Elementary Algebra Practice Questions — ACT — Page 4

Question 61Question →

A specialty coffee shop blends three types of coffee beans: Colombian ( $8.00$ dollars per pound), Ethiopian ( $11.00$ dollars per pound), and Sumatran ( $14.00$ dollars per pound). The shop manager wants to create a $50$ -pound blend that costs exactly $11.60$ dollars per pound. Due to supply constraints, the weight of the Ethiopian beans in the blend must be exactly $2$ pounds more than $\frac{1}{5}$ of the combined weight of the Colombian and Sumatran beans. What is the value of $2s - c$ , where $s$ is the amount of Sumatran beans, in pounds, and $c$ is the amount of Colombian beans, in pounds, in the blend?

Show answer & explanation

Answer: 35

Answer

The correct answer is

35

.

The correct answer of

35

is found by translating the given constraints into three linear equations, solving for

e = 10

using substitution of the sum

c+s = 50 - e

, and then solving the resulting two-variable system to find

s = 25

and

c = 15

. Evaluating

2s - c

yields

2(25) - 15 = 35

.

Step-by-Step Solution

1

Define variables and write the system of equations representing the constraints.

Let

c

,

e

, and

s

represent the weight in pounds of Colombian, Ethiopian, and Sumatran beans, respectively. The system is:
1)

c + e + s = 50

(total weight)
2)

8c + 11e + 14s = 50 \times 11.60 = 580

(total cost)
3)

e = \frac{1}{5}(c + s) + 2

(Ethiopian bean weight constraint)

This translates the word problem's conditions into algebraic expressions.

2

Solve for the variable

e

using substitution.

From equation (1), we have

c + s = 50 - e

. Substituting this expression into equation (3) yields:

e = \frac{1}{5}(50 - e) + 2

Multiply both sides by

5

:

5e = 50 - e + 10

6e = 60 \implies e = 10

Grouping

c + s

allows us to solve for

e

directly without dealing with three separate variable eliminations.

3

Substitute

e = 10

back into the first two equations to simplify the system to two variables.

Equation (1) becomes:

c + s = 40 \implies c = 40 - s

Equation (2) becomes:

8c + 11(10) + 14s = 580 \implies 8c + 14s = 470

This reduces the remaining problem to a standard system of two linear equations.

4

Solve for

c

and

s

by substitution.

Substitute

c = 40 - s

into the simplified cost equation:

8(40 - s) + 14s = 470

320 - 8s + 14s = 470

320 + 6s = 470 \implies 6s = 150 \implies s = 25

Then, find

c

:

c = 40 - 25 = 15

This gives the exact weight of both the Sumatran and Colombian beans.

5

Evaluate the expression

2s - c

requested by the prompt.

2s - c = 2(25) - 15 = 50 - 15 = 35

This satisfies the specific algebraic quantity requested in the question.

Key Concept

Translating word problems into systems of linear equations and solving them using algebraic substitution.

Question 62Question →

Which of the following is the completely factored form of the expression $x^3 - 9x$ ?

x(x - 3)^2

x(x^3 - 9)

x(x - 3)(x + 3)

9x(x - 1)(x + 1)

x(x + 3)^2

Show answer & explanation

Answer: $x(x - 3)(x + 3)$

Answer

x(x - 3)(x + 3)

The expression

x^3 - 9x

can be factored by first finding the greatest common factor of the terms. Since both terms share a factor of

x

, factoring out

x

yields

x(x^2 - 9)

. The binomial

x^2 - 9

is a difference of squares because it can be written as

x^2 - 3^2

. Applying the difference of squares formula,

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

, gives

(x - 3)(x + 3)

. Therefore, the completely factored form is

x(x - 3)(x + 3)

.

Step-by-Step Solution

1

Identify and factor out the greatest common factor (GCF) of the terms in the expression.

x(x^2 - 9)

Both

x^3

and

9x

share a common factor of

x

.

2

Factor the remaining binomial expression inside the parentheses using the difference of squares formula.

x(x - 3)(x + 3)

The expression

x^2 - 9

is a difference of squares, which factors as

(a - b)(a + b)

where

a = x

and

b = 3

.

Key Concept

Factoring out the greatest common factor and factoring a difference of squares.

Question 63Question →

If $x(x - 6) = 7$ , what is the positive difference between the two solutions to this equation?

6

8

10

14

0

Show answer & explanation

Answer: 8

Answer

The positive difference between the two solutions is

8

.

First, expand the left side of the equation to get

x^2 - 6x = 7

. Next, subtract

7

from both sides to write the equation in standard form:

x^2 - 6x - 7 = 0

. Factoring this quadratic expression gives

(x - 7)(x + 1) = 0

. Setting each factor to zero yields the solutions

x = 7

and

x = -1

. The positive difference between these two solutions is

7 - (-1) = 8

.

Step-by-Step Solution

1

Distribute the

x

on the left side of the equation.

x^2 - 6x = 7

Multiplying

x

by

(x - 6)

expands the left side to prepare for standard form.

2

Subtract

7

from both sides to set the quadratic equation to zero.

x^2 - 6x - 7 = 0

Setting the quadratic equation to standard form

ax^2 + bx + c = 0

is necessary before factoring.

3

Factor the quadratic trinomial.

(x - 7)(x + 1) = 0

We need two numbers that multiply to

-7

and add to

-6

, which are

-7

and

1

.

4

Set each factor to zero to find the solutions.

x = 7

and

x = -1

The zero product property states that if a product is zero, at least one factor must be zero.

5

Calculate the positive difference between the two solutions.

7 - (-1) = 8

The positive difference is found by subtracting the smaller solution from the larger solution.

Key Concept

Solving Quadratic Equations by Factoring

Estimated Time:1m 0s

Question 64Question →

The area of a rectangle is represented by the expression $18x^3 - 50x$ square inches. If the width of the rectangle is $2x$ inches, which of the following expressions represents the length of the rectangle, in inches?

(3x - 5)^2

9x - 25

9x^2 - 50x

(3x - 5)(3x + 5)

18x^3 - 52x

Show answer & explanation

Answer: $(3x - 5)(3x + 5)$

Answer

The length of the rectangle is represented by the expression

(3x - 5)(3x + 5)

.

The correct answer is the expression

(3x - 5)(3x + 5)

because the length is found by dividing the area,

18x^3 - 50x

, by the width,

2x

. Simplifying this quotient yields

9x^2 - 25

. The expression

9x^2 - 25

is a difference of squares and factors completely into

(3x - 5)(3x + 5)

.

Step-by-Step Solution

1

Set up the equation for the length of the rectangle by dividing the area by the width.

\text{Length} = \frac{18x^3 - 50x}{2x}

Since the area of a rectangle is the product of its length and width, the length can be found by dividing the area by the width.

2

Perform the polynomial division by dividing each term of the numerator by the denominator

2x

.

\frac{18x^3 - 50x}{2x} = \frac{18x^3}{2x} - \frac{50x}{2x} = 9x^2 - 25

Dividing a polynomial by a monomial requires dividing every term of the polynomial by that monomial.

3

Factor the resulting expression

9x^2 - 25

using the difference of squares identity,

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

.

9x^2 - 25 = (3x)^2 - 5^2 = (3x - 5)(3x + 5)

The quadratic expression is a difference of two perfect squares, which can be factored into a product of binomial conjugates.

Key Concept

Factoring a difference of squares after dividing a polynomial by a monomial

Estimated Time:1m 30s

Question 65Question →

For all non-zero real numbers $x$ and $y$ , the algebraic expression $\left(\frac{x^{a/2} y^{-1}}{x^{-2/3} y^{b/3}}\right)^{-6}$ is equivalent to $\frac{y^{12}}{x^{16}}$ , where $a$ and $b$ are integers. What is the value of $a + b$ ?

-14

1

7

9

11

Show answer & explanation

Answer: 7

Answer

7

The correct answer is 7. Simplifying the expression inside the parentheses using the quotient rule yields

x^{\frac{a}{2} + \frac{2}{3}} y^{-1 - \frac{b}{3}}

. Raising this expression to the power of

-6

results in

x^{-3a - 4} y^{6 + 2b}

. Equating this to

x^{-16} y^{12}

gives

a = 4

and

b = 3

, so

a + b = 7

.

Step-by-Step Solution

1

Simplify the expression inside the parentheses using the quotient rule for exponents,

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

.

x^{\frac{a}{2} - \left(-\frac{2}{3}\right)} y^{-1 - \frac{b}{3}} = x^{\frac{a}{2} + \frac{2}{3}} y^{-1 - \frac{b}{3}}

To combine the base

x

and base

y

terms inside the parentheses before applying the outer power.

2

Apply the outer exponent of

-6

to the simplified expression using the power of a power rule,

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

.

x^{-6\left(\frac{a}{2} + \frac{2}{3}\right)} y^{-6\left(-1 - \frac{b}{3}\right)} = x^{-3a - 4} y^{6 + 2b}

To distribute the negative power of

-6

to each factor in the product.

3

Equate the resulting exponents to the exponents of the given equivalent expression,

\frac{y^{12}}{x^{16}} = x^{-16} y^{12}

, and solve the equations for

a

and

b

.

For

x

:

-3a - 4 = -16 \implies -3a = -12 \implies a = 4

. For

y

:

6 + 2b = 12 \implies 2b = 6 \implies b = 3

.

Since the expressions are equivalent for all non-zero values of

x

and

y

, their respective exponents must be equal.

4

Calculate the sum of

a

and

b

.

a + b = 4 + 3 = 7

To find the final requested value.

Key Concept

Properties of exponents in algebraic expressions including the quotient rule, power rule, and operations with fractional/negative exponents.

Question 66Question →

What is the positive difference between the two real solutions to the equation $(2x - 1)^2 = x(3x - 5) + 7$ ?

1

3

5

6

7

Show answer & explanation

Answer: 5

Answer

The positive difference between the two real solutions is 5.

The correct answer is 5. By expanding the equation, we get

4x^2 - 4x + 1 = 3x^2 - 5x + 7

. Rearranging the terms to set the equation to zero yields

x^2 + x - 6 = 0

. Factoring this equation gives

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

, which has the solutions

x = -3

and

x = 2

. The positive difference between these two solutions is

|2 - (-3)| = 5

.

Step-by-Step Solution

1

Expand both sides of the equation.

4x^2 - 4x + 1 = 3x^2 - 5x + 7

Expanding the squared term on the left side and distributing the

x

on the right side allows us to write the equation in polynomial form.

2

Rearrange the equation by moving all terms to the left side to set the right side to zero.

x^2 + x - 6 = 0

Subtracting

3x^2

, adding

5x

, and subtracting

7

from both sides simplifies the equation into standard quadratic form,

ax^2 + bx + c = 0

.

3

Factor the quadratic trinomial.

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

Finding two integers that multiply to

-6

and add to

1

gives

3

and

-2

, allowing us to factor the equation over the integers.

4

Solve for

x

by setting each linear factor to zero.

x = -3

and

x = 2

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

5

Calculate the positive difference between the two solutions.

|2 - (-3)| = 5

Subtracting the smaller solution from the larger solution gives the positive distance between them on the number line.

Key Concept

Solving quadratic equations by expanding, rearranging into standard form, and factoring over the integers.

Estimated Time:2m 0s

Question 67Question →

What is the positive difference between the two real solutions to the equation $(x - 1)^2 = 5x - 5$ ?

1

5

7

6

4

Show answer & explanation

Answer: 5

Answer

The positive difference between the two real solutions is 5.

Expanding the left side of

(x - 1)^2 = 5x - 5

yields

x^2 - 2x + 1 = 5x - 5

. Moving all terms to the left side by subtracting

5x

and adding

5

gives the standard quadratic equation

x^2 - 7x + 6 = 0

. Factoring this expression gives

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

, which yields the solutions

1

and

6

. The positive difference between these solutions is

6 - 1 = 5

.

Step-by-Step Solution

1

Expand the squared binomial on the left side of the equation.

x^2 - 2x + 1 = 5x - 5

Before factoring a quadratic equation, all terms must be expanded and moved to one side to set the equation equal to zero.

2

Subtract

5x

and add

5

to both sides to rearrange the equation into standard quadratic form.

x^2 - 7x + 6 = 0

This sets the equation equal to zero, which is a prerequisite for using the zero product property.

3

Factor the quadratic expression by finding two numbers that multiply to

6

and add to

-7

.

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

Factoring allows us to split the quadratic equation into two linear equations.

4

Set each factor to zero to solve for

x

.

x = 1

and

x = 6

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

5

Subtract the smaller solution from the larger solution to find the positive difference.

6 - 1 = 5

The question asks for the positive difference between the two solutions.

Key Concept

Solving quadratic equations by expanding, rearranging into standard form, and factoring over the integers.

Question 68Question →

A rental car company charges a flat fee of $30$ dollars plus $0.20$ dollars per mile driven. If a customer's total rental bill is $56$ dollars, how many miles did they drive?

Show answer & explanation

Answer: 130

Answer

130 miles

Subtracting the flat fee of

30

dollars from the total bill of

56

dollars leaves

26

dollars representing the cost of the miles driven. Dividing this remaining cost of

26

dollars by the rate of

0.20

dollars per mile yields a total of

130

miles driven.

Step-by-Step Solution

1

Set up a linear equation for the total cost

30 + 0.20m = 56

The total cost of

56

dollars is the sum of the flat fee of

30

dollars and the variable cost of

0.20

dollars per mile multiplied by the number of miles

m

.

2

Subtract the flat fee from both sides of the equation

0.20m = 26

Isolating the variable term shows that the total amount spent on the mileage portion of the trip is

26

dollars.

3

Divide by the rate per mile to find the total miles

m = 130

Dividing the mileage portion of the cost by the cost per mile gives the total distance driven.

Key Concept

Translating and Solving Linear Word Problems

Question 69Question →

A local community theater group sells tickets for their upcoming play. A student ticket costs $8$ dollars, which is $4$ dollars less than half the price of an adult ticket. If $a$ represents the cost of an adult ticket in dollars, what is the value of $a$ ?

6

8

12

20

24

Show answer & explanation

Answer: $24$

Answer

The cost of the adult ticket is

24

dollars.

The correct answer is

24

. According to the problem, the student ticket price of

8

dollars is equal to

4

dollars less than half the price of the adult ticket (

a

). This relationship can be expressed algebraically as

8 = \frac{1}{2}a - 4

. Adding

4

to both sides of the equation results in

12 = \frac{1}{2}a

. Multiplying both sides by

2

yields

a = 24

.

Step-by-Step Solution

1

Translate the word problem into a linear equation.

8 = \frac{1}{2}a - 4

The student ticket price (

8

dollars) is equal to

4

dollars subtracted from half the price of the adult ticket (

a

).

2

Isolate the variable term by adding

4

to both sides of the equation.

12 = \frac{1}{2}a

To solve for

a

, we first eliminate the constant subtraction on the variable's side.

3

Solve for

a

by multiplying both sides of the equation by

2

.

a = 24

Multiplying by the reciprocal of

\frac{1}{2}

isolates the variable

a

.

Key Concept

Translating verbal statements into two-step linear equations and solving for the unknown variable.

Question 70Question →

If the expression $6x^2 - 11x - 10$ is factored completely into the form $(ax + b)(cx - d)$ , where $a, b, c,$ and $d$ are positive integers, what is the value of $a + b + c + d$ ?

Show answer & explanation

Answer: 12

Answer

The value of the sum of the coefficients and constants

a + b + c + d

is

12

.

The factored form of

6x^2 - 11x - 10

is

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

. Matching this with

(ax + b)(cx - d)

where

a, b, c,

and

d

are positive integers results in

a = 3

,

b = 2

,

c = 2

, and

d = 5

. The sum of these values is

3 + 2 + 2 + 5 = 12

.

Step-by-Step Solution

1

Factor the quadratic trinomial

6x^2 - 11x - 10

using the grouping method.

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

Factoring splits the quadratic expression into its constituent linear binomial factors.

2

Equate the factored expression to the given form

(ax + b)(cx - d)

to find the values of

a, b, c,

and

d

.

a = 3

,

b = 2

,

c = 2

, and

d = 5

Since the variables represent positive integers, we match the positive constant term to

b

and the negative constant term to

-d

.

3

Sum the values of

a, b, c,

and

d

.

12

Calculating the final sum answers the target mathematical question.

Key Concept

Factoring quadratic polynomials with a leading coefficient greater than 1 using the grouping method.

Question 71Question →

A scientist is mixing two solutions in a laboratory. Solution A contains $\frac{1}{4}$ active ingredient by volume, and Solution B contains $\frac{3}{4}$ active ingredient by volume. The scientist needs to mix these to create a $20$ -liter solution containing exactly $\frac{11}{20}$ active ingredient by volume.

Let

x

represent the volume, in liters, of Solution A used. The correct relationship is represented by the linear equation:

\frac{1}{4}x + \frac{3}{4}(20 - x) = 11

An assistant translates the problem description incorrectly, writing the equation as:

\frac{1}{4}x + \frac{3}{4}(x - 20) = 11

What is the absolute difference, in liters, between the value of $x$ obtained from the assistant's incorrect equation and the value of $x$ obtained from the correct equation?

8

30

18

26

44

Show answer & explanation

Answer: 18

Answer

The correct answer is 18, which is the absolute difference between the two solutions.

The correct answer is 18. Solving the correct equation

\frac{1}{4}x + \frac{3}{4}(20 - x) = 11

by multiplying by 4 gives

x + 60 - 3x = 44

, which simplifies to

-2x = -16

and yields

x = 8

. Solving the assistant's incorrect equation

\frac{1}{4}x + \frac{3}{4}(x - 20) = 11

in a similar manner gives

x + 3x - 60 = 44

, which simplifies to

4x = 104

and yields

x = 26

. The absolute difference between these two values is

|26 - 8| = 18

.

Step-by-Step Solution

1

Solve the correct equation for

x

.

Multiply the entire equation by 4 to clear the denominators:

x + 3(20 - x) = 44

. Distribute the 3:

x + 60 - 3x = 44

. Combine like terms:

-2x + 60 = 44

. Subtract 60 from both sides:

-2x = -16

. Divide by -2:

x = 8

.

This determines the correct volume of Solution A needed.

2

Solve the assistant's incorrect equation for

x

.

Multiply the entire equation by 4 to clear the denominators:

x + 3(x - 20) = 44

. Distribute the 3:

x + 3x - 60 = 44

. Combine like terms:

4x - 60 = 44

. Add 60 to both sides:

4x = 104

. Divide by 4:

x = 26

.

This determines the incorrect volume of Solution A obtained by the assistant.

3

Calculate the absolute difference between the two solutions.

|26 - 8| = 18

.

The question asks for the absolute difference between the two values of

x

.

Key Concept

Solving linear equations involving fractions, parentheses, and algebraic manipulation.

Question 72Question →

Match each algebraic expression on the left with its fully simplified equivalent expression on the right for all real values of $m$ and $n$ .

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches are: (1)

m(m^2 - 3mn) - 2n(m^2 - n^2) - (m^3 - mn^2)

matches with

-5m^2n + mn^2 + 2n^3

; (2)

(m - n)^3 - m(m^2 - 3n^2) + n^3

matches with

-3m^2n + 6mn^2

; (3)

3mn(m - n) - 2m(n^2 - mn) - (m^2n - 5mn^2)

matches with

4m^2n

; and (4)

m^2(2m - n) - n(m^2 - n^2) - (m^3 + n^3)

matches with

m^3 - 2m^2n

.

Each expression is simplified by systematically expanding parenthetical groups and collecting like terms with identical variable powers.

Step-by-Step Solution

1

Simplify the expression

m(m^2 - 3mn) - 2n(m^2 - n^2) - (m^3 - mn^2)

-5m^2n + mn^2 + 2n^3

Distribute each term across the parenthetical expressions:

m^3 - 3m^2n - 2m^2n + 2n^3 - m^3 + mn^2

. Group the like terms:

(m^3 - m^3) + (-3m^2n - 2m^2n) + mn^2 + 2n^3

, which simplifies to

-5m^2n + mn^2 + 2n^3

.

2

Simplify the expression

(m - n)^3 - m(m^2 - 3n^2) + n^3

-3m^2n + 6mn^2

Use the binomial expansion formula to expand

(m - n)^3 = m^3 - 3m^2n + 3mn^2 - n^3

. Distribute the

-m

term to obtain

-m^3 + 3mn^2

. Sum all the expressions and combine like terms:

(m^3 - m^3) - 3m^2n + (3mn^2 + 3mn^2) + (-n^3 + n^3) = -3m^2n + 6mn^2

.

3

Simplify the expression

3mn(m - n) - 2m(n^2 - mn) - (m^2n - 5mn^2)

4m^2n

Expand the terms by distributing the outer coefficients to get

3m^2n - 3mn^2 - 2mn^2 + 2m^2n - m^2n + 5mn^2

. Grouping similar variables yields

(3 + 2 - 1)m^2n + (-3 - 2 + 5)mn^2 = 4m^2n + 0 = 4m^2n

.

4

Simplify the expression

m^2(2m - n) - n(m^2 - n^2) - (m^3 + n^3)

m^3 - 2m^2n

Expand the terms by distributing the multiplication:

2m^3 - m^2n - m^2n + n^3 - m^3 - n^3

. Grouping like terms yields

(2m^3 - m^3) + (-m^2n - m^2n) + (n^3 - n^3) = m^3 - 2m^2n

.

Key Concept

Simplifying multivariable expressions by distributing terms (including negative signs) and combining like terms.

Question 73Question →

For all real values of $x$ and $y$ , the expression $5x^3y - 2x^2(3xy - y) - 4x^2y$ can be simplified. Which of the following is equivalent to this simplified expression?

-x^3y - 6x^2y

5x^3y - 8x^2y

-x^3y - 2x^2y

-3x^5y^2

-x^3y + 6x^2y

Show answer & explanation

Answer: $-x^3y - 2x^2y$

Answer

-x^3y - 2x^2y

Distributing

-2x^2

across the parentheses

(3xy - y)

yields

-6x^3y + 2x^2y

. Substituting this back gives

5x^3y - 6x^3y + 2x^2y - 4x^2y

. Combining the

x^3y

terms (

5x^3y - 6x^3y = -x^3y

) and the

x^2y

terms (

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

) yields the simplified expression

-x^3y - 2x^2y

.

Step-by-Step Solution

1

Distribute the term

-2x^2

to both terms inside the parentheses

(3xy - y)

.

-6x^3y + 2x^2y

Multiply

-2x^2

by

3xy

(yielding

-6x^3y

) and multiply

-2x^2

by

-y

(yielding

+2x^2y

).

2

Substitute the expanded terms back into the original expression.

5x^3y - 6x^3y + 2x^2y - 4x^2y

Replace

-2x^2(3xy - y)

with the expanded terms

-6x^3y + 2x^2y

.

3

Combine like terms by grouping the

x^3y

terms and the

x^2y

terms.

-x^3y - 2x^2y

Combine

5x^3y - 6x^3y = -x^3y

and

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

.

Key Concept

Simplifying expressions by distributing terms and combining like terms.

Question 74Question →

Which of the following is equivalent to the expression $2a(3a - b)^2 - 3b(2a^2 - ab) - a^2(18a - 15b)$ for all real values of $a$ and $b$ ?

9a^2b + 5ab^2

-33a^2b + 5ab^2

-3a^2b - ab^2

-3a^2b + 5ab^2

-3a^2b + 5ab

Show answer & explanation

Answer: $-3a^2b + 5ab^2$

Answer

The simplified expression is

-3a^2b + 5ab^2

.

The correct expression is

-3a^2b + 5ab^2

. Expanding the three parts yields

18a^3 - 12a^2b + 2ab^2

,

-6a^2b + 3ab^2

, and

-18a^3 + 15a^2b

. Summing these parts together cancels out the cubic

a^3

terms (

18a^3 - 18a^3 = 0

), combines the

a^2b

terms (

-12a^2b - 6a^2b + 15a^2b = -3a^2b

), and combines the

ab^2

terms (

2ab^2 + 3ab^2 = 5ab^2

).

Step-by-Step Solution

1

Expand the first term

2a(3a - b)^2

18a^3 - 12a^2b + 2ab^2

First expand the squared binomial

(3a - b)^2 = 9a^2 - 6ab + b^2

, then distribute the

2a

to each term.

2

Expand the second term

-3b(2a^2 - ab)

-6a^2b + 3ab^2

Distribute

-3b

to both terms inside the parentheses, ensuring that the negative sign is applied to both terms.

3

Expand the third term

-a^2(18a - 15b)

-18a^3 + 15a^2b

Distribute

-a^2

to both terms inside the parentheses, changing the sign of both terms.

4

Combine the expanded expressions and group like terms

(18a^3 - 18a^3) + (-12a^2b - 6a^2b + 15a^2b) + (2ab^2 + 3ab^2)

Grouping terms with identical variable parts allows them to be added or subtracted.

5

Perform the final simplification

-3a^2b + 5ab^2

The

a^3

terms cancel out, the

a^2b

terms combine to

-3a^2b

, and the

ab^2

terms combine to

5ab^2

.

Key Concept

Simplifying algebraic expressions by expanding parentheses, applying exponent rules, and combining like terms.

Question 75Question →

A charity concert sells floor tickets and balcony tickets. The price of a floor ticket is $5$ dollars more than twice the price of a balcony ticket. If the concert organizers sell $80$ balcony tickets and $50$ floor tickets, their total revenue is $R$ dollars. Under a new promotional structure, the price of a balcony ticket is discounted by $25\%$ , the price of a floor ticket is increased by $10\%$ , and the organizers sell $120$ balcony tickets. In terms of the original price of a balcony ticket, $b$ , which of the following expressions represents the number of floor tickets they must sell under the new promotional structure to achieve the same total revenue, $R$ ?

\frac{1500b + 2500}{22b + 55}

\frac{900b + 2500}{22b - 55}

\frac{900b + 2500}{22b + 55}

\frac{2500 - 1100b}{55 - 22b}

\frac{900b + 5000}{22b + 110}

Show answer & explanation

Answer: $\frac{900b + 2500}{22b + 55}$

Answer

\frac{900b + 2500}{22b + 55}

The correct expression is derived by first writing the original revenue in terms of the balcony ticket price

b

, which yields

R = 180b + 250

. Under the new pricing, the balcony ticket costs

0.75b

and the floor ticket costs

1.1(2b + 5) = 2.2b + 5.5

. Selling

120

balcony tickets generates

90b

in revenue. Setting the new total revenue equal to

R

gives

90b + F(2.2b + 5.5) = 180b + 250

. Solving for the number of floor tickets,

F

, results in

F = \frac{90b + 250}{2.2b + 5.5}

. Multiplying the numerator and denominator by

10

to clear the decimals yields the correct expression.

Step-by-Step Solution

1

Express the original ticket prices and revenue in terms of

b

.

Balcony ticket price =

b

. Floor ticket price =

2b + 5

. Original revenue

R = 80b + 50(2b + 5) = 180b + 250

.

To set up the baseline total revenue equation using the algebraic descriptions.

2

Determine the new pricing for both balcony and floor tickets.

New balcony ticket price =

0.75b

. New floor ticket price =

1.1(2b + 5) = 2.2b + 5.5

.

To apply the

25\%

discount and

10\%

increase to the original ticket prices.

3

Set up the equation equating the new promotional revenue to the original revenue

R

.

New Revenue =

120(0.75b) + F(2.2b + 5.5) = 180b + 250

, where

F

is the number of floor tickets.

To represent the condition that the total revenue remains the same under the new structure.

4

Simplify the equation and isolate the variable

F

.

90b + F(2.2b + 5.5) = 180b + 250 \implies F(2.2b + 5.5) = 90b + 250 \implies F = \frac{90b + 250}{2.2b + 5.5}

.

To solve for the number of floor tickets algebraically.

5

Clear decimals from the rational expression.

F = \frac{10(90b + 250)}{10(2.2b + 5.5)} = \frac{900b + 2500}{22b + 55}

.

To match the standard fraction format of the options by multiplying the numerator and denominator by

10

.

Key Concept

Translating verbal descriptions into multi-step algebraic systems and isolating a target variable from rational equations.

Estimated Time:3m 0s

Question 76Question →

The cubic polynomial $6x^3 - 19x^2 + 11x + 6$ can be factored completely over the integers in the form $(x - a)(bx - c)(dx + e)$ , where $a$ , $b$ , $c$ , $d$ , and $e$ are positive integers. What is the value of $a + b + c + d + e$ ?

Show answer & explanation

Answer: 11

Answer

The sum of the coefficients and constants from the factored form is 11.

By applying the Rational Root Theorem, we find the root

x = 2

, which gives the factor

(x - 2)

. Dividing the original cubic expression by

(x - 2)

yields

6x^2 - 7x - 3

. Factoring this quadratic expression by grouping yields

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

. Writing the completely factored form as

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

and comparing it to

(x - a)(bx - c)(dx + e)

where

a, b, c, d, e

are positive integers results in

a = 2

,

b = 2

,

c = 3

,

d = 3

, and

e = 1

. The sum

a + b + c + d + e

is equal to 11.

Step-by-Step Solution

1

Identify a rational root of

6x^3 - 19x^2 + 11x + 6

x = 2

is a root, meaning

(x - 2)

is a factor.

Applying the Rational Root Theorem and testing potential integer root values.

2

Divide

6x^3 - 19x^2 + 11x + 6

by

(x - 2)

The quotient is the quadratic polynomial

6x^2 - 7x - 3

.

To reduce the cubic polynomial to a quadratic expression that can be factored using standard trinomial methods.

3

Factor the quadratic trinomial

6x^2 - 7x - 3

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

Finding two numbers that multiply to

-18

and add to

-7

(which are

-9

and

2

) and factoring by grouping.

4

Match the factored form

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

to

(x - a)(bx - c)(dx + e)

a = 2, b = 2, c = 3, d = 3, e = 1

Since

a, b, c, d,

and

e

must be positive integers, the constant terms and signs uniquely determine the mapping of each factor.

5

Calculate the sum of

a, b, c, d,

and

e

2 + 2 + 3 + 3 + 1 = 11

To find the final numeric value requested by the question.

Key Concept

Factoring cubic polynomials by finding rational roots and factoring quadratic trinomials by grouping.

Estimated Time:2m 30s

Question 77Question →

For what value of $a$ does the equation $\frac{3}{4}(x - 2) - \frac{1}{3}(2x + a) = \frac{1}{12}x - 5$ have infinitely many solutions for $x$ ?

Show answer & explanation

Answer: 10.5

Answer

10.5

Expanding the left side of the equation yields

\frac{3}{4}x - \frac{3}{2} - \frac{2}{3}x - \frac{a}{3}

. Combining the coefficients of

x

gives

(\frac{3}{4} - \frac{2}{3})x = \frac{1}{12}x

. The equation becomes

\frac{1}{12}x - (\frac{3}{2} + \frac{a}{3}) = \frac{1}{12}x - 5

. For a linear equation to have infinitely many solutions, the variable coefficients must be equal and the constant terms must also be equal. Therefore, we equate the constants:

-\frac{3}{2} - \frac{a}{3} = -5

. Multiplying all terms by

-6

to clear the denominators yields

9 + 2a = 30

, which simplifies to

2a = 21

, or

a = 10.5

.

Step-by-Step Solution

1

Expand and simplify the left side of the equation by distributing the fraction coefficients.

\frac{1}{12}x - \frac{3}{2} - \frac{a}{3}

Distributing

\frac{3}{4}

and

-\frac{1}{3}

across their respective parentheses and combining the

x

terms allows us to compare the coefficients on both sides.

2

Equate the constant terms from both sides of the equation.

-\frac{3}{2} - \frac{a}{3} = -5

A linear equation of the form

Ax + B = Cx + D

has infinitely many solutions if and only if

A = C

and

B = D

. Since both

A

and

C

are

\frac{1}{12}

, we set the constant terms equal.

3

Isolate the variable

a

and solve.

a = 10.5

Adding

\frac{3}{2}

to both sides gives

-\frac{a}{3} = -3.5

. Multiplying both sides by

-3

yields the final value.

Key Concept

Solving linear equations with infinitely many solutions by equating coefficients and constant terms.

Estimated Time:2m 0s

Question 78Question →

A chemist is preparing $120\text{ milliliters}$ of a chemical mixture with an overall acid concentration of $27.5\%$ by volume. To do this, she mixes Solution $X$ ( $10\%$ acid by volume), Solution $Y$ ( $25\%$ acid by volume), and Solution $Z$ ( $40\%$ acid by volume). The chemist decides that the volume of Solution $Y$ must be exactly $20\text{ milliliters}$ less than twice the volume of Solution $X$ used in the mixture. What is the positive difference, in milliliters, between the volume of Solution $Z$ and the volume of Solution $X$ in the final mixture?

Show answer & explanation

Answer: 20

Answer

The positive difference between the volume of Solution Z and Solution X is 20 milliliters.

The correct answer is 20, because solving the system of equations yields a volume of 30 milliliters for Solution X and 50 milliliters for Solution Z. The positive difference between these two volumes is 20 milliliters.

Step-by-Step Solution

1

Define variables for the volume of each solution used in the mixture.

Let

x

be the volume of Solution

X

,

y

be the volume of Solution

Y

, and

z

be the volume of Solution

Z

(all in milliliters).

Defining variables allows for translating the word problem's conditions into algebraic equations.

2

Translate the given information into a system of three linear equations.

Equation 1 (Total Volume):

x + y + z = 120

. Equation 2 (Total Acid):

0.10x + 0.25y + 0.40z = 33

(since

27.5\%

of

120\text{ ml}

is

33\text{ ml}

). Equation 3 (Volume Relation):

y = 2x - 20

.

These equations model the constraints and quantities described in the problem.

3

Express

z

in terms of

x

by substituting the expression for

y

into the total volume equation.

x + (2x - 20) + z = 120 \implies 3x - 20 + z = 120 \implies z = 140 - 3x

.

This substitution reduces the system to two variables (

x

and

z

), making it easier to solve.

4

Substitute the expressions for

y

and

z

in terms of

x

into the acid equation and solve for

x

.

0.10x + 0.25(2x - 20) + 0.40(140 - 3x) = 33 \implies 0.10x + 0.50x - 5 + 56 - 1.20x = 33 \implies -0.60x + 51 = 33 \implies -0.60x = -18 \implies x = 30

.

This isolates the single variable

x

so that its value can be calculated.

5

Determine the volume of Solution

Z

and calculate the final positive difference.

z = 140 - 3(30) = 50

. The positive difference between the volume of Solution

Z

and Solution

X

is

|50 - 30| = 20

.

This directly answers the question's requirement for the difference between the two solution volumes.

Key Concept

Translating word problems into a system of three linear equations and solving them using substitution.

Question 79Question →

An algebra student is asked to simplify a polynomial expression by factoring it completely. The student is given the expression $2x^6 - 32x^2$ . Which of the following is the completely factored form of the expression?

2x^2(x^2 - 4)(x^2 + 4)

2x^2(x^3 - 16)

2x^2(x - 2)(x + 2)(x^2 + 4)

32x^2(x - 2)(x + 2)(x^2 + 4)

2x^2(x - 2)^2(x^2 + 4)

Show answer & explanation

Answer: $2x^2(x - 2)(x + 2)(x^2 + 4)$

Answer

The completely factored form of the expression is

2x^2(x - 2)(x + 2)(x^2 + 4)

.

The correct answer is

2x^2(x - 2)(x + 2)(x^2 + 4)

. First, find the greatest common factor (GCF) of

2x^6

and

32x^2

, which is

2x^2

. Factoring this out gives

2x^2(x^4 - 16)

. Next, recognize that

x^4 - 16

is a difference of squares since

x^4 = (x^2)^2

and

16 = 4^2

. This factors into

(x^2 - 4)(x^2 + 4)

. Finally, the term

x^2 - 4

is also a difference of squares and factors into

(x - 2)(x + 2)

, while the sum of squares

x^2 + 4

cannot be factored further over the real numbers. Thus, the completely factored form is

2x^2(x - 2)(x + 2)(x^2 + 4)

.

Step-by-Step Solution

1

Identify and factor out the greatest common factor (GCF) of the two terms in

2x^6 - 32x^2

.

The GCF is

2x^2

, so the expression becomes

2x^2(x^4 - 16)

.

Factoring out the GCF simplifies the polynomial and reveals a difference of squares.

2

Recognize that

x^4 - 16

is a difference of squares and factor it.

Since

x^4 = (x^2)^2

and

16 = 4^2

, the expression factors as

2x^2(x^2 - 4)(x^2 + 4)

.

The difference of squares formula

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

applies directly to

x^4 - 16

.

3

Check if any remaining factors can be factored further, and factor them.

The term

x^2 - 4

is another difference of squares (

x^2 - 2^2

), which factors into

(x - 2)(x + 2)

. The term

x^2 + 4

is a sum of squares and cannot be factored over the real numbers. The completely factored expression is

2x^2(x - 2)(x + 2)(x^2 + 4)

.

To factor completely, all factorable terms must be broken down into their prime polynomial factors.

Key Concept

Factoring polynomials completely using the greatest common factor and the difference of squares.

Question 80Question →

What is the value of $x$ that satisfies the equation $\frac{2}{3}x - \frac{1}{4} = \frac{5}{12}$ ?

\frac{1}{4}

\frac{9}{16}

\frac{4}{9}

1

12

Show answer & explanation

Answer: $1$

Answer

1

The correct answer is 1. Adding

\frac{1}{4}

to both sides of the equation

\frac{2}{3}x - \frac{1}{4} = \frac{5}{12}

gives

\frac{2}{3}x = \frac{5}{12} + \frac{3}{12} = \frac{8}{12}

. Simplifying

\frac{8}{12}

yields

\frac{2}{3}

, so

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

. Multiplying both sides by

\frac{3}{2}

isolates

x

, giving

x = 1

.

Step-by-Step Solution

1

Add

\frac{1}{4}

to both sides of the equation to isolate the variable term.

\frac{2}{3}x = \frac{5}{12} + \frac{1}{4}

To solve for

x

, terms containing

x

must be isolated on one side of the equation.

2

Find a common denominator to add the fractions on the right side.

\frac{2}{3}x = \frac{5}{12} + \frac{3}{12} = \frac{8}{12} = \frac{2}{3}

Fractions must have the same denominator to be added.

3

Multiply both sides of the equation by the reciprocal of the coefficient of

x

, which is

\frac{3}{2}

.

x = \frac{2}{3} \cdot \frac{3}{2} = 1

Multiplying a coefficient by its reciprocal yields

1

, isolating the variable

x

.

Key Concept

Solving single-variable linear equations involving fractions by isolating the variable using inverse operations.

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