All practice questions

The functions $f$ and $g$ are defined for the integers $-2 \leq x \leq 3$ by the table below.

$x$	$f(x)$	$g(x)$
$-2$	$3$	$1$
$-1$	$2$	$3$
$0$	$-2$	$-1$
$1$	$0$	$2$
$2$	$-1$	$0$
$3$	$1$	$2$

What is the value of $x$ for which $g(f(x)) = -1$ ?

-2

-1

0

1

3

Show answer & explanation

Answer: $1$

Answer

The value of

x

is

1

.

To solve

g(f(x)) = -1

, we work from the outside in. First, we find the input to the outer function

g

that results in an output of

-1

. Looking at the table,

g(0) = -1

, which means the inner expression

f(x)

must equal

0

. Next, we find the input to

f

that results in an output of

0

. Looking at the table,

f(1) = 0

. Therefore, the value of

x

that satisfies the equation is

1

.

Step-by-Step Solution

1

Identify the outer function in the composition

g(f(x)) = -1

.

The outer function is

g

, so we must first find an input

u

such that

g(u) = -1

.

This allows us to break down the composition into two simpler function evaluation steps.

2

Look at the column for

g(x)

in the table to find where the output is

-1

.

The table shows

g(0) = -1

, meaning the input to

g

must be

0

. Therefore,

f(x) = 0

.

This establishes the required output value for the inner function

f(x)

.

3

Look at the column for

f(x)

in the table to find where the output is

0

.

The table shows

f(1) = 0

, meaning the input

x

must be

1

.

This provides the final value of the variable

x

that satisfies the original equation.

Key Concept

Evaluating composite functions using a table of values

Estimated Time:1m 0s

Question 2162Question →

Which of the following inequality expressions represents the complete solution set for $p$ in the inequality $15 - 3|2p + 5| < 6$ ?

p > -1

-4 < p < -1

p < -4

or

p > -1

-\frac{11}{4} < p < -\frac{9}{4}

p < -1

Show answer & explanation

Answer: $p < -4$ or $p > -1$

Answer

p < -4

or

p > -1

The correct solution is obtained by first isolating the absolute value expression. Subtracting 15 from both sides of the inequality

15 - 3|2p + 5| < 6

yields

-3|2p + 5| < -9

. Dividing both sides by

-3

and reversing the inequality sign gives

|2p + 5| > 3

. This absolute value inequality splits into two cases:

2p + 5 > 3

(which simplifies to

p > -1

) or

2p + 5 < -3

(which simplifies to

p < -4

). Combining these yields the complete solution set

p < -4

or

p > -1

.

Step-by-Step Solution

1

Subtract 15 from both sides of the inequality

15 - 3|2p + 5| < 6

.

-3|2p + 5| < -9

To isolate the absolute value term, first subtract the constant term from both sides.

2

Divide both sides of

-3|2p + 5| < -9

by

-3

and reverse the inequality sign.

|2p + 5| > 3

Dividing an inequality by a negative number requires reversing the direction of the inequality sign.

3

Solve the absolute value inequality

|2p + 5| > 3

by setting up two separate inequalities.

2p + 5 > 3

or

2p + 5 < -3

An absolute value inequality of the form

|u| > c

splits into

u > c

or

u < -c

.

4

Solve each linear inequality for

p

.

p > -1

or

p < -4

Subtract 5 from both sides and then divide by 2 for both inequalities to isolate

p

.

Key Concept

Solving absolute value inequalities involving algebraic manipulation and reversing the inequality sign when multiplying or dividing by a negative number.

Question 2163Question →

The table below shows corresponding values of $x$ and $y$ for a linear relationship in the standard $(x, y)$ coordinate plane.

$x$	$y$
$-3$	$13$
$1$	$5$
$5$	$a$
$b$	$-7$

What is the value of $a + b$ ?

4

8

-8

28

-2

Show answer & explanation

Answer: 4

Answer

4

The correct answer is

4

. The slope of the line is determined to be

-2

using the points

(-3, 13)

and

(1, 5)

. This gives the equation of the line as

y = -2x + 7

. Substituting

x = 5

yields

a = -3

, and substituting

y = -7

yields

b = 7

. Adding these values together gives

4

.

Step-by-Step Solution

1

Calculate the slope of the linear relationship using the points

(-3, 13)

and

(1, 5)

.

Slope

m = -2

Since the relationship is linear, the slope is constant and is given by

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

.

2

Determine the equation of the line using the point-slope formula with point

(1, 5)

and slope

-2

.

Equation is

y = -2x + 7

Using

y - y_1 = m(x - x_1)

allows us to find the relation between any

x

and

y

coordinate on this line.

3

Find the value of

a

by substituting

x = 5

into the linear equation.

a = -3

The table indicates that when

x = 5

, the

y

-value is

a

.

4

Find the value of

b

by substituting

y = -7

into the linear equation and solving for

x

.

b = 7

The table indicates that when

y = -7

, the

x

-value is

b

.

5

Sum the values of

a

and

b

.

a + b = 4

The question asks for the sum

a + b

.

Key Concept

Linear Equations and Graphing

Estimated Time:1m 30s

Question 2164Question →

If $x$ and $y$ are non-zero real numbers such that the expression $\frac{(x^2 y^a)^3}{x^5 y^{-2}}$ is equivalent to $x^b y^{14}$ for some integers $a$ and $b$ , what is the value of $a + b$ ?

Show answer & explanation

Answer: 5

Answer

The value of

a + b

is

5

.

Simplifying the expression

\frac{(x^2 y^a)^3}{x^5 y^{-2}}

using the exponent rules yields

x^{6-5} y^{3a-(-2)} = x^1 y^{3a+2}

. Equating this to

x^b y^{14}

shows that

b = 1

and

3a + 2 = 14

. Solving for

a

gives

a = 4

. Thus, the sum

a + b

is

4 + 1 = 5

.

Step-by-Step Solution

1

Simplify the numerator of the given expression.

x^6 y^{3a}

Applying the power of a product rule

(uv)^n = u^n v^n

and the power of a power rule

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

to

(x^2 y^a)^3

results in

(x^2)^3 (y^a)^3 = x^6 y^{3a}

.

2

Simplify the quotient by subtracting exponents with the same base.

x^1 y^{3a+2}

Using the quotient rule

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

, the base

x

term becomes

x^{6-5} = x^1

, and the base

y

term becomes

y^{3a - (-2)} = y^{3a+2}

.

3

Equate the exponents of like bases to find the values of

a

and

b

.

b = 1

and

a = 4

Comparing

x^1 y^{3a+2}

to

x^b y^{14}

gives

b = 1

and

3a + 2 = 14

. Solving

3a + 2 = 14

yields

3a = 12

, which simplifies to

a = 4

.

4

Sum the values of

a

and

b

.

5

Adding

a

and

b

yields

4 + 1 = 5

.

Key Concept

Applying properties of exponents, including the power of a product, power of a power, and quotient rules, to simplify algebraic expressions.

Estimated Time:1m 30s

Question 2165Question →

In 1912, German meteorologist Alfred Wegener proposed the theory of continental drift, arguing that Earth’s continents were once joined in a single landmass called Pangaea before drifting apart. To support this radical claim, Wegener constructed a multi-layered argument, sequencing distinct categories of geological and biological evidence.

Wegener began his argument with a simple, visual observation: the puzzle-like fit of continental coastlines. He pointed out that the eastern coast of South America and the western coast of Africa appeared as if they could lock together. Recognizing that coastal erosion made this fit imperfect, he refined his observation by comparing the edges of the stable continental shelves rather than the shoreline boundaries, revealing an even more precise match.

To demonstrate that this fit was not coincidental, Wegener next introduced geological evidence. He identified identical rock sequences, mountain ranges, and coal deposits on opposite sides of the Atlantic Ocean. For instance, the Appalachian mountain system of North America lined up perfectly with the Caledonian mountains of Scotland and Scandinavia. Wegener argued that these structures could only have formed continuously if the landmasses were connected.

Wegener then bolstered his theory by citing paleontological data. He pointed to fossil remains of identical species found in locations separated by vast oceans. The Mesosaurus, a freshwater reptile incapable of swimming across saltwater, was found only in eastern South America and western Africa. Wegener argued that these distributions proved the existence of land bridges or, more likely, contiguous continents, as the species could not have crossed the modern Atlantic.

Finally, Wegener incorporated paleoclimatic data, showing that ancient glacial deposits and tropical coal beds lay in regions that currently have incompatible climates. By showing that equatorial regions once experienced glaciation while northern regions were tropical, he argued that the continents had shifted relative to the equator. Although Wegener’s theory was initially rejected because he could not identify the physical mechanism driving the movement, his careful sequencing of diverse evidence laid the groundwork for modern plate tectonics.

Based on the passage, arrange the pieces of evidence Alfred Wegener uses to support his theory of continental drift in the order they are introduced to develop his argument.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence starts with the observation of the matching continental shelves, followed by matching mountain ranges and rock sequences, then the distribution of identical fossils, and concludes with the analysis of ancient climates.

The passage builds Alfred Wegener's argument by first presenting the visual fit of continental shelves in the second paragraph, then matching geological formations in the third paragraph, followed by identical fossil distributions in the fourth paragraph, and finally paleoclimatic evidence in the fifth paragraph.

Step-by-Step Solution

1

Locate the introduction of the first piece of evidence in the text.

The second paragraph describes Wegener beginning his argument with the puzzle-like fit of continental coastlines and shelves.

This establishes the starting point of the argument's evidence sequence.

2

Find the second category of evidence presented by the author.

The third paragraph introduces geological evidence, specifically identical rock sequences and matching mountain systems like the Appalachians and Caledonians.

This shows the next logical step in Wegener's argument to prove the fit was not coincidental.

3

Find the third category of evidence in the text.

The fourth paragraph describes the paleontological data, citing identical fossils of the Mesosaurus on opposite sides of the ocean.

This adds biological proof to the physical alignment of landmasses.

4

Identify the final category of evidence used to complete the argument.

The fifth paragraph details the paleoclimatic data, mentioning glacial deposits and tropical coal beds in unexpected locations.

This serves as the final supportive pillar of Wegener's theory of continental movement.

Key Concept

Identifying the organizational progression and sequencing of evidence within a passage's argument.

Question 2166Question →

In the standard $(x, y)$ coordinate plane, a line segment has endpoints at $(3, -4)$ and $(9, 8)$ . What is the $y$ -coordinate of the midpoint of this line segment?

Show answer & explanation

Answer: 2

Answer

The

y

-coordinate of the midpoint is

2

.

The

y

-coordinate of the midpoint is calculated by finding the average of the

y

-coordinates of the endpoints:

\frac{y_1 + y_2}{2}

. Substituting

y_1 = -4

and

y_2 = 8

gives

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

.

Step-by-Step Solution

1

Identify the

y

-coordinates of the two given endpoints

(3, -4)

and

(9, 8)

.

y_1 = -4

and

y_2 = 8

The midpoint formula relies on the coordinates of the endpoints.

2

Calculate the average of the

y

-coordinates using the formula

y_m = \frac{y_1 + y_2}{2}

.

y_m = \frac{-4 + 8}{2} = 2

The

y

-coordinate of a midpoint is the arithmetic mean of the

y

-coordinates of the endpoints.

Key Concept

Midpoint Formula

Estimated Time:45s

Question 2167Question →

A hyperbola in the standard $(x, y)$ coordinate plane is defined by the equation:

16x^2 - 9y^2 - 32x + 36y - 164 = 0

What are the coordinates of the foci of this hyperbola?

(1, 7)

and

(1, -3)

(6, -2)

and

(-4, -2)

(6, 2)

and

(-4, 2)

(4, -2)

and

(-6, -2)

(1 + \sqrt{7}, 2)

and

(1 - \sqrt{7}, 2)

Show answer & explanation

Answer: $(6, 2)$ and $(-4, 2)$

Answer

The foci of the hyperbola are

(6, 2)

and

(-4, 2)

.

The correct answer is the set of coordinates

(6, 2)

and

(-4, 2)

. After rewriting the hyperbola's equation in standard form by completing the square, we obtain

\frac{(x - 1)^2}{9} - \frac{(y - 2)^2}{16} = 1

. The center of the hyperbola is

(1, 2)

. Since the

x^2

term is positive, it has a horizontal transverse axis. The distance from the center to each focus,

c

, satisfies

c^2 = a^2 + b^2 = 9 + 16 = 25

, so

c = 5

. Adding and subtracting this focal distance from the

x

-coordinate of the center yields the foci at

(1 + 5, 2) = (6, 2)

and

(1 - 5, 2) = (-4, 2)

.

Step-by-Step Solution

1

Group the

x

-terms and

y

-terms and move the constant to the right side of the equation.

16(x^2 - 2x) - 9(y^2 - 4y) = 164

Grouping the terms allows us to prepare for completing the square for both variables.

2

Complete the square for

x^2 - 2x

by adding

1

inside the parentheses, and for

y^2 - 4y

by adding

4

inside the parentheses. Add the corresponding balanced quantities to the right side:

16(1) = 16

and

-9(4) = -36

.

16(x^2 - 2x + 1) - 9(y^2 - 4y + 4) = 164 + 16 - 36

16(x - 1)^2 - 9(y - 2)^2 = 144

This rewrites the quadratic expressions into perfect square binomials.

3

Divide both sides of the equation by

144

to express it in standard form.

\frac{(x - 1)^2}{9} - \frac{(y - 2)^2}{16} = 1

The standard form of a horizontal hyperbola is

\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1

, which lets us identify the center,

a

, and

b

directly.

4

Identify the center

(h, k)

,

a^2

, and

b^2

, then calculate the focal distance

c

using the relation

c^2 = a^2 + b^2

.

Center is

(1, 2)

.

a^2 = 9

and

b^2 = 16

.

c^2 = 9 + 16 = 25 \implies c = 5

.

Foci are located at a distance of

c

from the center along the transverse axis.

5

Determine the coordinates of the foci by shifting the

x

-coordinate of the center by

\pm c

since the transverse axis is horizontal.

Foci coordinates are

(1 \pm 5, 2)

, which gives

(6, 2)

and

(-4, 2)

.

Adding and subtracting

c

from the center's

x

-coordinate gives the locations of the two foci.

Key Concept

Rewriting a hyperbola equation in standard form by completing the square and finding its foci.

Question 2168Question →

In the standard $(x,y)$ coordinate plane, the set of all points equidistant from the line $3x - 4y = 8$ and the line $5x + 12y = 13$ consists of two perpendicular lines. What is the slope of the line in this set that has a positive slope?

-8

\frac{2}{19}

\frac{4}{7}

\frac{1}{8}

8

Show answer & explanation

Answer: $\frac{1}{8}$

Answer

The slope of the line with a positive slope is 1/8.

By using the point-to-line distance formula, the equidistant relationship

\frac{|3x - 4y - 8|}{5} = \frac{|5x + 12y - 13|}{13}

yields two linear equations:

14x - 112y - 39 = 0

and

64x + 8y - 169 = 0

. The slopes of these lines are 1/8 and -8, respectively. The line with the positive slope has a slope of 1/8.

Step-by-Step Solution

1

Set up the distance formula from a point

(x,y)

to both given lines.

The distance

d_1

to the first line is

\frac{|3x - 4y - 8|}{\sqrt{3^2 + (-4)^2}} = \frac{|3x - 4y - 8|}{5}

. The distance

d_2

to the second line is

\frac{|5x + 12y - 13|}{\sqrt{5^2 + 12^2}} = \frac{|5x + 12y - 13|}{13}

.

Points equidistant from both lines must satisfy

d_1 = d_2

.

2

Equate the two distance expressions to represent the equidistant relationship.

\frac{|3x - 4y - 8|}{5} = \frac{|5x + 12y - 13|}{13}

This represents the geometric condition of being equidistant from both lines.

3

Solve for Case 1 where the expressions inside the absolute values have the same sign.

13(3x - 4y - 8) = 5(5x + 12y - 13) \Rightarrow 39x - 52y - 104 = 25x + 60y - 65 \Rightarrow 14x - 112y - 39 = 0

.

One of the two bisecting lines is found when the signs match.

4

Solve for Case 2 where the expressions inside the absolute values have opposite signs.

13(3x - 4y - 8) = -5(5x + 12y - 13) \Rightarrow 39x - 52y - 104 = -25x - 60y + 65 \Rightarrow 64x + 8y - 169 = 0

.

The other bisecting line is found when the signs are opposite.

5

Find the slope of each resulting linear equation to identify the positive one.

For

14x - 112y - 39 = 0

, the slope is

-\frac{14}{-112} = \frac{1}{8}

. For

64x + 8y - 169 = 0

, the slope is

-\frac{64}{8} = -8

.

The slope of a line in the standard form

Ax + By + C = 0

is given by

-\frac{A}{B}

.

Key Concept

The set of points equidistant from two intersecting lines forms two perpendicular lines representing the angle bisectors of the original lines, which can be determined by equating their point-to-line distance formulas.

Alternative Method

Instead of using the distance formula, one could find the angle of inclination of each line using trigonometry:

\theta_1 = \arctan(3/4)

and

\theta_2 = \arctan(-5/12)

. The angle bisectors have inclinations at the average of these two angles plus or minus 90 degrees. Converting back to slopes using

m = \tan(\theta)

yields the same results.

Estimated Time:3m 0s

Question 2169Question →

A botanist is monitoring the heights of two bamboo plants. Plant A is $20$ inches tall and grows at a constant rate of $1.5$ inches per day. Plant B is $12$ inches tall and grows at a constant rate of $2.5$ inches per day. After how many days will Plant B be exactly $6$ inches taller than Plant A?

Show answer & explanation

Answer: 14

Answer

The correct answer is 14 days.

The correct answer is

14

days because when we translate the relationship, we get the equation

12 + 2.5d = 20 + 1.5d + 6

. Solving this equation yields

d = 14

.

Step-by-Step Solution

1

Define the variable

d

as the number of days and write the expressions for the heights of both plants.

Plant A's height is

20 + 1.5d

inches, and Plant B's height is

12 + 2.5d

inches.

To represent the growth of each plant algebraically over time.

2

Set up an equation representing that Plant B's height is

6

inches more than Plant A's height.

12 + 2.5d = (20 + 1.5d) + 6

To translate the verbal relationship into a mathematical equation.

3

Simplify the equation and solve for

d

.

d = 14

To find the number of days that satisfies the given condition.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:1m 15s

Question 2170Question →

A fitness tracker records a user's daily steps. On Monday, the user walked a certain number of steps. On Tuesday, they walked $1.5$ times the number of steps they walked on Monday. On Wednesday, they walked $2,500$ fewer steps than they did on Tuesday. If the user walked a total of $21,500$ steps over these three days, how many steps did they walk on Monday?

4,750

6,000

6,500

8,000

9,000

Show answer & explanation

Answer: 6,000

Answer

6,000

The correct answer is

6,000

. By letting

x

represent the steps on Monday, we express Tuesday's steps as

1.5x

and Wednesday's steps as

1.5x - 2,500

. Summing these expressions gives the equation

x + 1.5x + 1.5x - 2,500 = 21,500

. Combining like terms yields

4x - 2,500 = 21,500

. Adding

2,500

to both sides gives

4x = 24,000

, and dividing by

4

results in

x = 6,000

.

Step-by-Step Solution

1

Define the variable and write expressions for each day's steps.

Let

x

be the number of steps walked on Monday. Tuesday's steps are

1.5x

, and Wednesday's steps are

1.5x - 2,500

.

This translates the word problem statements into algebraic terms using a single variable.

2

Set up the linear equation representing the total steps.

x + 1.5x + (1.5x - 2,500) = 21,500

The sum of the steps over the three days must equal the given total of

21,500

steps.

3

Combine like terms and solve for

x

.

4x - 2,500 = 21,500 \Rightarrow 4x = 24,000 \Rightarrow x = 6,000

Simplifying the equation isolates the variable to find Monday's step count.

Key Concept

Formulating and solving a single-variable linear equation from a word problem context.

Estimated Time:1m 15s

Question 2171Question →

A hyperbola in the standard $(x, y)$ coordinate plane is defined by the equation $9x^2 - 16y^2 - 36x - 32y - 124 = 0$ . One of the foci of this hyperbola is located at the point $(f, -1)$ , where $f > 0$ . What is the value of $f$ ?

Show answer & explanation

Answer: 7

Answer

7

Completing the square transforms the equation into the standard form of a horizontal hyperbola,

\frac{(x - 2)^2}{16} - \frac{(y + 1)^2}{9} = 1

, which has its center at

(2, -1)

with

a^2 = 16

and

b^2 = 9

. The distance to the foci is

c = \sqrt{a^2 + b^2} = \sqrt{16 + 9} = 5

. Since the transverse axis is horizontal, the foci are located at

(2 \pm 5, -1)

, which are

(-3, -1)

and

(7, -1)

. Given the constraint that

f > 0

, the positive x-coordinate of the focus is 7.

Step-by-Step Solution

1

Group the terms and prepare to complete the square.

9(x^2 - 4x) - 16(y^2 + 2y) = 124

Grouping the variables helps isolate the quadratic expressions for completing the square.

2

Complete the square for both the

x

and

y

terms.

9(x - 2)^2 - 16(y + 1)^2 = 144

To complete the square for

x^2 - 4x

, add 4 inside the first parentheses, adding

9 \times 4 = 36

to the right side. To complete the square for

y^2 + 2y

, add 1 inside the second parentheses, which subtracts

16 \times 1 = 16

from the right side because of the leading negative coefficient. This leaves the right side as

124 + 36 - 16 = 144

.

3

Divide both sides of the equation by the constant to find the standard form.

\frac{(x - 2)^2}{16} - \frac{(y + 1)^2}{9} = 1

Dividing by 144 puts the equation in the standard horizontal hyperbola form:

\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1

.

4

Find the distance

c

from the center to the foci.

c = 5

For a hyperbola, the relationship between the semi-axes and the focal distance is

c = \sqrt{a^2 + b^2}

. Substituting

a^2 = 16

and

b^2 = 9

gives

c = \sqrt{16 + 9} = 5

.

5

Determine the coordinates of the foci and extract the value of

f

.

f = 7

The center of the hyperbola is

(h, k) = (2, -1)

. The foci are located at

(h \pm c, k) = (2 \pm 5, -1)

, which corresponds to the points

(-3, -1)

and

(7, -1)

. Since the problem states

f > 0

, the target focus must be

(7, -1)

, meaning

f = 7

.

Key Concept

Converting a general hyperbola equation into standard form to calculate focal points

Question 2172Question →

A triangle has a vertex located at the point with coordinates $(7, -2)$ . If the triangle is translated $4$ units to the left and $5$ units up, what are the coordinates of this vertex after the translation?

(3, 3)

(11, 3)

(3, -7)

(11, -7)

(-6, 12)

Show answer & explanation

Answer: $(3, 3)$

Answer

(3, 3)

Applying a translation of

4

units left means subtracting

4

from the

x

-coordinate of the point

(7, -2)

, resulting in

7 - 4 = 3

. Translating

5

units up means adding

5

to the

y

-coordinate, resulting in

-2 + 5 = 3

. Therefore, the new coordinates of the vertex are

(3, 3)

.

Step-by-Step Solution

1

Identify the horizontal translation and apply it to the

x

-coordinate.

Since the translation is

4

units to the left, we subtract

4

from the initial

x

-coordinate of

7

:

7 - 4 = 3

.

Moving left along the

x

-axis decreases the coordinate value.

2

Identify the vertical translation and apply it to the

y

-coordinate.

Since the translation is

5

units up, we add

5

to the initial

y

-coordinate of

-2

:

-2 + 5 = 3

.

Moving up along the

y

-axis increases the coordinate value.

3

Combine the new coordinates into an ordered pair.

The final coordinates are

(3, 3)

.

The coordinates

(x', y')

form the final position of the vertex.

Key Concept

Translating points in the coordinate plane by adding or subtracting values from their coordinates.

Estimated Time:45s

Question 2173Question →

For all positive real numbers $x$ and $y$ , which of the following expressions is equivalent to $\log_3(9x^4 y^{-2})$ ?

9 + 4\log_3(x) - 2\log_3(y)

2 + 4\log_3(x) + 2\log_3(y)

2 + 4\log_3(x) - 2\log_3(y)

\frac{2 + 4\log_3(x)}{2\log_3(y)}

2 + \log_3(4x) - \log_3(2y)

Show answer & explanation

Answer: $2 + 4\log_3(x) - 2\log_3(y)$

Answer

2 + 4\log_3(x) - 2\log_3(y)

The correct answer is correct because applying the product and power properties of logarithms allows us to expand the argument

9x^4 y^{-2}

. Specifically, the expression can be written as

\log_3(9) + \log_3(x^4) + \log_3(y^{-2})

. Since

3^2 = 9

, the term

\log_3(9)

evaluates to

2

. Applying the power rule

\log_b(M^k) = k\log_b(M)

to the remaining terms yields

4\log_3(x)

and

-2\log_3(y)

. Combining these terms results in the equivalent expression.

Step-by-Step Solution

1

Use the product rule of logarithms,

\log_b(MN) = \log_b(M) + \log_b(N)

, to separate the factors in the argument.

\log_3(9x^4 y^{-2}) = \log_3(9) + \log_3(x^4) + \log_3(y^{-2})

This allows the expression to be broken down into individual terms for each base and variable.

2

Evaluate the constant logarithm

\log_3(9)

. Since

3^2 = 9

, this simplifies to

2

.

\log_3(9) = 2

To simplify the numerical term to its integer value.

3

Use the power rule of logarithms,

\log_b(M^k) = k\log_b(M)

, to bring the exponents of the variables to the front of each logarithm.

\log_3(x^4) = 4\log_3(x)

and

\log_3(y^{-2}) = -2\log_3(y)

To isolate the variable terms inside simpler logarithmic expressions.

4

Combine all the simplified parts into a single expression.

2 + 4\log_3(x) - 2\log_3(y)

To write the final expanded equivalent expression.

Key Concept

Logarithmic properties, including product, power, and evaluation of basic log terms.

Alternative Method

Substitute test values for the variables. For example, let

x = 3

and

y = 3

. The original expression evaluates to

\log_3(9 \cdot 3^4 \cdot 3^{-2}) = \log_3(81) = 4

. Substituting

x = 3

and

y = 3

into the correct expression yields

2 + 4\log_3(3) - 2\log_3(3) = 2 + 4(1) - 2(1) = 4

, which matches the original expression's value.

Estimated Time:1m 0s

Question 2174Question →

A local gym charges a monthly membership fee of $12.50$ dollars plus a one-time registration fee of $b$ dollars. The total cost, $y$ , in dollars, for $x$ months is given by the linear equation $y = 12.50x + b$ . If a member paid a total of $100$ dollars for $6$ months of membership, what is the registration fee, $b$ , in dollars?

75.00

81.50

25.00

175.00

37.50

Show answer & explanation

Answer: $25.00$

Answer

25.00

The correct answer is found by substituting the total cost of

100

for

y

and the number of months,

6

, for

x

in the equation

y = 12.50x + b

, giving

100 = 12.50(6) + b

. Multiplying

12.50

by

6

yields

75

, and subtracting

75

from

100

gives the registration fee

b = 25

.

Step-by-Step Solution

1

Identify and substitute the given values into the linear equation.

Substitute

y = 100

(total cost) and

x = 6

(number of months) into

y = 12.50x + b

to get

100 = 12.50(6) + b

.

This sets up the equation with one variable, allowing us to solve for the unknown registration fee

b

.

2

Calculate the total cost of the monthly fees.

12.50 \times 6 = 75

. The equation becomes

100 = 75 + b

.

We multiply the monthly fee rate by the number of months to find the portion of the total cost spent on monthly fees.

3

Solve for the registration fee

b

.

b = 100 - 75 = 25

.

Subtracting the total monthly fees from the total cost isolates

b

to find the registration fee.

Key Concept

Solving linear equations in slope-intercept form by substituting known values to find the y-intercept.

Question 2175Question →

A certain radioactive isotope decays according to the formula $N(t) = N_0 \cdot 2^{-t/8}$ , where $N_0$ is the initial amount of the isotope and $t$ is the time in years. If a sample initially contains $120$ grams of the isotope, how many years will it take for the amount of the isotope to decay to $15$ grams?

Show answer & explanation

Answer: 24

Answer

24

The correct answer is 24 because substituting the initial value of 120 and the final value of 15 into the equation yields

15 = 120 \cdot 2^{-t/8}

. Dividing both sides by 120 gives

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

, which can be rewritten as

2^{-3} = 2^{-t/8}

. Setting the exponents equal to each other gives

-3 = -t/8

, and solving for

t

yields 24.

Step-by-Step Solution

1

Substitute the given values into the decay formula.

15 = 120 \cdot 2^{-t/8}

The initial amount

N_0

is

120

grams and the final amount

N(t)

is

15

grams.

2

Isolate the exponential term.

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

Divide both sides by

120

. Since

\frac{15}{120}

reduces to

\frac{1}{8}

, this isolates the base

2

term.

3

Write the fraction as a power with base 2.

2^{-3} = 2^{-t/8}

Using exponent rules,

\frac{1}{8} = \frac{1}{2^3} = 2^{-3}

.

4

Equate the exponents and solve for

t

.

t = 24

Since the bases are equal, the exponents must be equal, so

-3 = -\frac{t}{8}

which gives

t = 24

.

Key Concept

Solving exponential equations using a common base.

Question 2176Question →

In the standard $(x, y)$ coordinate plane, the line defined by the equation $3x - 4y = 24$ intersects the $x$ -axis at a certain point. What is the $x$ -coordinate of this intersection point?

Show answer & explanation

Answer: 8

Answer

The

x

-coordinate of the intersection point is

8

.

To find the

x

-intercept, set

y = 0

in the equation

3x - 4y = 24

. This simplifies to

3x = 24

. Dividing both sides of the equation by

3

gives

x = 8

.

Step-by-Step Solution

1

Set

y = 0

in the equation.

3x - 4(0) = 24 \implies 3x = 24

The intersection of any graph with the

x

-axis occurs where the

y

-coordinate is

0

.

2

Solve for

x

.

x = 8

Dividing both sides of

3x = 24

by

3

isolates the variable

x

.

Key Concept

Finding the

x

-intercept of a line by setting

y = 0

Alternative Method

Convert the standard form equation

3x - 4y = 24

into slope-intercept form:

-4y = -3x + 24 \implies y = \frac{3}{4}x - 6

. To find the

x

-intercept, set

y = 0

and solve the equation

0 = \frac{3}{4}x - 6 \implies 6 = \frac{3}{4}x \implies x = 8

.

Estimated Time:45s

Question 2177Question →

In $\triangle PQR$ , the measure of $\angle P$ is $50^\circ$ , and the measure of the exterior angle at vertex $Q$ is $110^\circ$ . If the bisector of $\angle PRQ$ intersects side $PQ$ at point $S$ , what is the measure, in degrees, of $\angle PRS$ ?

Show answer & explanation

Answer: 30

Answer

The measure of

\angle PRS

is

30^\circ

.

By using the linear pair relationship, the interior angle

\angle PQR

is found to be

70^\circ

. Applying the triangle angle sum theorem, the third interior angle

\angle PRQ

is

180^\circ - (50^\circ + 70^\circ) = 60^\circ

. The angle bisector

RS

divides this angle into two equal parts, resulting in a measure of

30^\circ

for

\angle PRS

.

Step-by-Step Solution

1

Find the interior angle at vertex

Q

\angle PQR = 70^\circ

The interior and exterior angles at a vertex are supplementary, summing to

180^\circ

.

2

Find the measure of interior angle

\angle PRQ

\angle PRQ = 60^\circ

The sum of the interior angles in any triangle is

180^\circ

.

3

Calculate the measure of the bisected angle

\angle PRS

\angle PRS = 30^\circ

An angle bisector divides the angle into two equal measures.

Key Concept

Triangle Angle Sum Theorem and Exterior Angle Relationships

Estimated Time:1m 30s

Question 2178Question →

The daily revenue $R$ , in dollars, of a manufacturing company is modeled by the quadratic function $R(x) = -0.2x^2 + kx - 1,200$ , where $x$ is the number of units produced and sold, and $k$ is a positive constant. If the maximum daily revenue the company can achieve is $800$ dollars, what is the value of $k$ ?

Show answer & explanation

Answer: 40

Answer

The value of the constant

k

is 40.

Setting the daily revenue function equal to 800 and rewriting it in standard form yields

-0.2x^2 + kx - 2000 = 0

. For a quadratic equation to have exactly one real solution, which represents the maximum vertex of the parabola, the discriminant must be equal to 0. Setting the discriminant

b^2 - 4ac = 0

gives

k^2 - 4(-0.2)(-2000) = 0

, which simplifies to

k^2 - 1600 = 0

. Solving for the positive constant

k

gives

k = 40

.

Step-by-Step Solution

1

Set the revenue function equal to the maximum daily revenue of 800 dollars.

-0.2x^2 + kx - 1,200 = 800

The maximum revenue is the highest point (vertex) on the parabola, where the line

y = 800

is tangent to the curve.

2

Convert the equation into standard quadratic form

ax^2 + bx + c = 0

.

-0.2x^2 + kx - 2,000 = 0

Standard form is required to identify the coefficients

a

,

b

, and

c

for the discriminant formula.

3

Set the discriminant equal to zero.

k^2 - 4(-0.2)(-2,000) = 0

Since the maximum daily revenue is achieved at exactly one point, the quadratic equation must have exactly one real solution, meaning its discriminant (

b^2 - 4ac

) must be zero.

4

Solve for the positive constant

k

.

k^2 - 1,600 = 0 \implies k = 40

Solving the equation yields

k = \pm 40

. Since the problem specifies that

k

is a positive constant, we select

k = 40

.

Key Concept

Using the discriminant of a quadratic equation to find the value of a parameter when there is exactly one real solution.

Estimated Time:2m 0s

Question 2179Question →

A digital graphic designer positions a control point of a logo at the coordinates $(-4, 1)$ on a computer screen's coordinate grid. The designer then translates the logo so that the control point moves $3$ units to the left and $6$ units up. What are the coordinates of the control point in its new position?

(-1, 7)

(-7, 7)

(-7, -5)

(-1, -5)

(-10, 4)

Show answer & explanation

Answer: $(-7, 7)$

Answer

The correct coordinates of the control point in its new position are

(-7, 7)

.

To find the coordinates of the image after a translation, adjust the original coordinates based on the direction of movement. Since the point starting at

(-4, 1)

is translated

3

units to the left, subtract

3

from the

x

-coordinate:

-4 - 3 = -7

. Since it is translated

6

units up, add

6

to the

y

-coordinate:

1 + 6 = 7

. This gives the final coordinates of

(-7, 7)

.

Step-by-Step Solution

1

Identify the initial coordinates of the point and the translation instructions.

Initial coordinates are

(x, y) = (-4, 1)

. The translation is

3

units to the left and

6

units up.

This establishes the starting point and the transformation rules that need to be applied.

2

Apply the horizontal translation to the

x

-coordinate.

x' = -4 - 3 = -7

Translating a point to the left on the coordinate plane decreases its

x

-value, so we subtract

3

from the initial

x

-coordinate.

3

Apply the vertical translation to the

y

-coordinate.

y' = 1 + 6 = 7

Translating a point upward on the coordinate plane increases its

y

-value, so we add

6

to the initial

y

-coordinate.

Key Concept

Applying translations to coordinate points by adding or subtracting units from the

x

- and

y

-coordinates depending on the direction of movement.

Question 2180Question →

In the standard $(x, y)$ coordinate plane, a line segment $AB$ has midpoint $M(3, 4)$ . If endpoint $A$ lies on the line $y = 2x - 7$ , and the length of segment $AB$ is $10$ units, what is the product of all possible $x$ -coordinates of endpoint $A$ ?

Show answer & explanation

Answer: 21

Answer

The product of all possible

x

-coordinates of endpoint

A

is

21

.

By determining that the distance from endpoint

A(x, 2x - 7)

to the midpoint

M(3, 4)

is half of the segment length

AB

(which is

5

), we set up the distance formula equation:

(x - 3)^2 + (2x - 11)^2 = 25

. Simplifying this equation leads to the quadratic expression

x^2 - 10x + 21 = 0

, which factors into

(x - 3)(x - 7) = 0

. The two possible

x

-coordinates are

3

and

7

. Multiplying these values yields the product

21

.

Step-by-Step Solution

1

Find the distance between endpoint

A

and midpoint

M

.

The distance

AM

is

5

.

Since

M

is the midpoint of segment

AB

of length

10

, the distance from either endpoint to the midpoint is half of the total length:

10 \div 2 = 5

.

2

Express the coordinates of endpoint

A

in terms of a single variable.

Endpoint

A

is represented as

(x, 2x - 7)

.

Endpoint

A

lies on the line

y = 2x - 7

.

3

Apply the distance formula to find the relationship for

x

.

(x - 3)^2 + (2x - 11)^2 = 25

The distance between

A(x, 2x - 7)

and

M(3, 4)

is

5

, so the square of the distance is

5^2 = 25

.

4

Simplify the quadratic equation.

x^2 - 10x + 21 = 0

Expanding

(x - 3)^2 + (2x - 11)^2 = 25

yields

x^2 - 6x + 9 + 4x^2 - 44x + 121 = 25

. Combining like terms gives

5x^2 - 50x + 130 = 25

. Subtracting

25

from both sides gives

5x^2 - 50x + 105 = 0

. Dividing the entire equation by

5

yields

x^2 - 10x + 21 = 0

.

5

Solve for the possible values of

x

.

x = 3

or

x = 7

Factoring the quadratic equation yields

(x - 3)(x - 7) = 0

, which gives the roots

x = 3

and

x = 7

.

6

Calculate the product of the possible

x

-coordinates.

21

The product of the two possible

x

-coordinates is

3 \times 7 = 21

.

Key Concept

Distance and Midpoint Formulas

Alternative Method

Instead of expanding the quadratic equation, one can use the geometric interpretation. The points

A

are the intersections of the circle

(x - 3)^2 + (y - 4)^2 = 25

and the line

y = 2x - 7

. Substituting

y = 2x - 7

directly into the circle equation and simplifying to

x^2 - 10x + 21 = 0

is the most direct approach.

Estimated Time:2m 0s

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