Elementary Algebra Practice Questions — ACT — Page 5

Question 81Question →

A manufacturer of custom planners determines that the setup cost for a production run is $100$ dollars, and each planner costs $6$ dollars to produce. The planners sell for $10$ dollars each, except for the first $10$ planners sold, which are discounted by $2$ dollars each. If the manufacturer wants to achieve a net profit of exactly $300$ dollars for a single production run, how many planners must they produce and sell?

Show answer & explanation

Answer: 105

Answer

The manufacturer must produce and sell

105

planners to achieve a net profit of

300

dollars.

The correct answer is found by setting up the profit equation:

\text{Profit} = \text{Total Revenue} - \text{Total Cost}

. The cost is

100 + 6x

and the revenue is

10(8) + 10(x-10) = 10x - 20

. Equating their difference to

300

yields

(10x - 20) - (100 + 6x) = 300

, which simplifies to

4x - 120 = 300

. Solving for

x

results in

105

.

Step-by-Step Solution

1

Define the variable for the number of planners.

Let

x

be the number of planners produced and sold, where

x \geq 10

.

Establishing the variable is necessary to set up algebraic expressions for cost and revenue.

2

Write the total cost expression.

\text{Total Cost} = 100 + 6x

The cost combines the fixed setup fee of

100

dollars and the variable cost of

6

dollars per planner.

3

Write the total revenue expression.

\text{Total Revenue} = 10(8) + 10(x - 10) = 10x - 20

The first

10

planners sell for

8

dollars each, and the remaining

x - 10

planners sell for the regular price of

10

dollars each.

4

Set up the profit equation and solve for

x

.

(10x - 20) - (100 + 6x) = 300 \implies 4x - 120 = 300 \implies 4x = 420 \implies x = 105

Profit is the difference between total revenue and total cost, which must equal the target profit of

300

dollars.

Key Concept

Translating real-world pricing and cost constraints into a single-variable linear equation.

Question 82Question →

For $x \neq 0$ , the expression $\frac{(x^2)^5}{x^k}$ simplifies to $x^6$ . What is the value of the exponent $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of the exponent

k

is 4.

Applying the power of a power rule to the numerator yields

(x^2)^5 = x^{10}

. Next, applying the quotient rule to divide

x^{10}

by

x^k

yields

x^{10-k}

. Setting this equal to the simplified term

x^6

leads to the exponent equation

10 - k = 6

. Solving this equation gives the final result

k = 4

.

Step-by-Step Solution

1

Simplify the numerator expression

(x^2)^5

x^{10}

Multiply the exponents when raising a power to another power:

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

.

2

Simplify the division of the two exponential expressions

\frac{x^{10}}{x^k}

x^{10-k}

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

\frac{x^a}{x^b} = x^{a-b}

.

3

Solve the linear equation for

k

using the target exponent 6

k = 4

Equating the exponent

10-k

to

6

gives

10-k=6

. Subtracting 10 from both sides gives

-k = -4

, so

k = 4

.

Key Concept

Applying properties of exponents in algebraic expressions, specifically the power of a power rule and the quotient rule.

Question 83Question →

For what value of the constant $a$ does the linear equation $\frac{a(x - 2)}{3} - \frac{3x - 1}{2} = -\frac{5x + 7}{6}$ have no real solution for $x$ ?

Show answer & explanation

Answer: 2

Answer

The constant

a

must be equal to

2

for the equation to have no solution.

A linear equation in the form

Ax + B = Cx + D

has no solution if the coefficients of the variable are equal (

A = C

) but the constant terms are different (

B \neq D

). Multiplying the given equation by the least common denominator, 6, clears the fractions and yields

2a(x - 2) - 3(3x - 1) = -(5x + 7)

. Expanding both sides results in

2ax - 4a - 9x + 3 = -5x - 7

, which simplifies to

(2a - 9)x + (3 - 4a) = -5x - 7

. Equating the coefficients of

x

gives

2a - 9 = -5

, which solves to

a = 2

. Substituting

a = 2

back into the constants yields a left-side constant of

-5

and a right-side constant of

-7

. Since

-5 \neq -7

, the variable terms cancel out while leaving an inequality, meaning the equation has no solution when

a = 2

.

Step-by-Step Solution

1

Clear the denominators by multiplying the entire equation by the least common denominator, which is 6.

2a(x - 2) - 3(3x - 1) = -(5x + 7)

Multiplying by the LCD simplifies the rational expressions into polynomial terms.

2

Distribute and expand the terms on both sides of the equation.

2ax - 4a - 9x + 3 = -5x - 7

Expanding the terms allows us to group variable terms and constant terms together.

3

Group the

x

terms and constant terms on the left side.

(2a - 9)x + (3 - 4a) = -5x - 7

Structuring the equation in the standard form

Ax + B = Cx + D

makes it easier to compare coefficients.

4

Set the coefficients of

x

on both sides equal to each other.

2a - 9 = -5

For a linear equation to have no solution, the variable terms must cancel out, meaning their coefficients must be identical.

5

Solve for the parameter

a

and verify the constant terms are unequal.

2a = 4 \implies a = 2

. Constant check:

3 - 4(2) = -5

, and

-5 \neq -7

.

If the constant terms were equal, the equation would have infinitely many solutions instead of no solution.

Key Concept

Identifying the parameter value that results in a linear equation having no solution by equating variable coefficients and ensuring constant terms are unequal.

Question 84Question →

For each of the given quadratic equations, solve for $x$ by factoring. Match each quadratic equation on the left to its correct solution set on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The equation

2x^2 + 5x = 3

matches the solution set

\left\{-3, \frac{1}{2}\right\}

; the equation

3x^2 - 10x + 8 = 0

matches the solution set

\left\{\frac{4}{3}, 2\right\}

; and the equation

x(x + 2) = 15

matches the solution set

\left\{-5, 3\right\}

.

Each equation matches its corresponding solution set through distributing terms if necessary, rewriting the equation in standard quadratic form

ax^2 + bx + c = 0

, factoring the quadratic trinomial over the integers, and then using the zero product property to solve for

x

.

Step-by-Step Solution

1

For the equation

2x^2 + 5x = 3

, rewrite in standard form

ax^2 + bx + c = 0

by subtracting 3 from both sides to get

2x^2 + 5x - 3 = 0

.

The equation is rewritten as

2x^2 + 5x - 3 = 0

.

Before factoring a quadratic equation, all terms must be moved to one side so the other side is equal to zero.

2

Factor the trinomial

2x^2 + 5x - 3

by grouping. Find two integers that multiply to

2 \times (-3) = -6

and add to

5

. These integers are

6

and

-1

. Rewrite the middle term and factor by grouping:

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

.

The equation becomes

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

.

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

3

Set each factor of

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

to zero and solve for

x

:

2x - 1 = 0 \Rightarrow x = \frac{1}{2}

and

x + 3 = 0 \Rightarrow x = -3

.

The solutions are

x = \frac{1}{2}

and

x = -3

, forming the solution set

\left\{-3, \frac{1}{2}\right\}

.

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

4

For the equation

3x^2 - 10x + 8 = 0

, factor the trinomial by finding two integers that multiply to

3 \times 8 = 24

and add to

-10

. These integers are

-6

and

-4

. Rewrite the middle term and factor by grouping:

3x^2 - 6x - 4x + 8 = 3x(x - 2) - 4(x - 2) = (3x - 4)(x - 2) = 0

.

The equation becomes

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

.

The equation is already in standard form, so we can directly proceed with factoring.

5

Set each factor of

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

to zero and solve for

x

:

3x - 4 = 0 \Rightarrow x = \frac{4}{3}

and

x - 2 = 0 \Rightarrow x = 2

.

The solutions are

x = \frac{4}{3}

and

x = 2

, forming the solution set

\left\{\frac{4}{3}, 2\right\}

.

Solving each linear factor yields the roots of the quadratic equation.

6

For the equation

x(x + 2) = 15

, first distribute

x

to get

x^2 + 2x = 15

, then subtract 15 from both sides to write in standard form:

x^2 + 2x - 15 = 0

.

The equation is rewritten as

x^2 + 2x - 15 = 0

.

Distributing and moving terms sets the quadratic to zero, which is necessary for factoring.

7

Factor the quadratic

x^2 + 2x - 15 = 0

by finding two integers that multiply to

-15

and add to

2

. These integers are

5

and

-3

, yielding

(x + 5)(x - 3) = 0

.

The equation becomes

(x + 5)(x - 3) = 0

.

Factoring a quadratic trinomial with a leading coefficient of 1 involves finding numbers that sum to the linear coefficient and multiply to the constant term.

8

Set each factor of

(x + 5)(x - 3) = 0

to zero and solve for

x

:

x + 5 = 0 \Rightarrow x = -5

and

x - 3 = 0 \Rightarrow x = 3

.

The solutions are

x = -5

and

x = 3

, forming the solution set

\left\{-5, 3\right\}

.

Solving the resulting linear equations gives the roots of the original quadratic equation.

Key Concept

Solving Quadratic Equations by Factoring

Alternative Method

Instead of factoring, the solutions to these quadratic equations can be verified by substituting the values in the solution sets back into the original equations, or by applying the quadratic formula:

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

after rewriting them in standard form.

Estimated Time:2m 0s

Question 85Question →

Which of the following is the completely factored form of the expression $9x^2 - 36$ ?

(3x - 6)^2

(3x - 6)(3x + 6)

9(x - 2)(x + 2)

9(x^2 - 4)

9(x - 2)^2

Show answer & explanation

Answer: $9(x - 2)(x + 2)$

Answer

The correct factored form is

9(x - 2)(x + 2)

The expression

9x^2 - 36

has a greatest common factor of

9

. Factoring out

9

yields

9(x^2 - 4)

. The term inside the parentheses,

x^2 - 4

, is a difference of squares that can be factored as

(x - 2)(x + 2)

. Combining these gives the completely factored form

9(x - 2)(x + 2)

.

Step-by-Step Solution

1

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

The GCF of

9x^2

and

36

is

9

. Factoring it out gives

9(x^2 - 4)

.

Factoring out the GCF simplifies the remaining polynomial and is the first step in completely factoring an expression.

2

Recognize and factor the quadratic expression inside the parentheses.

The term

x^2 - 4

is a difference of squares of the form

a^2 - b^2

, where

a = x

and

b = 2

. Factoring it yields

(x - 2)(x + 2)

.

A difference of squares

a^2 - b^2

always factors into

(a - b)(a + b)

.

3

Combine the factored terms to write the completely factored expression.

Combining the GCF and the factored binomials gives

9(x - 2)(x + 2)

.

This represents the expression written as a product of its prime polynomial factors.

Key Concept

Factoring polynomials by first extracting the greatest common factor (GCF) and then applying the difference of squares identity.

Alternative Method

Instead of factoring out the greatest common factor first, the expression can be factored as a difference of squares directly:

9x^2 - 36 = (3x)^2 - 6^2 = (3x - 6)(3x + 6)

. Then, a common factor of

3

can be factored out from each binomial:

3(x - 2) \cdot 3(x + 2) = 9(x - 2)(x + 2)

.

Estimated Time:45s

Question 86Question →

If the algebraic expression $\frac{(x^a y^2)^{-3} (x^2 y^b)^2}{(x^{-3} y^{-1})^{-2}}$ simplifies to $x^4 y^4$ for all non-zero real numbers $x$ and $y$ , where $a$ and $b$ are integers, what is the value of $a + b$ ?

Show answer & explanation

Answer: 4

Answer

The value of

a + b

is

4

.

Applying exponent rules to the expression

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

yields

\frac{x^{-3a}y^{-6} \cdot x^4 y^{2b}}{x^6 y^2} = \frac{x^{-3a+4} y^{2b-6}}{x^6 y^2} = x^{-3a-2} y^{2b-8}

. Equating these exponents to the target expression

x^4 y^4

gives the system

-3a - 2 = 4

and

2b - 8 = 4

. Solving these equations gives

a = -2

and

b = 6

, resulting in a sum of

a + b = 4

.

Step-by-Step Solution

1

Simplify the terms in the numerator.

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

and

(x^2 y^b)^2 = x^4 y^{2b}

Apply the power of a product rule:

(u^m v^n)^p = u^{mp} v^{np}

.

2

Multiply the simplified terms in the numerator.

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

Apply the product rule of exponents by adding exponents of like bases:

u^m \cdot u^n = u^{m+n}

.

3

Simplify the denominator.

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

Apply the power of a product rule.

4

Divide the numerator by the denominator.

x^{-3a-2} y^{2b-8}

Apply the quotient rule of exponents by subtracting denominator exponents from numerator exponents:

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

.

5

Set up equations by equating the simplified exponents to the exponents in the target expression

x^4 y^4

.

-3a - 2 = 4

and

2b - 8 = 4

For the expressions to be equivalent for all non-zero real numbers, the corresponding exponents of

x

and

y

must be equal.

6

Solve the linear equations for the integer constants

a

and

b

.

a = -2

and

b = 6

Isolate the variables:

-3a = 6 \implies a = -2

, and

2b = 12 \implies b = 6

.

7

Find the sum of

a

and

b

.

4

Add the values of the constants:

-2 + 6 = 4

.

Key Concept

Properties of exponents (product, quotient, and power rules) in multi-step algebraic simplification

Question 87Question →

For all real numbers $m$ and $n$ , the expression $4m(2m - 3n)^2 - n(5m^2 - 36mn)$ can be written in the form $am^3 + bm^2n + cmn^2$ , where $a$ , $b$ , and $c$ are real constants. What is the value of the coefficient $b$ ?

Show answer & explanation

Answer: -53

Answer

The value of the coefficient

b

is

-53

.

Expanding the entire expression yields

16m^3 - 53m^2n + 72mn^2

. Comparing this to the template form

am^3 + bm^2n + cmn^2

shows that

b = -53

.

Step-by-Step Solution

1

Expand the squared binomial

(2m - 3n)^2

4m^2 - 12mn + 9n^2

Apply the identity

(x - y)^2 = x^2 - 2xy + y^2

to expand the expression inside the parentheses.

2

Distribute

4m

through the trinomial

16m^3 - 48m^2n + 36mn^2

Multiply each term of

4m^2 - 12mn + 9n^2

by

4m

using properties of exponents.

3

Distribute

-n

across

(5m^2 - 36mn)

-5m^2n + 36mn^2

Multiply

-n

by both terms inside the parentheses, paying attention to sign rules.

4

Group and combine the like terms

16m^3 + (-48m^2n - 5m^2n) + (36mn^2 + 36mn^2)

Identify terms with the same variables and exponents to combine them.

5

Combine the coefficients of the like terms

16m^3 - 53m^2n + 72mn^2

Perform arithmetic on the coefficients:

-48 - 5 = -53

for the

m^2n

terms and

36 + 36 = 72

for the

mn^2

terms.

6

Identify the coefficient

b

-53

Compare the simplified polynomial to the form

am^3 + bm^2n + cmn^2

to find the value of

b

.

Key Concept

Simplifying algebraic expressions by expanding binomials, distributing variables and signs, and combining like terms.

Alternative Method

Instead of expanding the entire expression, focus only on terms that produce

m^2n

: from the first part,

4m \times (-12mn) = -48m^2n

, and from the second part,

-n \times 5m^2 = -5m^2n

. Adding these gives

-53m^2n

.

Estimated Time:1m 30s

Question 88Question →

Which of the following expressions is equivalent to $\frac{12x^8}{4x^2}$ for all $x \neq 0$ ?

8x^4

8x^6

3x^4

3x^6

3x^{10}

Show answer & explanation

Answer: $3x^6$

Answer

The expression

3x^6

To simplify the expression, divide the coefficients first:

\frac{12}{4} = 3

. Then, apply the quotient rule of exponents to the variable terms, which states that for any non-zero base,

\frac{x^a}{x^b} = x^{a-b}

. Subtracting the exponents gives

x^{8-2} = x^6

. Combining these results yields the correct expression

3x^6

.

Step-by-Step Solution

1

Divide the numerical coefficients of the terms.

\frac{12}{4} = 3

When simplifying a fraction with algebraic terms, the coefficients are divided normally.

2

Apply the quotient rule of exponents to simplify the variable terms.

\frac{x^8}{x^2} = x^{8-2} = x^6

According to the quotient rule of exponents, when dividing expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator.

3

Multiply the simplified coefficient and variable results together.

3x^6

Combining the divided coefficient and the simplified variable expression gives the final simplified result.

Key Concept

Quotient Rule of Exponents

Alternative Method

Alternatively, you can expand the exponent terms in the numerator and denominator:

\frac{12 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x}{4 \cdot x \cdot x}

. Simplifying the coefficients gives

3

, and canceling two pairs of

x

from both the numerator and denominator leaves six factors of

x

in the numerator, which simplifies to

3x^6

.

Estimated Time:45s

Question 89Question →

What is the sum of the solutions to the equation $3(x - 2)^2 = -6x + 36$ ?

2

-2

\frac{10}{3}

6

-6

Show answer & explanation

Answer: 2

Answer

The sum of the solutions is 2.

The correct answer is 2. Expanding the equation correctly and rearranging it into standard form yields

x^2 - 2x - 8 = 0

. Factoring this expression gives

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

, which results in the solutions

x = 4

and

x = -2

. The sum of these two solutions is

4 + (-2) = 2

.

Step-by-Step Solution

1

Divide both sides of the equation by 3 to simplify.

(x - 2)^2 = -2x + 12

Dividing both sides by the common factor of 3 simplifies the coefficients, making the algebraic manipulation and factoring steps easier.

2

Expand the squared binomial on the left side.

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

Expanding the binomial

(x - 2)^2

using the identity

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

is necessary to write the equation in standard form.

3

Move all terms to the left side to set the equation equal to zero.

x^2 - 2x - 8 = 0

A quadratic equation must be in the standard form

ax^2 + bx + c = 0

to solve it by factoring.

4

Factor the quadratic trinomial.

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

We find two numbers that multiply to

-8

and add to

-2

. These numbers are

-4

and

2

.

5

Set each factor to zero to solve for the roots.

x = 4

or

x = -2

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

6

Calculate the sum of the solutions.

4 + (-2) = 2

The question asks for the sum of the solutions, so we add the two calculated values of

x

.

Key Concept

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

Question 90Question →

For a real number $x$ , the equation $2(x + 5) = 16$ is true. What is the value of the expression $3x - 1$ ?

Show answer & explanation

Answer: 8

Answer

The value of the expression

3x - 1

is 8.

Solving the equation

2(x + 5) = 16

gives

x = 3

. Evaluating the expression

3x - 1

for

x = 3

yields

3(3) - 1 = 8

.

Step-by-Step Solution

1

Distribute the 2 on the left side of the equation.

2x + 10 = 16

To eliminate the parentheses using the distributive property.

2

Subtract 10 from both sides of the equation.

2x = 6

To isolate the variable term on one side of the equation.

3

Divide both sides of the equation by 2.

x = 3

To find the value of

x

.

4

Substitute the value of

x

into the expression

3x - 1

.

3(3) - 1 = 8

To evaluate the final expression as requested by the question.

Key Concept

Solving multi-step linear equations and evaluating algebraic expressions.

Estimated Time:45s

Question 91Question →

The cubic polynomial $2x^3 + 5x^2 - 8x - 20$ can be factored completely over the integers into the form $(x - a)(x + b)(cx + d)$ , where $a$ , $b$ , $c$ , and $d$ are positive integers. What is the value of $a + b + c + d$ ?

Show answer & explanation

Answer: 11

Answer

The value of

a + b + c + d

is 11.

The polynomial

2x^3 + 5x^2 - 8x - 20

can be factored completely by first grouping the terms as

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

. Factoring the difference of squares

x^2 - 4

yields

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

. Matching this to the given form

(x - a)(x + b)(cx + d)

gives the positive integers

a = 2

,

b = 2

,

c = 2

, and

d = 5

. The sum of these values is

2 + 2 + 2 + 5 = 11

.

Step-by-Step Solution

1

Group the terms of the polynomial

2x^3 + 5x^2 - 8x - 20

.

(2x^3 + 5x^2) - (8x + 20)

Grouping the terms allows us to look for common factors within each pair of terms.

2

Factor out the greatest common factor (GCF) from each group.

x^2(2x + 5) - 4(2x + 5)

The GCF of the first group

2x^3 + 5x^2

is

x^2

, and the GCF of the second group

8x + 20

is

4

.

3

Factor out the common binomial factor

(2x + 5)

.

(x^2 - 4)(2x + 5)

Both terms share the binomial factor

(2x + 5)

.

4

Factor the difference of squares

x^2 - 4

.

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

The term

x^2 - 4

is a difference of squares, which factors as

(x - 2)(x + 2)

.

5

Compare the factored expression with the template

(x - a)(x + b)(cx + d)

to determine the values of

a

,

b

,

c

, and

d

.

a = 2

,

b = 2

,

c = 2

, and

d = 5

Comparing the terms yields

x - a = x - 2 \implies a = 2

,

x + b = x + 2 \implies b = 2

, and

cx + d = 2x + 5 \implies c = 2, d = 5

. All values are positive integers as required.

6

Calculate the sum

a + b + c + d

.

11

Substituting the values of the variables into the expression gives

2 + 2 + 2 + 5 = 11

.

Key Concept

Factoring a cubic polynomial by grouping and then factoring the resulting difference of squares.

Alternative Method

Instead of factoring by grouping, we can use the Rational Root Theorem to find rational roots of the polynomial. The possible rational roots of

2x^3 + 5x^2 - 8x - 20 = 0

are of the form

\pm \frac{p}{q}

, where

p

is a factor of

20

and

q

is a factor of

2

. Testing values shows that

x = 2

and

x = -2

are roots, which corresponds to the linear factors

(x - 2)

and

(x + 2)

. Dividing the original cubic by their product,

(x^2 - 4)

, yields the remaining linear factor

(2x + 5)

.

Estimated Time:1m 30s

Question 92Question →

A gardener starts with $15$ flowers already planted in a garden. She plans to plant additional flowers at a constant rate of $8$ flowers per hour. How many hours will it take the gardener to have a total of $79$ flowers planted?

Show answer & explanation

Answer: 8

Answer

It will take the gardener

8

hours to have a total of

79

flowers planted.

The total number of flowers planted can be modeled by the linear equation

15 + 8h = 79

, where

h

is the number of hours. Subtracting

15

from both sides of the equation gives

8h = 64

. Dividing both sides by

8

reveals that

h = 8

hours.

Step-by-Step Solution

1

Set up the linear equation representing the total flowers planted over time.

15 + 8h = 79

, where

h

is the number of hours.

The gardener begins with

15

flowers and adds

8

flowers for each hour

h

, with the final goal of

79

total flowers.

2

Isolate the variable term by subtracting the initial number of flowers from the total.

8h = 64

Subtracting

15

from both sides of the equation isolates the term containing the variable

h

.

3

Divide by the rate to solve for the number of hours.

h = 8

Dividing the remaining flowers to be planted (

64

) by the planting rate (

8

flowers per hour) yields the total number of hours required.

Key Concept

Translating a real-world scenario into a linear equation and solving for the unknown variable.

Question 93Question →

If $x$ and $y$ are real numbers, the expression $3x(x - 2y)^2 - x^2(3x - 10y) - y^2(2x - y)$ can be simplified to which of the following?

-22x^2y + 10xy^2 - y^3

-2x^2y + 10xy^2 - y^3

-2x^2y + 10xy^2 + y^3

10x^2y + 10xy^2 + y^3

-2x^2y + 10xy^2 + y^2

Show answer & explanation

Answer: $-2x^2y + 10xy^2 + y^3$

Answer

The correct expression is

-2x^2y + 10xy^2 + y^3

.

The correct expression

-2x^2y + 10xy^2 + y^3

is obtained by correctly expanding the binomial

(x-2y)^2

into

x^2 - 4xy + 4y^2

, distributing the outer terms

3x

,

-x^2

, and

-y^2

to all terms inside their respective parentheses, and then combining the coefficients of the matching variable terms: the

x^3

terms cancel out, the

x^2y

terms combine to

-2x^2y

, the

xy^2

terms combine to

10xy^2

, and the

y^3

term remains.

Step-by-Step Solution

1

Expand the binomial expression

(x - 2y)^2

.

x^2 - 4xy + 4y^2

Before distributing

3x

, the squared binomial must be expanded using the identity

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

.

2

Multiply the expanded binomial by the coefficient

3x

.

3x^3 - 12x^2y + 12xy^2

Distribute

3x

to each of the three terms in the expanded expression:

3x(x^2) = 3x^3

,

3x(-4xy) = -12x^2y

, and

3x(4y^2) = 12xy^2

.

3

Distribute

-x^2

to

(3x - 10y)

.

-3x^3 + 10x^2y

Distribute

-x^2

to both terms, noting that multiplying two negative signs yields a positive term:

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

and

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

.

4

Distribute

-y^2

to

(2x - y)

.

-2xy^2 + y^3

Distribute

-y^2

to both terms, adding the exponents of

y

where appropriate:

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

and

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

.

5

Combine all parts and group the like terms together.

-2x^2y + 10xy^2 + y^3

Group and add the coefficients of identical variable terms:

(3x^3 - 3x^3) + (-12x^2y + 10x^2y) + (12xy^2 - 2xy^2) + y^3 = 0x^3 - 2x^2y + 10xy^2 + y^3

.

Key Concept

Simplifying polynomial expressions with multiple variables by expanding binomials, distributing coefficients (including negative signs), and combining like terms.

Question 94Question →

A company produces solar-powered chargers. The daily production cost, in dollars, is a linear function of the number of chargers produced. The setup cost is the cost when $0$ chargers are produced. If the company produces $15$ chargers, the total daily cost is $400$ dollars. If the company produces $25$ chargers, the total daily cost is $580$ dollars. The company updates its production process, which reduces the cost per charger by $20\%$ but increases the setup cost by $50$ dollars. Under the updated process, what is the total daily cost, in dollars, to produce $30$ chargers?

562

dollars

586

dollars

612

dollars

714

dollars

828

dollars

Show answer & explanation

Answer: $612$ dollars

Answer

The correct daily cost under the updated process is

612

dollars.

The correct answer is

612

dollars. First, the relationship between unit cost and setup cost is modeled as a linear equation. Subtracting the cost of producing

15

chargers (

400

dollars) from the cost of producing

25

chargers (

580

dollars) gives the cost of producing the

10

additional units, which is

180

dollars. This determines the unit cost is

18

dollars per charger. Substituting this value back shows the original setup cost is

130

dollars. The updated rate decreases the unit cost to

14.40

dollars (

80\%

of

18

) and increases the setup cost to

180

dollars (

130 + 50

). For

30

chargers, the total daily cost is

14.40(30) + 180 = 612

dollars.

Step-by-Step Solution

1

Set up a system of linear equations using the cost function

C(c) = mc + S

, where

m

is the cost per charger,

S

is the setup cost, and

c

is the number of chargers.

Equation 1:

15m + S = 400

Equation 2:

25m + S = 580

This establishes the relationship between production volume and total cost under the initial process.

2

Solve the system of equations for the unit cost

m

and the setup cost

S

.

Subtracting Equation 1 from Equation 2 yields

10m = 180 \implies m = 18

.
Substituting

m = 18

into Equation 1 yields

15(18) + S = 400 \implies 270 + S = 400 \implies S = 130

.

This determines the original pricing parameters.

3

Apply the updates to the cost parameters.

New unit cost:

18 \times (1 - 0.20) = 14.40

dollars.
New setup cost:

130 + 50 = 180

dollars.

This accounts for the

20\%

decrease in the variable cost and the

50

dollar increase in the fixed setup cost.

4

Evaluate the new linear cost function for

30

chargers.

C_{\text{new}}(30) = 14.40(30) + 180 = 432 + 180 = 612

dollars.

This calculates the total daily cost under the updated process.

Key Concept

Translating and Solving Algebraic Word Problems

Alternative Method

Instead of solving for the setup cost first, note that under the original process, the cost of

30

chargers would be the cost of

15

chargers plus the cost of

15

more units:

400 + 15(18) = 670

dollars. The updated process reduces the rate of each of the

30

units by

20\%

of

18

dollars (saving

3.60 \times 30 = 108

dollars) and increases the setup cost by

50

dollars. Thus, the new cost is

670 - 108 + 50 = 612

dollars.

Estimated Time:2m 0s

Question 95Question →

When the expression $2x(3x - y)^2 - 3y^2(x - 2y) - x^2(18x - 15y)$ is simplified by combining like terms, what is the coefficient of $x^2y$ ?

-27

-12

3

15

-15

Show answer & explanation

Answer: 3

Answer

The coefficient of

x^2y

is

3

.

The correct coefficient of

3

is obtained by expanding the expression step-by-step. First, the square of the binomial

(3x - y)^2

is

9x^2 - 6xy + y^2

. Distributing

2x

gives

18x^3 - 12x^2y + 2xy^2

. Next, distributing

-x^2

to

(18x - 15y)

gives

-18x^3 + 15x^2y

. Combining the

x^2y

terms gives

-12x^2y + 15x^2y = 3x^2y

, meaning the coefficient is

3

.

Step-by-Step Solution

1

Expand the squared binomial term

(3x - y)^2

using the algebraic identity

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

.

(3x - y)^2 = 9x^2 - 6xy + y^2

This is necessary to remove the parentheses before distributing the outer variable.

2

Multiply the term

2x

by each term inside the expanded binomial expression:

2x(9x^2 - 6xy + y^2)

.

18x^3 - 12x^2y + 2xy^2

Applying the distributive property expands the first part of the expression.

3

Distribute

-x^2

to the terms inside the parentheses

(18x - 15y)

, paying close attention to the signs.

-18x^3 + 15x^2y

This expands the third part of the expression and correctly distributes the negative sign.

4

Identify and combine the like terms of the form

x^2y

from the expanded parts of the expression.

-12x^2y + 15x^2y = 3x^2y

To find the coefficient of

x^2y

, we only need to sum the coefficients of the terms that contain exactly

x^2y

.

Key Concept

Simplifying Expressions and Combining Like Terms

Estimated Time:1m 30s

Question 96Question →

An electronics retailer sells tablet computers. The retailer's weekly revenue from tablets, in thousands of dollars, is modeled by the expression $\frac{2}{5}(3x - 4)$ , where $x$ represents the average number of tablets sold per day. The weekly operating costs, in thousands of dollars, are modeled by the expression $\frac{1}{3}\left(2x - \frac{1}{2}\right)$ . The retailer's weekly profit is equal to the profit of a competitor, which is modeled by $\frac{1}{4}(x + 5) - \frac{11}{6}$ thousand dollars. If the retailer and the competitor earn the same weekly profit, what is the value of $12x - 5$ ?

31

3

\frac{767}{17}

51

\frac{367}{97}

Show answer & explanation

Answer: 31

Answer

31

The correct answer is 31. By equating the profit models, we get

\frac{2}{5}(3x - 4) - \frac{1}{3}(2x - 0.5) = \frac{1}{4}(x + 5) - \frac{11}{6}

. Multiplying the entire equation by the least common multiple of the denominators (60) yields

24(3x - 4) - 20(2x - 0.5) = 15(x + 5) - 110

. Expanding the terms gives

72x - 96 - 40x + 10 = 15x + 75 - 110

, which simplifies to

32x - 86 = 15x - 35

. Solving for

x

gives

17x = 51

, or

x = 3

. Evaluating the required expression

12x - 5

for

x = 3

gives

12(3) - 5 = 31

.

Step-by-Step Solution

1

Set up the equation equating the retailer's profit (revenue minus cost) to the competitor's profit.

\frac{2}{5}(3x - 4) - \frac{1}{3}\left(2x - \frac{1}{2}\right) = \frac{1}{4}(x + 5) - \frac{11}{6}

Profit is calculated as revenue minus operating costs. Since the retailer and the competitor earn the same profit, their profit models are equal.

2

Multiply the entire equation by 60 to eliminate all fractional denominators.

24(3x - 4) - 20\left(2x - \frac{1}{2}\right) = 15(x + 5) - 10(11)

The least common multiple (LCM) of 5, 3, 4, and 6 is 60. Multiplying both sides by 60 simplifies the equation into integer terms.

3

Distribute the coefficients to remove parentheses, taking care with the negative signs.

72x - 96 - 40x + 10 = 15x + 75 - 110

Distributing

-20

across

\left(2x - \frac{1}{2}\right)

yields

-40x + 10

, and distributing

24

across

(3x - 4)

yields

72x - 96

.

4

Combine like terms on both sides of the equation.

32x - 86 = 15x - 35

On the left side,

72x - 40x = 32x

and

-96 + 10 = -86

. On the right side,

75 - 110 = -35

.

5

Isolate the variable term

x

on one side and the constants on the other.

17x = 51 \implies x = 3

Subtracting

15x

from both sides gives

17x - 86 = -35

. Adding

86

to both sides gives

17x = 51

. Dividing by 17 yields

x = 3

.

6

Substitute

x = 3

into the requested expression

12x - 5

to find the final value.

12(3) - 5 = 31

The question asks for the value of the expression

12x - 5

rather than just the variable

x

.

Key Concept

Solving multi-step linear equations containing fractions by clearing denominators and distributing terms correctly.

Question 97Question →

For all non-zero real numbers $x$ and $y$ , which of the following is equivalent to the expression $\frac{\left( 2x^{-1} y^2 + \frac{1}{2} x^{-1} y^2 \right)^{-3}}{(x^2 y^{-3})^{-2}}$ ?

\frac{5x^7}{2y^{12}}

\frac{8x^7}{125y^{12}}

\frac{65x^7}{8y^{12}}

\frac{8}{125x^{12}y^{36}}

\frac{x^7}{y^{12}}

Show answer & explanation

Answer: $\frac{8x^7}{125y^{12}}$

Answer

\frac{8x^7}{125y^{12}}

To find the equivalent expression, we first combine the like terms inside the parentheses in the numerator to get

\frac{5}{2} x^{-1} y^2

. Raising this product to the power of

-3

yields

\left(\frac{5}{2}\right)^{-3} (x^{-1})^{-3} (y^2)^{-3} = \frac{8}{125} x^3 y^{-6}

. Next, the denominator simplifies to

(x^2 y^{-3})^{-2} = x^{-4} y^6

. Dividing the numerator by the denominator requires subtracting the exponents of like bases: for

x

, we have

3 - (-4) = 7

, and for

y

, we have

-6 - 6 = -12

. This results in

\frac{8}{125} x^7 y^{-12}

, which is equivalent to the correct expression

\frac{8x^7}{125y^{12}}

.

Step-by-Step Solution

1

Combine the like terms inside the parentheses in the numerator.

2x^{-1} y^2 + \frac{1}{2} x^{-1} y^2 = \left(2 + \frac{1}{2}\right) x^{-1} y^2 = \frac{5}{2} x^{-1} y^2

Before applying the outer negative exponent, it is mathematically simpler to combine the like terms inside the grouping.

2

Apply the power of

-3

to the term in the numerator.

\left(\frac{5}{2} x^{-1} y^2\right)^{-3} = \left(\frac{5}{2}\right)^{-3} (x^{-1})^{-3} (y^2)^{-3} = \frac{8}{125} x^3 y^{-6}

The power of a product rule

(ab)^n = a^n b^n

and the power of a power rule

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

are applied to expand the term.

3

Simplify the denominator by applying the power of

-2

.

(x^2 y^{-3})^{-2} = (x^2)^{-2} (y^{-3})^{-2} = x^{-4} y^6

The power of a product rule is applied to the denominator to resolve the outer exponent.

4

Divide the simplified numerator by the simplified denominator using the quotient rule for exponents.

\frac{\frac{8}{125} x^3 y^{-6}}{x^{-4} y^6} = \frac{8}{125} x^{3 - (-4)} y^{-6 - 6} = \frac{8}{125} x^7 y^{-12} = \frac{8x^7}{125y^{12}}

The quotient rule

a^m / a^n = a^{m-n}

is used to subtract the exponents of the corresponding variables, and negative exponents are rewritten in the denominator.

Key Concept

Properties of Exponents in Algebraic Expressions

Alternative Method

Alternatively, you can write out all variables with positive exponents before simplifying. Rewrite the term in the numerator as

\frac{2y^2}{x} + \frac{y^2}{2x} = \frac{5y^2}{2x}

. Raising this to the

-3

power flips the fraction and cubes it, yielding

\left(\frac{2x}{5y^2}\right)^3 = \frac{8x^3}{125y^6}

. Simplifying the denominator yields

\frac{1}{(x^2 y^{-3})^2} = \frac{1}{x^4 y^{-6}} = \frac{y^6}{x^4}

. Dividing the numerator by the denominator yields

\frac{8x^3}{125y^6} \div \frac{y^6}{x^4} = \frac{8x^3}{125y^6} \cdot \frac{x^4}{y^6} = \frac{8x^7}{125y^{12}}

.

Estimated Time:2m 0s

Question 98Question →

A dog-walking service charges a flat fee of $10$ dollars per visit plus $20$ dollars for each dog walked. If a client was charged a total of $90$ dollars for a single visit, how many dogs were walked during that visit?

9

5

4

7

60

Show answer & explanation

Answer: 4

Answer

4

The correct answer is

4

. The relation between the total cost and the number of dogs walked is

10 + 20d = 90

. Subtracting

10

from both sides yields

20d = 80

, and dividing by

20

yields

d = 4

.

Step-by-Step Solution

1

Set up the equation based on the scenario details.

Let

d

be the number of dogs walked. The total cost is represented by the equation

10 + 20d = 90

.

The total cost of

90

dollars consists of a flat fee of

10

dollars plus

20

dollars per dog walked.

2

Subtract the flat fee from both sides of the equation.

20d = 80

Subtracting

10

from both sides isolates the cost of walking the dogs.

3

Divide both sides by the per-dog rate to solve for

d

.

d = 4

Dividing by

20

yields the number of dogs walked.

Key Concept

Translating verbal descriptions of multi-step scenarios into linear equations and solving them.

Question 99Question →

For all non-zero real numbers $x$ , the expression $x^4 \cdot (x^3)^k$ is equivalent to $x^{10}$ . What is the value of the integer $k$ ?

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

First, use the power of a power property to write

(x^3)^k

as

x^{3k}

. The expression then becomes

x^4 \cdot x^{3k}

. Next, use the product of powers property to combine the terms into

x^{4+3k}

. Since the expression is equivalent to

x^{10}

, set the exponents equal:

4 + 3k = 10

. Solving this equation gives

3k = 6

, which simplifies to

k = 2

.

Step-by-Step Solution

1

Apply the power of a power rule

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

to simplify

(x^3)^k

.

x^{3k}

To raise a power to another power, multiply the exponents.

2

Apply the product of powers rule

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

to combine the terms

x^4 \cdot x^{3k}

.

x^{4+3k}

When multiplying exponential terms with the same base, add their exponents.

3

Set the combined exponent

4+3k

equal to the target exponent

10

and solve for

k

.

k = 2

Since the bases are equal and non-zero, their exponents must be equal.

Key Concept

Properties of exponents in algebraic expressions (power of a power rule and product of powers rule)

Estimated Time:45s

Question 100Question →

For all non-zero real numbers $x$ and $y$ , the expression

\frac{(x^2 y^{-3})^{-2} (x^{-1} y^4)^3}{(x^3 y^{-2})^d}

can be written in the form $x^p y^q$ , where $p$ and $q$ are integers. If $q = 2p$ , what is the value of $d$ ?

Show answer & explanation

Answer: -4

Answer

-4

Applying the rules of exponents yields the simplified expression

x^{-7-3d} y^{18+2d}

. Setting the exponent of

y

equal to twice the exponent of

x

gives the equation

18+2d = 2(-7-3d)

, which solves to

d = -4

.

Step-by-Step Solution

1

Apply the power of a power rule to the terms in the numerator.

(x^2 y^{-3})^{-2} = x^{-4} y^6

and

(x^{-1} y^4)^3 = x^{-3} y^{12}

To raise a power to another power, multiply the exponents:

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

.

2

Multiply the simplified terms in the numerator together.

x^{-4} y^6 \cdot x^{-3} y^{12} = x^{-7} y^{18}

To multiply powers with the same base, add the exponents:

u^m \cdot u^n = u^{m+n}

.

3

Apply the power of a power rule to the denominator.

(x^3 y^{-2})^d = x^{3d} y^{-2d}

Distribute the exponent

d

to both variables inside the parentheses by multiplying the exponents.

4

Divide the numerator by the denominator.

\frac{x^{-7} y^{18}}{x^{3d} y^{-2d}} = x^{-7-3d} y^{18-(-2d)} = x^{-7-3d} y^{18+2d}

To divide powers with the same base, subtract the exponent of the denominator from the exponent of the numerator:

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

.

5

Set up the linear equation for

d

using

q = 2p

and solve.

18+2d = 2(-7-3d) \implies 18+2d = -14-6d \implies 8d = -32 \implies d = -4

The problem states the relationship between the final exponents is

q = 2p

, where

p = -7-3d

and

q = 18+2d

.

Key Concept

Properties of exponents (product, quotient, and power rules) combined with solving a linear equation.

Elementary 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

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

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

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

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

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes