Intermediate Algebra Practice Questions — ACT — Page 6

Question 101Question →

A community theater sells adult tickets for $\$15$ each and student tickets for $\$10$ each. The number of tickets sold for the evening performances on Friday and Saturday is represented by the matrix $A$ :

A = \begin{bmatrix} 80 & 45 \\ 110 & 60 \end{bmatrix}

where the rows represent Friday and Saturday, respectively, and the columns represent adult and student tickets, respectively.

The ticket prices are represented by the matrix $B$ :

B = \begin{bmatrix} 15 \\ 10 \end{bmatrix}

The product matrix $C = AB$ represents the total ticket revenue for each night. What is the element in the second row and first column of matrix $C$ , which represents the total ticket revenue, in dollars, for Saturday's performance?

Show answer & explanation

Answer: 2250

Answer

The correct answer is

2250

dollars, which represents the ticket revenue for Saturday's performance.

The correct answer is

2250

dollars. The element in the second row and first column of the product matrix

C = AB

is obtained by multiplying the elements of the second row of matrix

A

(Saturday's ticket sales:

110

adult,

60

student) by the corresponding elements of the first column of matrix

B

(ticket prices:

\$15

adult,

\$10

student) and summing the products:

110 \times 15 + 60 \times 10 = 1650 + 600 = 2250

.

Step-by-Step Solution

1

Identify the row of matrix

A

representing Saturday's sales and the column of matrix

B

representing the prices.

The second row of matrix

A

is

\begin{bmatrix} 110 & 60 \end{bmatrix}

(representing

110

adult tickets and

60

student tickets sold on Saturday). Matrix

B

is

\begin{bmatrix} 15 \\ 10 \end{bmatrix}

(representing the prices of

\$15

and

\$10

respectively).

Since the question asks for Saturday's performance (the second row of matrix

A

) and the revenue is calculated by multiplying sales by prices, we use these specific components for the matrix multiplication.

2

Calculate the dot product of Saturday's sales row and the price column.

110 \times 15 + 60 \times 10 = 1650 + 600 = 2250

.

This calculation determines the sum of the revenue from adult tickets (

110 \times 15 = 1650

) and student tickets (

60 \times 10 = 600

) to yield the total revenue.

Key Concept

Matrix Multiplication

Question 102Question →

For the quadratic equation $3x^2 - 6x + 2 = 0$ , what is the value of the discriminant?

Show answer & explanation

Answer: 12

Answer

The discriminant of the quadratic equation is 12.

The discriminant of a quadratic equation

ax^2 + bx + c = 0

is determined by the expression

b^2 - 4ac

. By substituting the coefficients

a = 3

,

b = -6

, and

c = 2

from the given equation

3x^2 - 6x + 2 = 0

, we calculate

(-6)^2 - 4(3)(2) = 36 - 24 = 12

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation

a = 3

,

b = -6

, and

c = 2

The standard form of a quadratic equation is

ax^2 + bx + c = 0

.

2

Substitute the coefficients into the discriminant formula

D = (-6)^2 - 4(3)(2)

The discriminant formula is

D = b^2 - 4ac

.

3

Simplify the expression to find the final value

36 - 24 = 12

Squaring

-6

gives

36

and multiplying

4 \times 3 \times 2

gives

24

. Subtracting

24

from

36

gives

12

.

Key Concept

Calculating the discriminant of a quadratic equation

Estimated Time:45s

Question 103Question →

A system of equations consists of a parabola with the equation $y = x^2$ and a line with the equation $y = 3x - 2$ . The line and the parabola intersect at two points, $(x_1, y_1)$ and $(x_2, y_2)$ . What is the product of the $y$ -coordinates, $y_1 \cdot y_2$ , of these two points?

Show answer & explanation

Answer: 4

Answer

The product of the y-coordinates of the intersection points is 4.

Solving the system of equations by setting

x^2 = 3x - 2

leads to the quadratic equation

x^2 - 3x + 2 = 0

, which factors as

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

. The x-coordinates of the intersection points are

x = 1

and

x = 2

. Substituting these into

y = x^2

gives the corresponding y-coordinates as

y = 1^2 = 1

and

y = 2^2 = 4

. The product of these y-coordinates is

1 \cdot 4 = 4

.

Step-by-Step Solution

1

Equate the two equations to find the x-coordinates of the intersection points.

x^2 = 3x - 2

Since both equations are solved for y, setting them equal to each other allows us to find the x-coordinates where they intersect.

2

Move all terms to one side to set the quadratic equation to zero.

x^2 - 3x + 2 = 0

This puts the equation in the standard form

ax^2 + bx + c = 0

, which can be solved by factoring.

3

Factor the quadratic equation.

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

, so

x_1 = 1

and

x_2 = 2

Factoring shows that the values of x that satisfy the equation are the roots of the quadratic.

4

Substitute the x-coordinates back into the equation of the parabola to find the y-coordinates.

For

x_1 = 1

,

y_1 = 1^2 = 1

. For

x_2 = 2

,

y_2 = 2^2 = 4

.

Substituting x into either of the original equations gives the corresponding y-coordinates of the intersection points.

5

Calculate the product of the y-coordinates.

1 \cdot 4 = 4

The question asks for the product of the y-coordinates, so we multiply

y_1

and

y_2

.

Key Concept

Solving systems of linear and quadratic equations by substitution.

Question 104Question →

What is the set of all real numbers $y$ that make the inequality $2 - 3|1 - 2y| \ge -10$ a true statement?

-2.5 \le y \le 1.5

y \le -1.5

or

y \ge 2.5

-1.5 \le y \le 2.5

y \ge -1.5

-4.5 \le y \le 5.5

Show answer & explanation

Answer: $-1.5 \le y \le 2.5$

Answer

-1.5 \le y \le 2.5

To solve the inequality

2 - 3|1 - 2y| \ge -10

, we first isolate the absolute value term by subtracting 2 from both sides to get

-3|1 - 2y| \ge -12

, and then dividing both sides by

-3

. Since we divide by a negative number, the inequality sign reverses, giving

|1 - 2y| \le 4

. We rewrite this as the compound inequality

-4 \le 1 - 2y \le 4

. Subtracting 1 from all parts yields

-5 \le -2y \le 3

. Finally, dividing by

-2

and reversing the inequality signs gives

2.5 \ge y \ge -1.5

, which is rewritten from least to greatest as

-1.5 \le y \le 2.5

.

Step-by-Step Solution

1

Subtract 2 from both sides of the inequality to begin isolating the absolute value term.

-3|1 - 2y| \ge -12

Isolating the absolute value expression allows us to rewrite it as a standard inequality.

2

Divide both sides by

-3

and reverse the inequality sign because of division by a negative number.

|1 - 2y| \le 4

Dividing by a negative value requires reversing the inequality direction to preserve the truth of the statement.

3

Rewrite the absolute value inequality as a compound inequality.

-4 \le 1 - 2y \le 4

An inequality of the form

|u| \le c

(where

c > 0

) is equivalent to the compound inequality

-c \le u \le c

.

4

Subtract 1 from all parts of the compound inequality.

-5 \le -2y \le 3

This is the next step to isolate the variable

y

in the middle.

5

Divide all parts of the compound inequality by

-2

and reverse the inequality signs.

2.5 \ge y \ge -1.5

, which is equivalent to

-1.5 \le y \le 2.5

Dividing by the negative coefficient

-2

requires reversing the direction of all inequality signs.

Key Concept

Solving absolute value inequalities involving negative coefficients by isolating the absolute value and reversing inequality signs when multiplying or dividing by negative numbers.

Estimated Time:1m 30s

Question 105Question →

For what positive integer value of $x$ is the rational expression $\frac{2x - 5}{3x^2 - 10x - 8}$ undefined?

Fill in the blanks below

The expression is undefined when

x

is equal to .

Show answer & explanation

Answer

The expression is undefined when the positive integer x is equal to 4.

The rational expression is undefined when its denominator is 0. Setting the denominator

3x^2 - 10x - 8

equal to 0 and factoring gives

(3x + 2)(x - 4) = 0

. The roots of this equation are

x = -\frac{2}{3}

and

x = 4

. Since the question restricts the target value to a positive integer, the correct value is 4.

Step-by-Step Solution

1

Set the denominator of the rational expression equal to 0.

3x^2 - 10x - 8 = 0

A rational expression is undefined when its denominator is equal to 0 because division by zero is undefined.

2

Factor the quadratic equation

3x^2 - 10x - 8 = 0

.

(3x + 2)(x - 4) = 0

Factoring the quadratic trinomial helps identify the values of x that satisfy the equation.

3

Solve for the roots by setting each factor to 0.

x = -\frac{2}{3}

or

x = 4

According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero.

4

Filter the solutions based on the given constraint.

x = 4

The question specifies that x must be a positive integer. Since

-\frac{2}{3}

is a negative fraction, we reject it and select 4.

Key Concept

Finding values that make a rational expression undefined by setting the denominator to zero and solving the resulting quadratic equation.

Estimated Time:1m 30s

Question 106Question →

The matrices $A$ and $B$ are defined as follows:

A = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 1 \\ 0 & 2 \end{bmatrix}

What is the product matrix $AB$ ?

\begin{bmatrix} 6 & 2 \\ 3 & 7 \end{bmatrix}

\begin{bmatrix} 6 & 0 \\ 0 & 6 \end{bmatrix}

\begin{bmatrix} 7 & 3 \\ 2 & 6 \end{bmatrix}

\begin{bmatrix} 5 & 1 \\ 1 & 5 \end{bmatrix}

\begin{bmatrix} 6 & 0 \\ 6 & 6 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 6 & 2 \\ 3 & 7 \end{bmatrix}$

Answer

\begin{bmatrix} 6 & 2 \\ 3 & 7 \end{bmatrix}

The correct matrix is obtained by performing row-by-column matrix multiplication, which yields row 1 entries of 6 and 2, and row 2 entries of 3 and 7.

Step-by-Step Solution

1

Multiply the first row of

A

by the first column of

B

to find the element in the first row, first column of the product matrix.

2(3) + 0(0) = 6

The row-by-column rule dictates that the entry at row

i

, column

j

of the product is the dot product of row

i

of the first matrix and column

j

of the second matrix.

2

Multiply the first row of

A

by the second column of

B

to find the element in the first row, second column.

2(1) + 0(2) = 2

This determines the top-right entry of the resulting matrix.

3

Multiply the second row of

A

by the first column of

B

to find the element in the second row, first column.

1(3) + 3(0) = 3

This determines the bottom-left entry of the resulting matrix.

4

Multiply the second row of

A

by the second column of

B

to find the element in the second row, second column.

1(1) + 3(2) = 1 + 6 = 7

This determines the bottom-right entry of the resulting matrix.

Key Concept

Matrix multiplication involves computing the dot product of the rows of the first matrix with the columns of the second matrix.

Question 107Question →

What are the solutions to the quadratic equation $x^2 - \frac{5}{2}x + 1 = 0$ ?

x = -\frac{1}{2}

and

x = -2

x = 1

and

x = \frac{1}{4}

x = \frac{1}{2}

and

x = 2

x = \frac{5}{4} + \frac{3}{4}i

and

x = \frac{5}{4} - \frac{3}{4}i

x = \frac{3}{4}

and

x = -\frac{3}{4}

Show answer & explanation

Answer: $x = \frac{1}{2}$ and $x = 2$

Answer

The solutions are

x = \frac{1}{2}

and

x = 2

.

The correct answer provides the exact solutions to the equation. Multiplying the equation by

2

yields

2x^2 - 5x + 2 = 0

. Factoring this expression gives

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

, which yields the solutions

x = \frac{1}{2}

and

x = 2

.

Step-by-Step Solution

1

Multiply the entire equation by

2

to clear the fraction.

2x^2 - 5x + 2 = 0

Converting the equation to integer coefficients simplifies factoring or applying the quadratic formula.

2

Identify the coefficients

a

,

b

, and

c

to use in the quadratic formula.

a = 2

,

b = -5

,

c = 2

These coefficients correspond to the standard quadratic form

ax^2 + bx + c = 0

.

3

Calculate the discriminant

b^2 - 4ac

.

(-5)^2 - 4(2)(2) = 25 - 16 = 9

The discriminant determines the number and type of solutions.

4

Substitute the coefficients and the discriminant into the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

.

x = \frac{-(-5) \pm \sqrt{9}}{2(2)} = \frac{5 \pm 3}{4}

This formula provides the exact solutions to any quadratic equation.

5

Evaluate the two cases for the plus/minus sign.

x = \frac{5 + 3}{4} = 2

and

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

Splitting the formula into positive and negative branches yields the two distinct real roots.

Key Concept

Solving quadratic equations using the quadratic formula

Question 108Question →

A retail electronics store sells two models of wireless headphones: Solo, which sells for $80, and Studio, which sells for$ 150. During a holiday sale, the number of headphones sold at the Downtown and Uptown store locations is represented by the matrix $N$ below, where the first row represents the Downtown store, the second row represents the Uptown store, the first column represents Solo headphones, and the second column represents Studio headphones:

N = \begin{bmatrix} 40 & 25 \\ 30 & 35 \end{bmatrix}

The prices of the headphones are represented by the matrix $P$ below:

P = \begin{bmatrix} 80 \\ 150 \end{bmatrix}

Which of the following matrices represents the total revenue, in dollars, generated from headphone sales at each of the two stores, where the first row represents the Downtown store and the second row represents the Uptown store?

\begin{bmatrix} 8,000 \\ 7,300 \end{bmatrix}

\begin{bmatrix} 5,600 \\ 9,000 \end{bmatrix}

\begin{bmatrix} 6,950 \\ 7,650 \end{bmatrix}

\begin{bmatrix} 6,850 \\ 7,650 \end{bmatrix}

\begin{bmatrix} 3,200 & 3,750 \\ 2,400 & 5,250 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 6,950 \\ 7,650 \end{bmatrix}$

Answer

The column matrix with first row 6,950 and second row 7,650

The total revenue for each store is found by calculating the product of the sales matrix

N

and the price matrix

P

. This multiplication is performed by taking the dot product of each row in

N

with the column in

P

. For the first row (Downtown store), the calculation is

40 \times 80 + 25 \times 150 = 3,200 + 3,750 = 6,950

. For the second row (Uptown store), the calculation is

30 \times 80 + 35 \times 150 = 2,400 + 5,250 = 7,650

. Combining these results yields the column matrix containing 6,950 and 7,650.

Step-by-Step Solution

1

Set up the matrix multiplication to calculate the revenue, multiplying the sales matrix

N

by the price matrix

P

.

NP = \begin{bmatrix} 40 & 25 \\ 30 & 35 \end{bmatrix} \begin{bmatrix} 80 \\ 150 \end{bmatrix}

To find the total revenue for each store, the number of units sold (organized by store in rows and product type in columns) must be multiplied by the unit prices (organized by product type in rows).

2

Calculate the revenue for the Downtown store by taking the dot product of the first row of

N

and the column of

P

.

40 \times 80 + 25 \times 150 = 3,200 + 3,750 = 6,950

This combines the units sold of each headphone model at the Downtown store with their respective prices to find the store's total revenue.

3

Calculate the revenue for the Uptown store by taking the dot product of the second row of

N

and the column of

P

.

30 \times 80 + 35 \times 150 = 2,400 + 5,250 = 7,650

This combines the units sold of each headphone model at the Uptown store with their respective prices to find the store's total revenue.

4

Combine the results into the final product matrix.

\begin{bmatrix} 6,950 \\ 7,650 \end{bmatrix}

The resulting matrix contains the total revenues for the Downtown store in the first row and the Uptown store in the second row.

Key Concept

Matrix Multiplication and Modeling

Alternative Method

Instead of setting up the entire matrix multiplication, calculate the total revenue for the first store (Downtown) individually:

40 \times 80 + 25 \times 150 = 6,950

. This single calculation is enough to eliminate the options that do not have 6,950 in the first row.

Estimated Time:2m 0s

Question 109Question →

Matrices $A$ , $B$ , and $C$ are defined such that $A = \begin{bmatrix} x & 3 \\ -2 & 4 \end{bmatrix}$ , $B = \begin{bmatrix} 2 & -1 \\ y & 5 \end{bmatrix}$ , and $C = \begin{bmatrix} 14 & 11 \\ 4 & 22 \end{bmatrix}$ . If $AB = C$ , what is the value of $x + y$ ?

Show answer & explanation

Answer: 6

Answer

The value of

x + y

is 6.

The correct answer is 6 because performing the matrix multiplication

AB

yields the matrix

\begin{bmatrix} 2x + 3y & -x + 15 \\ -4 + 4y & 22 \end{bmatrix}

. Equating this to matrix

C

gives the equations

-x + 15 = 11

, which yields

x = 4

, and

-4 + 4y = 4

, which yields

y = 2

. The sum of these values is

4 + 2 = 6

. This is verified by checking the remaining equation

2(4) + 3(2) = 14

.

Step-by-Step Solution

1

Multiply matrix

A

and matrix

B

to express the product

AB

algebraically.

AB = \begin{bmatrix} x(2) + 3(y) & x(-1) + 3(5) \\ -2(2) + 4(y) & -2(-1) + 4(5) \end{bmatrix} = \begin{bmatrix} 2x + 3y & -x + 15 \\ -4 + 4y & 22 \end{bmatrix}

To solve for the variables, we must first determine the resulting matrix from the multiplication of

A

and

B

.

2

Set up equations by equating the entries of

AB

with the corresponding entries of matrix

C

.

2x + 3y = 14

,

-x + 15 = 11

, and

-4 + 4y = 4

Two matrices are equal if and only if all of their corresponding elements are equal.

3

Solve the single-variable equation from the first row, second column for

x

.

-x + 15 = 11 \implies -x = -4 \implies x = 4

Isolating

x

gives its numeric value.

4

Solve the single-variable equation from the second row, first column for

y

.

-4 + 4y = 4 \implies 4y = 8 \implies y = 2

Isolating

y

gives its numeric value.

5

Verify consistency with the remaining equation and calculate the final sum

x + y

.

2(4) + 3(2) = 8 + 6 = 14

(consistent), and

x + y = 4 + 2 = 6

.

Checking the first row, first column confirms the values are correct, and adding them yields the requested sum.

Key Concept

Matrix multiplication and matrix equality

Question 110Question →

In the standard $(x, y)$ coordinate plane, the line defined by the equation $3x - 4y = k$ is tangent to the circle defined by the equation $x^2 + y^2 - 2x - 4y = 4$ . If $k > 0$ , what is the value of $k$ ?

Show answer & explanation

Answer: 10

Answer

10

The correct answer is 10. Completing the square for the circle's equation gives

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

, showing the center is

(1, 2)

and the radius is

3

. The distance from

(1, 2)

to the line

3x - 4y - k = 0

is

\frac{|3(1) - 4(2) - k|}{\sqrt{3^2 + (-4)^2}} = \frac{|k + 5|}{5}

. For tangency, this distance must equal the radius:

\frac{|k + 5|}{5} = 3

, which gives

|k + 5| = 15

. Solving this absolute value equation gives

k = 10

or

k = -20

. Since

k

must be positive, the value is 10.

Step-by-Step Solution

1

Complete the square for the circle's equation

x^2 + y^2 - 2x - 4y = 4

.

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

, which represents a circle with center

(1, 2)

and radius

R = 3

.

To find the center and radius of the circle, which are needed to use the distance formula.

2

Express the distance

d

from the center

(1, 2)

to the line

3x - 4y - k = 0

using the formula

d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}

.

d = \frac{|3(1) - 4(2) - k|}{\sqrt{3^2 + (-4)^2}} = \frac{|-5 - k|}{5} = \frac{|k + 5|}{5}

.

A line is tangent to a circle if and only if the distance from the center of the circle to the line equals the radius.

3

Set the distance equal to the radius (

3

) and solve for

k

.

\frac{|k + 5|}{5} = 3 \Rightarrow |k + 5| = 15

, which yields

k + 5 = 15 \Rightarrow k = 10

, or

k + 5 = -15 \Rightarrow k = -20

.

To find the values of

k

that make the line tangent to the circle.

4

Select the positive value of

k

.

k = 10

.

The problem specifies that

k > 0

.

Key Concept

Systems of Linear and Non-Linear Equations

Estimated Time:2m 30s

Question 111Question →

For a certain geometric sequence, the first term is $2^3$ and the common ratio is $2^2$ . Which of the following expressions represents the 3rd term of this sequence?

2^5

2^9

2^7

2^{10}

2^{12}

Show answer & explanation

Answer: $2^7$

Answer

The 3rd term of the sequence is

2^7

.

The correct answer is

2^7

. The

n

-th term of a geometric sequence is given by

a_n = a_1 \cdot r^{n-1}

. For a first term

a_1 = 2^3

and a common ratio

r = 2^2

, the 3rd term is

a_3 = 2^3 \cdot (2^2)^{3-1} = 2^3 \cdot 2^4

. Applying the exponent addition rule for multiplying bases of the same value yields

2^{3+4} = 2^7

.

Step-by-Step Solution

1

Identify the formula for the

n

-th term of a geometric sequence.

a_n = a_1 \cdot r^{n-1}

To find any specific term in a geometric sequence, the general term formula is used.

2

Substitute the given values (

a_1 = 2^3

,

r = 2^2

, and

n = 3

) into the formula.

a_3 = 2^3 \cdot (2^2)^{3-1} = 2^3 \cdot (2^2)^2

This sets up the calculation for the 3rd term of the sequence.

3

Simplify the expression using exponent rules: first compute

(2^2)^2

, then multiply by

2^3

.

a_3 = 2^3 \cdot 2^4 = 2^{3+4} = 2^7

Power of a power rule states

(x^a)^b = x^{ab}

, and product of powers rule states

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

.

Key Concept

Finding a specific term of a geometric sequence using the general term formula

a_n = a_1 \cdot r^{n-1}

and applying laws of exponents.

Estimated Time:45s

Question 112Question →

The functions $f$ and $g$ are defined for all real numbers by $f(x) = 2x - 5$ and $g(x) = (x + 3)^2$ . What is the value of $f(g(-1))$ ?

15

16

3

27

-28

Show answer & explanation

Answer: 3

Answer

3

To evaluate the composite function

f(g(-1))

, first evaluate the inner function

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

. Then, substitute this result into the outer function to get

f(4) = 2(4) - 5 = 3

.

Step-by-Step Solution

1

Evaluate the inner function

g(x)

at

x = -1

.

g(-1) = 4

In the composition

f(g(-1))

, the inner function must be evaluated first. Substituting

-1

into

g(x) = (x + 3)^2

gives

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

.

2

Evaluate the outer function

f(x)

at the output of the inner function.

f(4) = 3

Substitute the output

g(-1) = 4

into

f(x) = 2x - 5

, which yields

f(4) = 2(4) - 5 = 8 - 5 = 3

.

Key Concept

Function Composition and Evaluation

Estimated Time:1m 0s

Question 113Question →

A clothing boutique sells two types of shirts: t-shirts and polo shirts. The number of shirts sold at the boutique's two locations, Downtown and Uptown, on a weekend is represented by matrix

S

:

S = \begin{pmatrix} 40 & 30 \\ 50 & 20 \\end{pmatrix}

where the rows represent the locations (Downtown and Uptown, respectively) and the columns represent the shirt types (t-shirts and polo shirts, respectively).

The selling price and the production cost per shirt (in dollars) are represented by matrix

C

:

C = \begin{pmatrix} 15 & 6 \\ 25 & 10 \\end{pmatrix}

where the rows represent the shirt types (t-shirts and polo shirts, respectively) and the columns represent the selling price and production cost, respectively.

Which of the following matrices represents the total selling revenue and total production cost for the two locations?

\begin{pmatrix} 1350 & 540 \\ 1250 & 500 \\end{pmatrix}

\begin{pmatrix} 600 & 180 \\ 1250 & 200 \\end{pmatrix}

\begin{pmatrix} 900 & 570 \\ 1500 & 950 \\end{pmatrix}

\begin{pmatrix} 1850 & 740 \\ 950 & 380 \\end{pmatrix}

\begin{pmatrix} 780 & 1300 \\ 870 & 1450 \\end{pmatrix}

Show answer & explanation

Answer: \begin{pmatrix} 1350 & 540 \\ 1250 & 500 \\end{pmatrix}

Answer

The matrix representing the total selling revenue and total production cost for the two locations is the matrix with top row 1350 and 540, and bottom row 1250 and 500.

The correct matrix is obtained by multiplying the sales matrix

S

by the price/cost matrix

C

. Standard matrix multiplication pairs the quantity of each shirt type sold at each store with its respective selling price and production cost, yielding the correct total revenue and total production cost for both locations.

Step-by-Step Solution

1

Identify the dimensions and layout of the sales matrix

S

and the price/cost matrix

C

.

Matrix

S

is a

2 \times 2

matrix representing sales. Matrix

C

is a

2 \times 2

matrix representing price and cost. The product

SC

is defined because the number of columns in

S

(2) equals the number of rows in

C

(2).

Before multiplying, we must verify that the dimensions are compatible and that the product yields the desired real-world quantities.

2

Calculate the entries for the first row of the resulting matrix

SC

(Downtown location) by taking the dot product of the first row of

S

with the columns of

C

.

Revenue (Row 1, Column 1):

40 \times 15 + 30 \times 25 = 600 + 750 = 1350

.
Cost (Row 1, Column 2):

40 \times 6 + 30 \times 10 = 240 + 300 = 540

.

This yields the total revenue and production cost for the Downtown location by combining sales and prices/costs for both shirt types.

3

Calculate the entries for the second row of the resulting matrix

SC

(Uptown location) by taking the dot product of the second row of

S

with the columns of

C

.

Revenue (Row 2, Column 1):

50 \times 15 + 20 \times 25 = 750 + 500 = 1250

.
Cost (Row 2, Column 2):

50 \times 6 + 20 \times 10 = 300 + 200 = 500

.

This yields the total revenue and production cost for the Uptown location.

Key Concept

Matrix multiplication

Question 114Question →

An infinite geometric series of positive terms has a sum of $9$ . The sum of the first two terms of the series is $8$ . What is the first term of this series?

2

3

4

6

12

Show answer & explanation

Answer: 6

Answer

The first term of the series is 6.

To find the first term of the geometric series, we set up a system of equations using the formulas for the sum of the first two terms,

S_2 = a(1+r) = 8

, and the sum of an infinite geometric series,

S_\infty = \frac{a}{1-r} = 9

. Expressing the first term as

a = 9(1-r)

and substituting this into the first equation yields

9(1-r)(1+r) = 8

, which simplifies to

9(1-r^2) = 8

. Solving for the ratio yields

r^2 = \frac{1}{9}

, which means

r = \frac{1}{3}

since all terms must be positive. Substituting

r = \frac{1}{3}

back into

a = 9(1-r)

gives the first term as

6

.

Step-by-Step Solution

1

Define variables and identify the given formulas.

Let the first term of the geometric series be

a

and the common ratio be

r

. Since the series has positive terms, we require

a > 0

and

0 < r < 1

.

This sets up the system of equations using standard geometric sequence notation.

2

Translate the given problem conditions into algebraic equations.

Equation 1 (sum of the first two terms):

S_2 = a + ar = a(1+r) = 8

.
Equation 2 (sum to infinity):

S_\infty = \frac{a}{1-r} = 9

.

We must express the two mathematical relationships given in the problem statement.

3

Express the first term

a

in terms of the common ratio

r

using Equation 2.

a = 9(1-r)

This allows for substitution into Equation 1 to solve for

r

.

4

Substitute the expression for

a

into Equation 1 and simplify.

9(1-r)(1+r) = 8 \implies 9(1-r^2) = 8 \implies 9 - 9r^2 = 8

.

Substituting simplifies the system from two variables to a single quadratic variable in terms of

r

.

5

Solve the quadratic equation for

r

.

9r^2 = 1 \implies r^2 = \frac{1}{9} \implies r = \frac{1}{3}

(since

r > 0

).

Finding the value of the common ratio is the final step before calculating the first term.

6

Substitute the value of

r

back into the expression for

a

.

a = 9\left(1 - \frac{1}{3}\right) = 9\left(\frac{2}{3}\right) = 6

.

This yields the value of the first term

a

to complete the problem.

Key Concept

Solving systems of non-linear equations using geometric sequence term and infinite sum formulas.

Alternative Method

Instead of substituting

a = 9(1-r)

into

a(1+r) = 8

, we can divide the two equations:

\frac{a(1+r)}{a/(1-r)} = \frac{8}{9} \implies (1-r)(1+r) = \frac{8}{9} \implies 1 - r^2 = \frac{8}{9}

. This directly yields

r^2 = \frac{1}{9}

without needing to isolate

a

first.

Estimated Time:2m 0s

Question 115Question →

Consider the function $f$ defined on the domain $[\frac{2}{3}, \infty)$ by the equation $f(x) = 2 + \sqrt{3x - 2}$ . If $f^{-1}$ represents the inverse of $f$ , what is the only real value of $x$ for which $f(x) = f^{-1}(x)$ ?

Show answer & explanation

Answer: 6

Answer

The only real value of

x

for which

f(x) = f^{-1}(x)

is 6.

For a strictly increasing function, the intersection of

f(x)

and

f^{-1}(x)

must occur on the line

y = x

. Equating

f(x) = x

gives

2 + \sqrt{3x - 2} = x

. Isolating the radical term yields

\sqrt{3x - 2} = x - 2

. Squaring both sides produces

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

, which simplifies to the quadratic equation

x^2 - 7x + 6 = 0

. Factoring this equation gives

(x - 6)(x - 1) = 0

, yielding the solutions

x = 6

and

x = 1

. Substituting these back into the original equation shows that

x = 6

is a valid solution (

2 + \sqrt{16} = 6

), whereas

x = 1

is extraneous because it results in

2 + 1 = 3 \neq 1

. Furthermore, the domain of

f^{-1}

is the range of

f

, which is

[2, \infty)

, meaning

f^{-1}(1)

is undefined. Thus, the only real solution is 6.

Step-by-Step Solution

1

Equate the function to

x

using properties of increasing functions and their inverses.

Since

f(x)

is strictly increasing on its domain

[\frac{2}{3}, \infty)

, the graph of

f(x)

and the graph of its inverse

f^{-1}(x)

can only intersect on the line of symmetry

y = x

. Therefore, the equation

f(x) = f^{-1}(x)

is equivalent to

f(x) = x

.

This simplifies the relation by eliminating the need to solve a high-degree polynomial equation derived from direct composition or substitution.

2

Set up the equation

f(x) = x

and isolate the radical term.

2 + \sqrt{3x - 2} = x \implies \sqrt{3x - 2} = x - 2

Isolating the square root term is a necessary prerequisite step before squaring both sides.

3

Square both sides and rewrite the equation as a standard quadratic equation.

3x - 2 = (x - 2)^2 \implies 3x - 2 = x^2 - 4x + 4 \implies x^2 - 7x + 6 = 0

Squaring eliminates the radical to yield a standard quadratic equation that can be solved analytically.

4

Solve the quadratic equation by factoring.

(x - 6)(x - 1) = 0 \implies x = 6 \text{ or } x = 1

Factoring the quadratic trinomial yields the two candidate values for

x

.

5

Check the candidate solutions in the original equation to eliminate extraneous roots.

For

x = 1

:

2 + \sqrt{3(1)-2} = 3 \neq 1

(extraneous). For

x = 6

:

2 + \sqrt{3(6)-2} = 2 + 4 = 6

(valid). Also, the domain of

f^{-1}

is the range of

f

, which is

[2, \infty)

, thus excluding

x = 1

.

Squaring both sides can introduce extraneous roots. We must verify that the solutions satisfy the original radical equation and fall within the domains of both

f

and

f^{-1}

.

Key Concept

Applying the symmetry of inverse functions about the line

y=x

to solve composition-based equations, while rigorously accounting for domain restrictions and extraneous roots.

Alternative Method

Find the algebraic formula for

f^{-1}(x)

by setting

y = 2 + \sqrt{3x-2}

. Subtracting 2 and squaring both sides gives

(y - 2)^2 = 3x - 2

for

y \geq 2

. Solving for

x

yields

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

for

x \geq 2

. Equating

f(x) = f^{-1}(x)

results in

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

. Multiplying by 3 and isolating the radical term gives

3\sqrt{3x - 2} = x^2 - 4x

. Squaring both sides results in a fourth-degree polynomial equation:

9(3x - 2) = (x^2 - 4x)^2 \implies x^4 - 8x^3 + 16x^2 - 27x + 18 = 0

. This factors into

(x - 6)(x - 1)(x^2 - x + 3) = 0

. Since the domain of

f^{-1}(x)

is restricted to

x \geq 2

, the root

x = 1

is rejected, and the quadratic factor

x^2 - x + 3 = 0

has no real roots, leaving

x = 6

as the unique real solution.

Estimated Time:3m 0s

Question 116Question →

A right triangle has an area of $6$ square units. The lengths of its legs, $a$ and $b$ , satisfy the quadratic equation $a^2 - 2.75ab + 1.5b^2 = 0$ . If the length of the hypotenuse is an integer, what is the perimeter of the triangle?

9

10

12

14

15

Show answer & explanation

Answer: 12

Answer

The perimeter of the triangle is 12.

The correct answer is 12. By solving the quadratic relationship between the legs, we find two possible ratios: one where one leg is twice the other, and one where one leg is 0.75 times the other. Using the area of 6, the first case yields non-integer side lengths, while the second case yields legs of length 3 and 4. This results in an integer hypotenuse of 5, giving a perimeter of 3 + 4 + 5 = 12.

Step-by-Step Solution

1

Express the relation

a^2 - 2.75ab + 1.5b^2 = 0

as a quadratic in terms of the ratio

r = \frac{a}{b}

.

The equation becomes

r^2 - 2.75r + 1.5 = 0

.

To find the relationship between the two legs of the triangle by solving for their ratio.

2

Apply the quadratic formula to solve for

r

.

r = 2

or

r = 0.75

, meaning either

a = 2b

or

a = 0.75b

.

To determine the two possible linear relationships between the legs of the right triangle.

3

Substitute each ratio case into the area formula

\text{Area} = \frac{1}{2}ab = 6

, which simplifies to

ab = 12

.

For

a = 2b

, we get

b = \sqrt{6}

and

a = 2\sqrt{6}

. For

a = 0.75b

, we get

b = 4

and

a = 3

.

To calculate the actual leg lengths for both geometric cases.

4

Calculate the hypotenuse

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

for both cases to check which yields an integer value.

The first case yields

c = \sqrt{30}

, which is not an integer. The second case yields

c = \sqrt{3^2 + 4^2} = 5

, which is an integer.

To satisfy the constraint that the hypotenuse must be an integer, identifying the correct leg lengths as

3

and

4

.

5

Calculate the perimeter of the triangle for the valid case.

The perimeter is

3 + 4 + 5 = 12

.

To find the final requested value.

Key Concept

Solving quadratic relationships and applying the quadratic formula in geometric constraints.

Estimated Time:3m 0s

Question 117Question →

If $\sqrt{3x + 19} - x = 3$ , what is the value of $x$ ?

-10

-5

-2

2

5

Show answer & explanation

Answer: 2

Answer

The value of

x

is 2.

To solve the equation, isolate the radical to obtain

\sqrt{3x + 19} = x + 3

. Squaring both sides yields

3x + 19 = x^2 + 6x + 9

. Rearranging terms results in the quadratic equation

x^2 + 3x - 10 = 0

, which factors into

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

. The potential solutions are

x = 2

and

x = -5

. Testing these in the original equation shows that

2

is a valid solution because

\sqrt{3(2) + 19} - 2 = 3

. Testing

-5

results in

\sqrt{3(-5) + 19} - (-5) = 7

, which does not equal

3

. Thus, the only real solution is

2

.

Step-by-Step Solution

1

Isolate the radical expression on one side of the equation.

\sqrt{3x + 19} = x + 3

Isolating the radical allows us to eliminate it by squaring both sides.

2

Square both sides of the equation.

3x + 19 = x^2 + 6x + 9

Squaring a square root removes the radical. Remember to expand

(x+3)^2

correctly as

x^2 + 6x + 9

.

3

Rearrange the terms into standard quadratic form,

ax^2 + bx + c = 0

.

x^2 + 3x - 10 = 0

Moving all terms to one side sets up the quadratic equation for factoring.

4

Factor the quadratic expression.

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

Factoring allows us to find the potential values of

x

by setting each factor to zero.

5

Solve for the potential values of

x

.

x = -5

or

x = 2

Setting

x + 5 = 0

gives

x = -5

, and setting

x - 2 = 0

gives

x = 2

.

6

Check both potential solutions in the original equation to identify any extraneous solutions.

For

x = 2

:

\sqrt{3(2) + 19} - 2 = \sqrt{25} - 2 = 5 - 2 = 3

(valid). For

x = -5

:

\sqrt{3(-5) + 19} - (-5) = \sqrt{4} + 5 = 2 + 5 = 7 \neq 3

(extraneous).

Squaring both sides of an equation can introduce extraneous solutions that do not satisfy the original equation.

Key Concept

Solving radical equations and checking for extraneous solutions.

Question 118Question →

Let matrix $X = \begin{bmatrix} 5 & -3 \\ 2 & 1 \end{bmatrix}$ and matrix $Y = \begin{bmatrix} 1 & 4 \\ -2 & 3 \end{bmatrix}$ . If $Z = X - Y$ , what is the value of the element in the first row and second column of $Z$ ?

Show answer & explanation

Answer: -7

Answer

The correct answer is

-7

.

To find the element in the first row and second column of matrix

Z = X - Y

, we subtract the element in the first row and second column of matrix

Y

from the corresponding element in matrix

X

. The element in the first row and second column of

X

is

-3

, and the element in the first row and second column of

Y

is

4

. Subtracting these values gives

-3 - 4 = -7

.

Step-by-Step Solution

1

Identify the elements in the first row and second column for both matrices.

x_{1,2} = -3

and

y_{1,2} = 4

To find the element in the first row and second column of the resulting matrix

Z

, we must use the corresponding elements from matrices

X

and

Y

.

2

Subtract the element of

Y

from the element of

X

.

-3 - 4 = -7

Since

Z = X - Y

, each element

z_{i,j}

of the resulting matrix is calculated as

x_{i,j} - y_{i,j}

.

Key Concept

Matrix Subtraction

Question 119Question →

What is the sum of the two solutions to the quadratic equation $2x^2 - 7x - 4 = 0$ ?

Show answer & explanation

Answer: 3.5

Answer

The sum of the solutions is

3.5

.

The sum of the solutions of the quadratic equation

2x^2 - 7x - 4 = 0

is

3.5

. According to Vieta's formulas, the sum of the roots of a quadratic equation in the form

ax^2 + bx + c = 0

is given by

-\frac{b}{a}

. Here,

a = 2

and

b = -7

, so the sum is

-\frac{-7}{2} = 3.5

. Alternatively, solving the equation using the quadratic formula yields the roots

4

and

-0.5

, which sum to

3.5

.

Step-by-Step Solution

1

Identify the coefficients from the quadratic equation

2x^2 - 7x - 4 = 0

.

a = 2

,

b = -7

, and

c = -4

.

A quadratic equation in standard form is written as

ax^2 + bx + c = 0

.

2

Apply the sum of roots formula

-\frac{b}{a}

.

Sum

= -\frac{-7}{2} = 3.5

.

By Vieta's formulas, the sum of the roots of

ax^2 + bx + c = 0

is

-\frac{b}{a}

.

Key Concept

Sum of roots of a quadratic equation using Vieta's formulas

Alternative Method

Solve the quadratic equation by factoring or using the quadratic formula. Factoring

2x^2 - 7x - 4 = 0

yields

(2x + 1)(x - 4) = 0

, which gives solutions

x = -0.5

and

x = 4

. Adding these solutions together gives

-0.5 + 4 = 3.5

.

Estimated Time:1m 0s

Question 120Question →

A system of equations is given below:

\begin{aligned} y &= (x - 1)^2 - 6 \\ y &= x - 1 \end{aligned}

If $(x, y)$ is a solution to this system in the first quadrant, what is the value of $x + y$ ?

5

-3

7

4

9

Show answer & explanation

Answer: 7

Answer

The sum of the coordinates of the first quadrant solution is 7.

To solve the system, substitute the expression for

y

from the linear equation into the quadratic equation to get

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

. Substituting

u = x - 1

yields

u^2 - u - 6 = 0

. Factoring the quadratic expression gives

(u - 3)(u + 2) = 0

, so

u = 3

or

u = -2

. Since

u = x - 1

, this means

x = 4

or

x = -1

. Evaluating both cases gives the intersection points

(4, 3)

and

(-1, -2)

. Only

(4, 3)

is in the first quadrant. Adding these coordinates together yields

4 + 3 = 7

.

Step-by-Step Solution

1

Substitute

y = x - 1

into the quadratic equation to set up an equation in terms of

x

.

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

To find the

x

-coordinates of the intersection points.

2

Solve the equation for

x

by substituting

u = x - 1

.

u = u^2 - 6 \Rightarrow u^2 - u - 6 = 0 \Rightarrow (u - 3)(u + 2) = 0

. This gives

u = 3

or

u = -2

.

To find the values of the substituted variable

u

.

3

Find the corresponding

x

and

y

values for both cases.

Case 1:

x - 1 = 3 \Rightarrow x = 4

, which gives

y = 3

. Point is

(4, 3)

. Case 2:

x - 1 = -2 \Rightarrow x = -1

, which gives

y = -2

. Point is

(-1, -2)

.

To determine the full coordinates of all intersection points.

4

Identify the first quadrant solution and calculate the sum of its coordinates.

The first quadrant solution is

(4, 3)

. The sum of the coordinates is

4 + 3 = 7

.

To answer the question's requirement for the sum of coordinates in the first quadrant.

Key Concept

Solving systems of linear and quadratic equations by substitution and identifying quadrant-specific solutions.

Intermediate Algebra

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