(x - 2)(x + k)

gives

f(0) = (-4)(-2)(k)

, which simplifies to

8k

.

3

Set the evaluated expression equal to the

y

-intercept value and solve for

k

k = 3

Equating

8k

to

24

and dividing both sides by

8

yields

k = 3

.

Key Concept

Evaluating a factored polynomial function at

x = 0

determines its

y

-intercept. Using a given

y

-intercept allows solving for unknown coefficients or constants within the factors of the polynomial.

Question 1798Question →

In the $xy$ -plane, the graph of the linear equation $kx - 3y = 18$ , where $k$ is a constant, has a $y$ -intercept of $(0, b)$ and an $x$ -intercept of $(a, 0)$ , where $a$ and $b$ are constants. If $a - b = 15$ , what is the value of $k$ ?

Show answer & explanation

Answer: 2

Answer

2

To find the value of

k

, we first determine the intercepts of the linear equation

kx - 3y = 18

. The

y

-intercept

(0, b)

occurs where

x = 0

. Substituting

x = 0

gives

-3b = 18

, which solves to

b = -6

. Next, we use the given relation

a - b = 15

. Substituting

b = -6

yields

a - (-6) = 15

, or

a + 6 = 15

, which gives

a = 9

. The

x

-intercept is therefore

(9, 0)

. Since this point lies on the line, we substitute

x = 9

and

y = 0

into the original equation:

k(9) - 3(0) = 18

. This simplifies to

9k = 18

, which gives

k = 2

.

Step-by-Step Solution

1

Find the

y

-coordinate of the

y

-intercept (

b

) by setting

x = 0

in the equation

kx - 3y = 18

.

b = -6

The

y

-intercept occurs where the graph crosses the

y

-axis, which corresponds to

x = 0

.

2

Use the equation

a - b = 15

and the value of

b = -6

to solve for

a

.

a = 9

Substituting

b = -6

into

a - b = 15

gives

a - (-6) = 15

, which simplifies to

a + 6 = 15

.

3

Find the value of

k

by substituting the

x

-intercept

(9, 0)

into the equation

kx - 3y = 18

.

k = 2

Since

(a, 0) = (9, 0)

is the

x

-intercept, it must satisfy the equation of the line.

Key Concept

Determining intercepts of a linear equation in two variables and using them to find unknown constants.

Question 1799Question →

In the quadratic equation $x^2 + kx + (k + 3) = 0$ , $k$ is a constant. If the equation has exactly one real solution, which of the following is a possible value of $k$ ?

-6

3

6

2

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6. For the quadratic equation

x^2 + kx + (k + 3) = 0

to have exactly one real solution, its discriminant must be equal to zero. Setting the discriminant

b^2 - 4ac

to zero gives

k^2 - 4(1)(k + 3) = 0

, which simplifies to

k^2 - 4k - 12 = 0

. Factoring this equation yields

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

, meaning the possible values of

k

are

6

and

-2

. Among the choices, 6 is the only possible value listed.

Step-by-Step Solution

1

Identify the coefficients of the given quadratic equation

x^2 + kx + (k + 3) = 0

.

The coefficients are

a = 1

,

b = k

, and

c = k + 3

.

To apply the discriminant formula, we need to identify the standard form coefficients

a

,

b

, and

c

.

2

Set the discriminant of the quadratic equation to zero.

The discriminant is

D = b^2 - 4ac

. Setting

D = 0

gives

k^2 - 4(1)(k + 3) = 0

, which simplifies to

k^2 - 4k - 12 = 0

.

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

3

Solve the quadratic equation

k^2 - 4k - 12 = 0

for

k

.

Factoring the quadratic yields

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

. Therefore, the possible values of

k

are

6

and

-2

.

Solving the equation reveals the values of the constant

k

that satisfy the condition.

Key Concept

Using the discriminant of a quadratic equation to determine the number of real solutions.

Alternative Method

Instead of factoring, the quadratic formula can be used to solve

k^2 - 4k - 12 = 0

, where

k = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-12)}}{2(1)} = \frac{4 \pm \sqrt{16 + 48}}{2} = \frac{4 \pm 8}{2}

, giving

k = 6

or

k = -2

.

Estimated Time:1m 30s

Question 1800Question →

In the equation below,

x

is a real number.

\sqrt{7 - 6x} = x + 8

What is the value of

x + 5

?

8

-14

2

-3

Show answer & explanation

Answer: 2

Answer

The correct answer is the value 2.

Substituting the only valid solution of the radical equation, which is

-3

, into the expression

x + 5

yields

2

.

Step-by-Step Solution

1

Square both sides of the equation to eliminate the radical sign.

7 - 6x = (x + 8)^2

, which expands to

7 - 6x = x^2 + 16x + 64

.

Squaring both sides of a radical equation allows it to be rewritten as a standard polynomial equation.

2

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

x^2 + 22x + 57 = 0

Grouping all terms on one side of the equation puts it in standard quadratic form.

3

Factor the quadratic equation to find potential solutions for

x

.

(x + 19)(x + 3) = 0

, which gives the potential solutions

x = -19

and

x = -3

.

Factoring allows us to solve the quadratic equation easily.

4

Test the potential solutions in the original equation to identify any extraneous solutions.

Substituting

x = -3

yields

\sqrt{7 - 6(-3)} = -3 + 8 \Rightarrow \sqrt{25} = 5

, which is true. Substituting

x = -19

yields

\sqrt{7 - 6(-19)} = -19 + 8 \Rightarrow \sqrt{121} = -11

, which is false because a principal square root must be non-negative. Therefore,

x = -3

is the only valid solution.

Squaring both sides can introduce extraneous solutions that must be discarded.

5

Substitute the valid solution

x = -3

into the target expression

x + 5

.

-3 + 5 = 2

The question asks for the value of the expression

x + 5

.

Key Concept

Solving radical equations by squaring and checking for extraneous solutions.

Answer

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Key Concept

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Common Mistakes

Alternative Method

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Common Mistakes

Alternative Method

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Alternative Method

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