Intermediate Algebra Practice Questions — ACT — Page 8

Question 141Question →

Consider the functions $f(x) = \frac{x+3}{x-1}$ , where $x \neq 1$ , and $g(x) = x^2 - x - 4$ . If $x$ is an integer such that the composite function evaluation $g(f(g(x))) = 16$ , what is the product of all such integer values of $x$ ?

Show answer & explanation

Answer: -6

Answer

The product of all integer values of

x

that satisfy the equation is

-6

.

Solving the nested composite equation

g(f(g(x))) = 16

systematically yields the integer solutions

x = 3

and

x = -2

. The product of these values is

-6

.

Step-by-Step Solution

1

Set

u = f(g(x))

and solve the outer quadratic equation

g(u) = 16

.

u = 5

or

u = -4

This simplifies the nested composition into a single-variable quadratic equation.

2

Set

v = g(x)

and solve the rational equations

f(v) = 5

and

f(v) = -4

.

v = 2

or

v = 0.2

This determines the required outputs of the function

g(x)

that will satisfy the composite equation.

3

Solve the quadratic equations

g(x) = 2

and

g(x) = 0.2

for

x

.

x = 3

,

x = -2

, or

x = \frac{5 \pm \sqrt{445}}{10}

This finds all real values of

x

that satisfy the composite equation.

4

Identify the integer values from the solution set and compute their product.

3 \times (-2) = -6

The question specifically requests the product of the integer values of

x

.

Key Concept

Function composition and multi-step equation solving

Question 142Question →

For a constant $k$ , the quadratic equation $\frac{1}{3}x^2 - \frac{k}{6}x + (k - 10) = 0$ has two real roots, $r_1$ and $r_2$ . If the sum of the reciprocals of the roots, $\frac{1}{r_1} + \frac{1}{r_2}$ , is equal to $-\frac{1}{2}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 7.5

Answer

The value of

k

is

7.5

.

By writing the sum of the reciprocals in terms of the sum and product of the roots, we find that

\frac{1}{r_1} + \frac{1}{r_2} = \frac{r_1 + r_2}{r_1 r_2}

. Substituting the values from Vieta's formulas (

r_1 + r_2 = \frac{k}{2}

and

r_1 r_2 = 3(k - 10)

) yields

\frac{k}{6(k - 10)} = -\frac{1}{2}

. Solving this equation for

k

results in

k = 7.5

.

Step-by-Step Solution

1

Identify the quadratic coefficients in terms of the constant

k

.

a = \frac{1}{3}

,

b = -\frac{k}{6}

, and

c = k - 10

Applying Vieta's formulas requires the coefficients from the standard form

ax^2 + bx + c = 0

.

2

Determine the sum and the product of the roots using Vieta's formulas.

r_1 + r_2 = \frac{k}{2}

and

r_1 r_2 = 3(k - 10)

Vieta's formulas state that the sum of the roots is

-\frac{b}{a}

and the product of the roots is

\frac{c}{a}

.

3

Express the sum of the reciprocals of the roots in terms of

k

.

\frac{1}{r_1} + \frac{1}{r_2} = \frac{r_1 + r_2}{r_1 r_2} = \frac{k}{6(k - 10)}

Finding a common denominator allows the sum of the reciprocals to be written as the ratio of the sum of the roots to the product of the roots.

4

Equate the expression to

-\frac{1}{2}

and solve the resulting equation for

k

.

k = 7.5

Solving

\frac{k}{6(k - 10)} = -\frac{1}{2}

gives

2k = -6(k - 10) \implies 8k = 60 \implies k = 7.5

.

Key Concept

Vieta's Formulas and Algebraic Relationships of Roots

Estimated Time:2m 30s

Question 143Question →

If $\log_5 x = 3$ , what is the value of $x$ ?

8

15

25

125

243

Show answer & explanation

Answer: 125

Answer

125

To solve the logarithmic equation

\log_5 x = 3

, we apply the fundamental definition of a logarithm. A logarithmic equation of the form

\log_b x = y

can be rewritten in exponential form as

b^y = x

. In this equation, the base

b

is 5 and the exponent

y

is 3. Rewriting gives

5^3 = x

. Evaluating

5^3

yields

5 \times 5 \times 5 = 125

. Therefore, the correct value of

x

is 125.

Step-by-Step Solution

1

Apply the definition of a logarithm to rewrite the logarithmic equation in its equivalent exponential form.

5^3 = x

By definition,

\log_b x = y

is equivalent to

b^y = x

, where

b

is the base,

y

is the exponent, and

x

is the argument.

2

Evaluate the exponential expression

5^3

to solve for

x

.

x = 125

Cubing 5 means multiplying it by itself three times:

5 \times 5 \times 5 = 125

.

Key Concept

Definition of Logarithms

Question 144Question →

The table below shows some values of the functions $f$ and $g$ for several integer values of $x$ .

$x$	$f(x)$	$g(x)$
$-2$	$3$	$1$
$-1$	$4$	$-2$
$0$	$-1$	$3$
$1$	$2$	$0$
$2$	$-2$	$-1$
$3$	$1$	$2$

What is the value of $f(f(3)) + g(g(-1))$ ?

Show answer & explanation

Answer: 3

Answer

The value of the expression is

3

.

To evaluate

f(f(3)) + g(g(-1))

, we start by evaluating the innermost expressions. Looking at the table for

x = 3

, we find

f(3) = 1

. Looking at the table for

x = -1

, we find

g(-1) = -2

. Substituting these values into the outer functions gives

f(1) + g(-2)

. Using the table again, we look up

x = 1

to find

f(1) = 2

, and look up

x = -2

to find

g(-2) = 1

. Adding these two results yields

2 + 1 = 3

.

Step-by-Step Solution

1

Find the inner function values from the given table.

f(3) = 1

and

g(-1) = -2

Before evaluating the composite functions, we must find the outputs of the innermost functions.

2

Evaluate the outer functions using the results from the first step.

f(f(3)) = f(1) = 2

and

g(g(-1)) = g(-2) = 1

Substitute the inner outputs into the outer functions and look up the new inputs in the table.

3

Add the two resulting values together.

2 + 1 = 3

Combine the two terms to find the total sum requested by the question.

Key Concept

Evaluating composite functions using tables of values

Estimated Time:1m 0s

Question 145Question →

A company sells two types of gift baskets: Standard and Premium. The sales at the company's North and South branches are represented by matrix $S$ :

S = \begin{bmatrix} 12 & 8 \\ 15 & 10 \end{bmatrix}

where the rows represent the North and South branches, respectively, and the columns represent the number of Standard and Premium baskets sold, respectively. The cost of the items inside each basket is represented by matrix $C$ :

C = \begin{bmatrix} x & 24 \\ 30 & y \end{bmatrix}

where the rows represent Standard and Premium baskets, respectively, and the columns represent the cost of food items and the cost of gift items (in dollars), respectively. If the total food item cost at the North branch is $600$ and the total gift item cost at the South branch is $760$ , what is the value of $x + y$ ?

Show answer & explanation

Answer: 70

Answer

The value of

x + y

is 70.

To find the total costs, the sales matrix

S

is multiplied by the unit cost matrix

C

. The product matrix

R = SC

is a

2 \times 2

matrix where the entry in Row 1, Column 1 represents the total food cost at the North branch, and the entry in Row 2, Column 2 represents the total gift cost at the South branch. Setting up the equations

12x + 240 = 600

and

360 + 10y = 760

yields

x = 30

and

y = 40

, which sum to 70.

Step-by-Step Solution

1

Set up the matrix multiplication

R = SC

to represent the total costs.

R = \begin{bmatrix} 12x + 240 & 288 + 8y \\ 15x + 300 & 360 + 10y \end{bmatrix}

The product of the sales matrix and the unit cost matrix yields the total cost matrix, where rows represent the branches and columns represent the cost categories.

2

Identify the expression for the total food item cost at the North branch and solve for

x

.

x = 30

The total food item cost at the North branch is the entry in Row 1, Column 1 of the product matrix, which is

12x + 8(30) = 12x + 240

. Setting this equal to

600

gives

12x + 240 = 600

, which simplifies to

12x = 360

, so

x = 30

.

3

Identify the expression for the total gift item cost at the South branch and solve for

y

.

y = 40

The total gift item cost at the South branch is the entry in Row 2, Column 2 of the product matrix, which is

15(24) + 10y = 360 + 10y

. Setting this equal to

760

gives

360 + 10y = 760

, which simplifies to

10y = 400

, so

y = 40

.

4

Calculate the sum of the variables

x

and

y

.

x + y = 70

Substitute

x = 30

and

y = 40

into the expression

x + y

to find the final answer.

Key Concept

Matrix multiplication and translating real-world scenarios into matrix equations.

Estimated Time:3m 0s

Question 146Question →

Consider the system of equations below:

y = \sqrt{x + 37}

y = x - 5

If

(x, y)

is a real solution to this system, what is the value of

xy

?

-12

-4

6

24

84

Show answer & explanation

Answer: 84

Answer

84

The correct answer is 84 because solving the system yields the quadratic equation

x^2 - 11x - 12 = 0

, which has roots

x = 12

and

x = -1

. Substituting these back into the original equations shows that only

x = 12

is valid (giving

y = 7

), while

x = -1

is extraneous (since

\sqrt{36} = 6 \neq -6

). The product of the coordinates of the valid solution is

12 \times 7 = 84

.

Step-by-Step Solution

1

Set the two expressions for

y

equal to each other to solve for

x

.

\sqrt{x + 37} = x - 5

Since both equations are solved for

y

, their right-hand sides must be equal at the point of intersection.

2

Square both sides of the equation to eliminate the radical.

x + 37 = (x - 5)^2 \Rightarrow x + 37 = x^2 - 10x + 25

Squaring is the inverse operation of the square root. The binomial on the right must be expanded fully using

(a-b)^2 = a^2 - 2ab + b^2

.

3

Rearrange the quadratic equation into standard form.

x^2 - 11x - 12 = 0

Subtracting

x

and

37

from both sides sets the quadratic equation to zero so it can be solved.

4

Factor the quadratic equation to find the algebraic solutions.

(x - 12)(x + 1) = 0 \Rightarrow x = 12 \text{ or } x = -1

Factoring the trinomial allows us to find the roots by setting each linear factor to zero.

5

Substitute each solution back into the original system to check for extraneous solutions.

For

x = 12

:

y = 12 - 5 = 7

and

y = \sqrt{12 + 37} = 7

(Valid). For

x = -1

:

y = -1 - 5 = -6

, but

y = \sqrt{-1 + 37} = 6 \neq -6

(Extraneous).

Squaring both sides of an equation can introduce extraneous solutions that do not satisfy the original radical relationship because the principal square root must be non-negative.

6

Calculate the product

xy

of the coordinates of the valid solution.

xy = 12 \times 7 = 84

The question asks for the product of

x

and

y

for the real solution

(12, 7)

.

Key Concept

Solving systems containing radical equations and verifying for extraneous solutions.

Question 147Question →

Matrices $A$ and $B$ are defined as:

A = \begin{bmatrix} x & 4 \\ -3 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & -2 \\ 5 & 3 \end{bmatrix}

Let $C$ represent the product matrix $AB$ , where:

C = \begin{bmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{bmatrix}

If the element $c_{12}$ is equal to $2$ , what is the value of $x$ ?

Show answer & explanation

Answer: 5

Answer

5

To find the entry

c_{12}

in the first row and second column of the product matrix

C = AB

, we calculate the dot product of the first row of matrix

A

and the second column of matrix

B

. The first row of

A

is

[x, 4]

and the second column of

B

is

[-2, 3]^T

. Thus,

c_{12} = x(-2) + 4(3) = -2x + 12

. Setting this equal to the given value of

2

gives the equation

-2x + 12 = 2

. Subtracting

12

from both sides results in

-2x = -10

. Dividing by

-2

yields

x = 5

.

Step-by-Step Solution

1

Identify the formula for the element

c_{12}

in the product matrix

C = AB

.

c_{12} = -2x + 12

The element

c_{12}

is located in the first row and second column of the product matrix, so it is the product of the first row of

A

, which is

[x, 4]

, and the second column of

B

, which is

[-2, 3]^T

.

2

Set the expression for

c_{12}

equal to the given value of

2

and solve the linear equation for

x

.

x = 5

Setting

-2x + 12 = 2

leads to

-2x = -10

, and dividing both sides by

-2

yields

x = 5

.

Key Concept

Matrix Multiplication and Element-wise Operations

Question 148Question →

If $x$ and $y$ are positive real numbers greater than $1$ such that $\log_y(x) + 6\log_x(y) = 5$ and $xy = 64$ , what is the sum of all possible values of $x$ ?

16 + 16\sqrt{2}

4 + 2\sqrt{2}

4\sqrt{2}

80

528

Show answer & explanation

Answer: $16 + 16\sqrt{2}$

Answer

16 + 16\sqrt{2}

Using the change of base property

\log_x(y) = \frac{1}{\log_y(x)}

, the equation can be written in terms of

u = \log_y(x)

as

u + \frac{6}{u} = 5

. Solving this quadratic equation gives

u = 2

or

u = 3

. If

\log_y(x) = 2

, then

x = y^2

. Substituting into

xy = 64

yields

y^3 = 64 \implies y = 4

, which gives

x = 16

. If

\log_y(x) = 3

, then

x = y^3

. Substituting into

xy = 64

yields

y^4 = 64 \implies y = 2\sqrt{2}

, which gives

x = 16\sqrt{2}

. The sum of these values is

16 + 16\sqrt{2}

.

Step-by-Step Solution

1

Apply the change of base formula to express the equation in terms of a single logarithmic base.

Since

\log_x(y) = \frac{1}{\log_y(x)}

, the equation

\log_y(x) + 6\log_x(y) = 5

becomes

\log_y(x) + \frac{6}{\log_y(x)} = 5

.

This allows us to solve for the log expression using substitution.

2

Substitute

u = \log_y(x)

and solve the resulting quadratic equation for

u

.

u + \frac{6}{u} = 5 \implies u^2 - 5u + 6 = 0 \implies (u-2)(u-3) = 0

, so

u = 2

or

u = 3

.

Solving the quadratic equation gives the possible relationships between

x

and

y

.

3

Analyze the first case where

u = 2

and solve for

x

.

If

\log_y(x) = 2

, then

x = y^2

. Substituting this into

xy = 64

gives

y^3 = 64

, which yields

y = 4

. Thus,

x = 4^2 = 16

.

This determines the first possible value of

x

.

4

Analyze the second case where

u = 3

and solve for

x

.

If

\log_y(x) = 3

, then

x = y^3

. Substituting this into

xy = 64

gives

y^4 = 64

, which yields

y = 64^{1/4} = (2^6)^{1/4} = 2^{3/2} = 2\sqrt{2}

. Thus,

x = (2\sqrt{2})^3 = 16\sqrt{2}

.

This determines the second possible value of

x

.

5

Sum the possible values of

x

.

16 + 16\sqrt{2}

The question asks for the sum of all possible values of

x

.

Key Concept

Solving systems of exponential and logarithmic equations using base-change properties and substitution

Alternative Method

Instead of using substitution directly, you can write both equations in terms of base 2 or natural logs: let

\log_y(x) = k

, which means

x = y^k

. We then have

k + 6/k = 5

giving

k = 2

or

k = 3

. This leads directly to

x = y^2

and

x = y^3

, which can then be substituted into the second equation.

Estimated Time:3m 0s

Question 149Question →

A community theater group sells student tickets and adult tickets for their weekend plays. The number of tickets sold for Friday and Saturday night shows is represented by the matrix $T$ :

T = \begin{bmatrix} 80 & 120 \\ 100 & 150 \end{bmatrix}

where the first and second rows represent Friday and Saturday, respectively, and the first and second columns represent student and adult tickets, respectively. The price of each ticket is represented by the matrix $P$ :

P = \begin{bmatrix} 6 \\ 10 \end{bmatrix}

where the first row represents the price of a student ticket ( $6) and the second row represents the price of an adult ticket ($ 10). Which of the following matrices represents the total ticket sales revenue, in dollars, for Friday and Saturday, respectively?

\begin{bmatrix} 480 & 1,200 \\ 600 & 1,500 \end{bmatrix}

\begin{bmatrix} 2,100 \\ 1,680 \end{bmatrix}

\begin{bmatrix} 1,680 \\ 2,100 \end{bmatrix}

\begin{bmatrix} 1,580 \\ 2,100 \end{bmatrix}

\begin{bmatrix} 1,520 \\ 1,900 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 1,680 \\ 2,100 \end{bmatrix}$

Answer

The matrix with entries 1,680 and 2,100

The correct answer is the matrix with entries 1,680 and 2,100. This is found by multiplying the sales matrix by the price matrix. The revenue for Friday is calculated as

80 \times 6 + 120 \times 10 = 480 + 1,200 = 1,680

. The revenue for Saturday is

100 \times 6 + 150 \times 10 = 600 + 1,500 = 2,100

. Placing these in a column matrix matching the order of the days yields the correct matrix.

Step-by-Step Solution

1

Set up the matrix multiplication of the ticket sales matrix

T

and the ticket price matrix

P

.

TP = \begin{bmatrix} 80 & 120 \\ 100 & 150 \end{bmatrix} \begin{bmatrix} 6 \\ 10 \end{bmatrix}

To find the total revenue for Friday and Saturday, we must multiply the ticket sales by their respective prices.

2

Perform the multiplication by taking the dot product of each row of

T

with the column of

P

.

\begin{bmatrix} 80(6) + 120(10) \\ 100(6) + 150(10) \end{bmatrix} = \begin{bmatrix} 480 + 1200 \\ 600 + 1500 \end{bmatrix}

The matrix product of a

2 \times 2

matrix and a

2 \times 1

matrix results in a

2 \times 1

matrix where the entries are the sums of the products of corresponding elements.

3

Simplify the sums in the resulting matrix.

\begin{bmatrix} 1,680 \\ 2,100 \end{bmatrix}

Adding the products gives the final total revenues for Friday and Saturday, respectively.

Key Concept

Matrix Multiplication in Word Problems

Alternative Method

Instead of formal matrix multiplication, you can calculate the scalar totals for each day directly (Friday:

80 \times 6 + 120 \times 10 = 1,680

; Saturday:

100 \times 6 + 150 \times 10 = 2,100

) and match them with the corresponding rows of the resulting

2 \times 1

matrix.

Estimated Time:1m 30s

Question 150Question →

If $x$ is a positive real number such that $\log_2(x) + \log_4(x) + \log_{16}(x) = 7$ , what is the value of $x$ ?

Show answer & explanation

Answer: 16

Answer

The value of

x

is

16

.

By converting all logarithms to base 2, we write

\log_4(x)

as

\frac{1}{2}\log_2(x)

and

\log_{16}(x)

as

\frac{1}{4}\log_2(x)

. Summing these gives

\frac{7}{4}\log_2(x) = 7

, which simplifies to

\log_2(x) = 4

. Converting to exponential form, we find

x = 2^4 = 16

.

Step-by-Step Solution

1

Apply the change of base formula to express the logarithms with a common base of 2.

\log_4(x) = \frac{1}{2}\log_2(x)

and

\log_{16}(x) = \frac{1}{4}\log_2(x)

, giving the equation

\log_2(x) + \frac{1}{2}\log_2(x) + \frac{1}{4}\log_2(x) = 7

.

Rewriting the terms with a common base allows them to be combined algebraically.

2

Combine the coefficients of the like terms on the left-hand side.

\frac{7}{4}\log_2(x) = 7

.

The sum of the coefficients is

1 + \frac{1}{2} + \frac{1}{4} = \frac{4}{4} + \frac{2}{4} + \frac{1}{4} = \frac{7}{4}

.

3

Isolate the logarithm term by dividing or multiplying by the reciprocal coefficient.

\log_2(x) = 4

.

Multiplying both sides by

\frac{4}{7}

solves for the value of

\log_2(x)

.

4

Convert the equation from logarithmic form to its equivalent exponential form.

x = 2^4 = 16

.

By definition,

\log_b(a) = c

is equivalent to

b^c = a

.

Key Concept

Change of Base Formula for Logarithms

Question 151Question →

Let the functions $f$ and $g$ be defined for all real numbers by $f(x) = |2x - 6| - 16$ and $g(x) = (x - 3)^2 - 5$ . What is the sum of all real values of $x$ that satisfy the equation $f(g(x)) = 0$ ?

0

6

9

10

12

Show answer & explanation

Answer: 9

Answer

9

The correct answer is 9. First, substitute the inner function into the outer function to get

f(g(x)) = |2g(x) - 6| - 16 = 0

. This simplifies to the absolute value equation

|2g(x) - 6| = 16

. Splitting this into its two possible cases gives

2g(x) - 6 = 16 \implies g(x) = 11

, and

2g(x) - 6 = -16 \implies g(x) = -5

. Solving the first case,

(x-3)^2 - 5 = 11 \implies (x-3)^2 = 16

. Taking both square roots yields

x - 3 = 4 \implies x = 7

and

x - 3 = -4 \implies x = -1

. Solving the second case,

(x-3)^2 - 5 = -5 \implies (x-3)^2 = 0 \implies x - 3 = 0 \implies x = 3

. The sum of all unique real values of

x

that satisfy the original equation is

7 + (-1) + 3 = 9

.

Step-by-Step Solution

1

Set up the composite function equation using the outer function definition.

|2g(x) - 6| - 16 = 0

Substitute the expression of

g(x)

as the input variable into

f(x)

.

2

Isolate the absolute value term and split the equation into two cases.

|2g(x) - 6| = 16 \implies 2g(x) - 6 = 16

or

2g(x) - 6 = -16

An absolute value equation of the form

|A| = B

where

B > 0

has two solutions:

A = B

and

A = -B

.

3

Solve for the target values of

g(x)

in each case.

g(x) = 11

and

g(x) = -5

Isolate the function

g(x)

by adding 6 and dividing by 2 on both sides of each equation.

4

Solve the first case

g(x) = 11

for the variable

x

.

(x-3)^2 - 5 = 11 \implies (x-3)^2 = 16 \implies x - 3 = \pm 4

, yielding

x = 7

and

x = -1

Substitute the algebraic rule for

g(x)

, isolate the squared expression, and extract the square roots.

5

Solve the second case

g(x) = -5

for the variable

x

.

(x-3)^2 - 5 = -5 \implies (x-3)^2 = 0 \implies x - 3 = 0

, yielding

x = 3

Substitute the algebraic rule for

g(x)

, isolate the squared expression, and solve for

x

.

6

Sum all unique real solutions found.

7 + (-1) + 3 = 9

Add the distinct solutions

x = 7

,

x = -1

, and

x = 3

to find the total sum.

Key Concept

Evaluating and solving equations involving composite functions, absolute values, and quadratic expressions.

Question 152Question →

If $\log_2(a) + \log_2(b) = 5$ and $\log_2(a^2) - \log_2(b) = 4$ for positive real numbers $a$ and $b$ , what is the value of $a + b$ ?

5

10

12

16

32

Show answer & explanation

Answer: 12

Answer

12

The correct answer is 12. We can simplify the system of equations by using the power rule of logarithms, which allows us to rewrite

\log_2(a^2)

as

2\log_2(a)

. Letting

x = \log_2(a)

and

y = \log_2(b)

gives us the system

x + y = 5

and

2x - y = 4

. Adding these equations gives

3x = 9

, which means

x = 3

. Substituting this back gives

y = 2

. Converting back from logarithmic form to exponential form, we get

a = 2^3 = 8

and

b = 2^2 = 4

. Therefore,

a + b = 8 + 4 = 12

.

Step-by-Step Solution

1

Use the power property of logarithms,

\log_b(x^k) = k \log_b(x)

, to rewrite the second equation.

The equation

\log_2(a^2) - \log_2(b) = 4

becomes

2\log_2(a) - \log_2(b) = 4

.

This simplifies the term

\log_2(a^2)

so that it is linear in terms of

\log_2(a)

.

2

Substitute variables to simplify solving the system of equations. Let

x = \log_2(a)

and

y = \log_2(b)

.

The system of equations becomes:
1)

x + y = 5

2)

2x - y = 4

Variable substitution reduces the logarithmic system to a standard system of linear equations.

3

Solve the linear system by adding the two equations together.

Adding the equations yields

(x + y) + (2x - y) = 5 + 4

, which simplifies to

3x = 9

, so

x = 3

. Substituting

x = 3

back into the first equation gives

3 + y = 5

, so

y = 2

.

Addition eliminates the variable

y

, allowing us to solve for

x

and then find

y

.

4

Convert the solved values of

x

and

y

back into

a

and

b

using the exponential form definition of a logarithm.

Since

x = \log_2(a) = 3

, we have

a = 2^3 = 8

. Since

y = \log_2(b) = 2

, we have

b = 2^2 = 4

.

The definition of a logarithm

\log_b(z) = w

is equivalent to

z = b^w

.

5

Calculate the sum of

a

and

b

.

a + b = 8 + 4 = 12

.

To find the final value requested by the question stem.

Key Concept

Solving systems of logarithmic equations using logarithm properties and exponential conversions

Question 153Question →

A system of equations consists of the circle defined by $x^2 + (y - 4)^2 = 10$ and the line defined by $y = 2x - 1$ . The two points of intersection of this system and the origin, $(0, 0)$ , form the vertices of a triangle in the standard $(x, y)$ coordinate plane. What is the area of this triangle?

2

\sqrt{7}

1

4

5

Show answer & explanation

Answer: 1

Answer

The correct area of the triangle is 1.

The correct answer of 1 is found by substituting the linear equation into the circle's equation, solving the resulting quadratic equation to find the coordinates of the two intersection points,

(1, 1)

and

(3, 5)

, and then applying the coordinate area formula for a triangle with a vertex at the origin.

Step-by-Step Solution

1

Substitute the linear equation into the circle's equation to form a quadratic equation in terms of

x

.

x^2 + (2x - 5)^2 = 10

This allows us to solve for the x-coordinates of the intersection points by replacing

y

with

2x - 1

in

(y - 4)^2

to get

(2x - 5)^2

.

2

Expand the squared binomial and simplify the quadratic equation.

5x^2 - 20x + 15 = 0

Expanding

(2x - 5)^2

yields

4x^2 - 20x + 25

. Adding

x^2

and subtracting

10

from both sides simplifies the equation to standard quadratic form.

3

Divide the quadratic equation by 5 and solve for

x

by factoring.

x = 1

and

x = 3

Dividing the equation by 5 yields

x^2 - 4x + 3 = 0

, which factors as

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

.

4

Find the corresponding y-coordinates by substituting the

x

-values back into the linear equation

y = 2x - 1

.

The intersection points are

(1, 1)

and

(3, 5)

.

For

x = 1

,

y = 2(1) - 1 = 1

. For

x = 3

,

y = 2(3) - 1 = 5

.

5

Calculate the area of the triangle with vertices at the origin

(0, 0)

and the two intersection points

(1, 1)

and

(3, 5)

.

Area = 1

Using the coordinate area formula for a triangle with one vertex at the origin,

\text{Area} = \frac{1}{2} |x_1 y_2 - x_2 y_1| = \frac{1}{2} |1(5) - 3(1)| = \frac{1}{2} |2| = 1

.

Key Concept

Solving a system of linear and non-linear equations and using coordinate geometry to find the area of a triangle.

Alternative Method

Instead of using the coordinate area formula, we can find the distance between the two intersection points to serve as the base of the triangle (

b = \sqrt{(3-1)^2 + (5-1)^2} = 2\sqrt{5}

), and find the perpendicular distance from the origin to the line

2x - y - 1 = 0

to serve as the height (

h = \frac{|-1|}{\sqrt{2^2 + (-1)^2}} = \frac{1}{\sqrt{5}}

). The area is then

\frac{1}{2} \times 2\sqrt{5} \times \frac{1}{\sqrt{5}} = 1

.

Estimated Time:2m 30s

Question 154Question →

In the inequality $7 - 3|2z + 1| \ge -8$ , which of the following inequality expressions represents the complete solution set for $z$ ?

z \le -3

or

z \ge 2

z \le 2

-3 \le z \le 2

-1.5 \le z \le 0.5

z \ge 2

Show answer & explanation

Answer: $-3 \le z \le 2$

Answer

-3 \le z \le 2

Subtracting 7 from both sides of

7 - 3|2z + 1| \ge -8

yields

-3|2z + 1| \ge -15

. Dividing both sides by

-3

and reversing the inequality sign results in

|2z + 1| \le 5

. Writing this as the compound inequality

-5 \le 2z + 1 \le 5

, then subtracting 1 and dividing by 2 yields the correct solution interval

-3 \le z \le 2

.

Step-by-Step Solution

1

Subtract 7 from both sides to begin isolating the absolute value term.

-3|2z + 1| \ge -15

To solve an absolute value inequality, we must first isolate the absolute value term on one side.

2

Divide both sides by -3 and reverse the direction of the inequality sign.

|2z + 1| \le 5

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

3

Rewrite the absolute value inequality as a compound inequality.

-5 \le 2z + 1 \le 5

An inequality of the form

|u| \le a

(where

a \ge 0

) is equivalent to the compound inequality

-a \le u \le a

.

4

Subtract 1 from all three parts of the compound inequality.

-6 \le 2z \le 4

This is the first step to isolate the variable

z

in the middle.

5

Divide all three parts by 2.

-3 \le z \le 2

This fully isolates

z

, giving the final solution interval.

Key Concept

Solving multi-step absolute value inequalities, including isolating the absolute value expression, reversing the inequality sign when dividing by a negative number, and expressing the solution as a compound inequality.

Question 155Question →

The length of a rectangle is $3$ inches greater than its width. If the area of the rectangle is $10$ square inches, what is the width of the rectangle, in inches?

1

2

3.5

5

7

Show answer & explanation

Answer: $2$

Answer

The width of the rectangle is

2

inches.

The correct answer is the option stating that the width is

2

inches. This is obtained by expressing the length as

w+3

and setting the area equation to

w(w+3) = 10

. Expanding and setting it to zero gives the quadratic equation

w^2 + 3w - 10 = 0

, which factors to

(w+5)(w-2) = 0

. Since width must be positive, the only valid solution is

2

.

Step-by-Step Solution

1

Define variables for the dimensions of the rectangle based on the problem statement.

Let the width of the rectangle be

w

inches. The length of the rectangle is

w + 3

inches.

The problem states that the length is

3

inches greater than the width.

2

Set up an equation representing the area of the rectangle.

The area is the product of width and length:

w(w + 3) = 10

.

The area of a rectangle is calculated as width times length, and the given area is

10

square inches.

3

Rewrite the equation in standard quadratic form

aw^2 + bw + c = 0

.

Distribute

w

to get

w^2 + 3w = 10

, then subtract

10

from both sides:

w^2 + 3w - 10 = 0

.

Standard quadratic form is required to solve the equation by factoring or using the quadratic formula.

4

Factor the quadratic equation.

(w + 5)(w - 2) = 0

.

Finding factors that multiply to

-10

and add to

3

allows us to find the roots.

5

Solve for

w

and apply real-world constraints.

w = -5

or

w = 2

. Since a physical width must be positive, discard

-5

, leaving

w = 2

.

A dimension of a geometric shape cannot be negative.

Key Concept

Solving quadratic equations derived from geometric word problems by factoring or using the quadratic formula.

Question 156Question →

Two infinite geometric series, Series A and Series B, are defined as follows:

* Series A has a first term of $2^x$ and a common ratio of $\frac{1}{2}$ .
* Series B has a first term of $2^{x+3}$ and a common ratio of $\frac{3}{4}$ .

If the sum of Series A and Series B is 136, what is the value of $x$ ?

0.5

1

2

3

4

Show answer & explanation

Answer: $2$

Answer

2

The correct answer is the value 2. Substituting

x = 2

into the sum of Series A gives

2^{2+1} = 8

. Substituting

x = 2

into the sum of Series B gives

2^{2+5} = 128

. The total sum is

8 + 128 = 136

, which matches the given sum.

Step-by-Step Solution

1

Express the sum of Series A, denoted as

S_A

, using the infinite geometric series sum formula

S = \frac{a_1}{1-r}

.

S_A = \frac{2^x}{1 - 1/2} = \frac{2^x}{1/2} = 2 \cdot 2^x = 2^{x+1}

Since the common ratio

r = 1/2

satisfies

|r| < 1

, the infinite sum exists and can be simplified using exponent properties.

2

Express the sum of Series B, denoted as

S_B

, using the infinite geometric series sum formula.

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

Since the common ratio

r = 3/4

satisfies

|r| < 1

, the infinite sum exists and can be simplified using exponent rules.

3

Set up the equation for the sum of both series and solve for

x

.

2^{x+1} + 2^{x+5} = 136 \implies 2 \cdot 2^x + 32 \cdot 2^x = 136 \implies 34 \cdot 2^x = 136 \implies 2^x = 4 \implies x = 2

Factoring out

2^x

from the terms allows us to isolate the exponential expression and find the value of

x

.

Key Concept

Sum of an Infinite Geometric Series and Exponential Properties

Question 157Question →

The first, third, and eleventh terms of a non-constant arithmetic sequence are the first, second, and third terms, respectively, of a geometric sequence. If the first term of the arithmetic sequence is $6$ , what is the sum of the first $4$ terms of the geometric sequence?

78

384

510

1,554

4,920

Show answer & explanation

Answer: 510

Answer

The sum of the first 4 terms of the geometric sequence is 510.

By writing the first, third, and eleventh terms of the arithmetic sequence as

6

,

6+2d

, and

6+10d

, we can set up the geometric sequence relation

(6+2d)^2 = 6(6+10d)

. Solving this quadratic equation for

d

yields

d=9

(since the sequence is non-constant,

d \neq 0

). Substituting

d=9

back gives the first three geometric terms as

6

,

24

, and

96

, which means the common ratio

r

is

4

. The sum of the first

4

terms of this geometric sequence is calculated as

6(4^4-1)/(4-1) = 510

.

Step-by-Step Solution

1

Define the terms of the arithmetic and geometric sequences.

Let the arithmetic sequence have first term

a_1 = 6

and common difference

d

. The first, third, and eleventh terms are

a_1 = 6

,

a_3 = 6 + 2d

, and

a_{11} = 6 + 10d

. These are the first three terms of the geometric sequence:

g_1 = 6

,

g_2 = 6 + 2d

, and

g_3 = 6 + 10d

.

This establishes algebraic expressions for the terms based on their positions in the sequences.

2

Set up a relation using the constant ratio of the geometric sequence and solve for

d

.

Since

g_1

,

g_2

, and

g_3

form a geometric sequence,

(g_2)^2 = g_1 \cdot g_3

. Substituting the expressions gives

(6 + 2d)^2 = 6(6 + 10d) \implies 36 + 24d + 4d^2 = 36 + 60d \implies 4d^2 - 36d = 0

. Since the sequence is non-constant (

d \neq 0

), we divide by

4d

to get

d = 9

.

Solving the equation yields the common difference of the arithmetic sequence.

3

Determine the terms and common ratio of the geometric sequence.

Using

d = 9

, the first two terms of the geometric sequence are

g_1 = 6

and

g_2 = 6 + 2(9) = 24

. The common ratio is

r = \frac{24}{6} = 4

.

Finding the common ratio allows the use of the geometric series sum formula.

4

Compute the sum of the first 4 terms of the geometric sequence.

S_4 = g_1 \frac{r^4 - 1}{r - 1} = 6 \frac{4^4 - 1}{4 - 1} = 6 \frac{256 - 1}{3} = 2(255) = 510

.

This calculates the final required sum.

Key Concept

Arithmetic and Geometric Sequences and Series

Question 158Question →

If $\log_3(2x - 1) = 2$ , what is the value of $x$ ?

Show answer & explanation

Answer: 5

Answer

The value of

x

is

5

.

Converting the logarithmic equation

\log_3(2x - 1) = 2

to its equivalent exponential form yields

3^2 = 2x - 1

. Simplifying the exponent gives

9 = 2x - 1

. Adding

1

to both sides results in

10 = 2x

, and dividing by

2

gives

x = 5

.

Step-by-Step Solution

1

Rewrite the logarithmic equation in exponential form.

2x - 1 = 3^2

By definition,

\log_b(y) = z

is equivalent to

b^z = y

.

2

Evaluate the exponent

3^2

.

2x - 1 = 9

Calculating

3

squared yields

9

.

3

Solve the linear equation for

x

.

2x = 10

, which simplifies to

x = 5

Adding

1

to both sides and then dividing by

2

isolates

x

.

Key Concept

Converting logarithmic equations to exponential equations

Estimated Time:45s

Question 159Question →

Let the functions $f$ and $g$ be defined for all real numbers by $f(x) = 2x - 3$ and $g(x) = x^2 - 5$ . If $g(f(x)) = 11$ and $x < 0$ , what is the value of $x$ ?

\frac{7}{2}

-\sqrt{12}

-\frac{\sqrt{7}}{2}

-\frac{1}{2}

-\frac{7}{2}

Show answer & explanation

Answer: $-\frac{1}{2}$

Answer

-\frac{1}{2}

The correct answer is

-\frac{1}{2}

. First, the composite function

g(f(x))

is found by substituting

f(x) = 2x - 3

into

g(x) = x^2 - 5

, giving

g(f(x)) = (2x - 3)^2 - 5

. Setting this expression equal to

11

yields the equation

(2x - 3)^2 - 5 = 11

. Adding

5

to both sides results in

(2x - 3)^2 = 16

. Taking the square root of both sides gives

2x - 3 = \pm 4

. Since the problem specifies that

x < 0

, the expression

2x - 3

must be negative because

2x < 0

and subtracting

3

makes the result less than

-3

. Thus, we set

2x - 3 = -4

. Adding

3

to both sides yields

2x = -1

, and dividing by

2

gives

x = -\frac{1}{2}

.

Step-by-Step Solution

1

Substitute the expression for

f(x)

into the function

g(x)

to obtain the composite function

g(f(x))

.

g(f(x)) = g(2x - 3) = (2x - 3)^2 - 5

To find

g(f(x))

, we substitute the entire function

f(x)

in place of the input variable in

g(x)

.

2

Set the composite function equal to

11

and isolate the squared term.

(2x - 3)^2 - 5 = 11 \Rightarrow (2x - 3)^2 = 16

We are given that

g(f(x)) = 11

, and adding

5

to both sides isolates the squared binomial.

3

Solve for

2x - 3

by taking the square root of both sides, applying the constraint

x < 0

.

2x - 3 = -4

Since

x < 0

, it follows that

2x < 0

, which means

2x - 3 < -3

. Because

2x-3

must be negative, we take the negative square root of

16

.

4

Solve the linear equation

2x - 3 = -4

for

x

.

2x = -1 \Rightarrow x = -\frac{1}{2}

Adding

3

to both sides gives

2x = -1

, and dividing by

2

yields the final value of

x

.

Key Concept

Function Composition and Solving Quadratic/Linear Equations

Estimated Time:1m 30s

Question 160Question →

If $\log_4 x = \frac{3}{2}$ , what is the value of $x$ ?

6

8

12

64

\frac{81}{16}

Show answer & explanation

Answer: $8$

Answer

The correct answer is

8

.

To solve the equation

\log_4 x = \frac{3}{2}

, we apply the definition of a logarithm to rewrite it in exponential form:

x = 4^{3/2}

. We then evaluate the exponent by first taking the square root of

4

, which is

2

, and then cubing it to get

2^3 = 8

. Thus, the value of

x

is

8

.

Step-by-Step Solution

1

Rewrite the logarithmic equation in its equivalent exponential form.

x = 4^{3/2}

By definition, a logarithmic equation of the form

\log_b a = c

is equivalent to the exponential equation

b^c = a

.

2

Evaluate the exponential expression

4^{3/2}

.

x = 8

The fractional exponent can be simplified by taking the square root of the base first, which is

\sqrt{4} = 2

, and then raising the result to the power of the numerator, giving

2^3 = 8

.

Key Concept

Converting logarithmic equations to exponential form and evaluating fractional exponents

Alternative Method

You can solve this by substituting the answer choices back into the equation. For example, testing the value

8

gives

\log_4 8 = \log_4 (2^3) = 3 \log_4 2

. Since

2

is the square root of

4

,

\log_4 2 = \frac{1}{2}

. Therefore,

3 \times \frac{1}{2} = \frac{3}{2}

, confirming that

8

is the correct solution.

Estimated Time:45s

Intermediate Algebra

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes