Elementary Algebra Practice Questions — ACT — Page 8

Question 141Question →

A chemist is preparing a mixture. The volume of acid, $V$ in liters, required for a specific reaction satisfies the equation:

\frac{3}{5}(2V - 7) - \frac{1}{2}(V + 4) = \frac{1}{10}(3V - 2)

What is the value of the expression $4V + 3$ ?

15

23

27

47

63

Show answer & explanation

Answer: 63

Answer

The value of the expression is 63.

The correct answer is 63. Multiplying both sides of the equation by the least common denominator of 10 eliminates the fractions and yields

6(2V - 7) - 5(V + 4) = 3V - 2

. Distributing the factors results in

12V - 42 - 5V - 20 = 3V - 2

, which simplifies to

7V - 62 = 3V - 2

. Subtracting

3V

and adding 62 to both sides produces

4V = 60

, which gives

V = 15

. Substituting 15 into the expression

4V + 3

results in

4(15) + 3 = 63

.

Step-by-Step Solution

1

Multiply the entire equation by the least common denominator (LCD) to eliminate the fractions.

6(2V - 7) - 5(V + 4) = 1(3V - 2)

The denominators are 5, 2, and 10, so the LCD is 10.

2

Distribute the constants on the left side of the equation.

12V - 42 - 5V - 20 = 3V - 2

Distributing 6 to

(2V - 7)

yields

12V - 42

, and distributing

-5

to

(V + 4)

yields

-5V - 20

.

3

Combine like terms on the left side of the equation.

7V - 62 = 3V - 2

Combining the variable terms gives

12V - 5V = 7V

, and combining the constant terms gives

-42 - 20 = -62

.

4

Isolate the variable term by performing inverse operations.

4V = 60

Subtract

3V

from both sides to get

4V

, and add 62 to both sides to get 60.

5

Solve for

V

by dividing both sides of the equation.

V = 15

Dividing 60 by 4 yields 15.

6

Substitute the value of

V

into the requested expression.

4(15) + 3 = 63

The question asks for the value of the expression

4V + 3

, not the value of

V

.

Key Concept

Solving multi-step linear equations containing fractions by finding a common denominator, distributing terms correctly, and evaluating algebraic expressions.

Estimated Time:2m 0s

Question 142Question →

For all real numbers $p$ and $q$ , which of the following is equivalent to the expression $2p^2(p - 3q) - 3q(p^2 - q^2) - (2p^3 - 5p^2q)$ ?

-14p^2q + 3q^3

-4p^2q - 3q^3

-4p^2q + 3q^3

-4p^2q + 3q^2

-14p^2q - 3q^3

Show answer & explanation

Answer: $-4p^2q + 3q^3$

Answer

-4p^2q + 3q^3

The correct answer is obtained by distributing all factors and negative signs, then combining the coefficients of the like terms. This yields

-4p^2q + 3q^3

.

Step-by-Step Solution

1

Distribute the term

2p^2

into the first parenthesis,

-3q

into the second parenthesis, and the negative sign into the third parenthesis.

2p^2(p - 3q) = 2p^3 - 6p^2q

-3q(p^2 - q^2) = -3p^2q + 3q^3

-(2p^3 - 5p^2q) = -2p^3 + 5p^2q

Expanding the terms removes the parentheses and prepares the expression for combining like terms.

2

Combine the expanded parts into a single expression and group the like terms together.

(2p^3 - 2p^3) + (-6p^2q - 3p^2q + 5p^2q) + 3q^3

Grouping terms with identical variable parts makes combining coefficients straightforward.

3

Simplify the coefficients for each group of like terms.

0p^3 + (-6 - 3 + 5)p^2q + 3q^3 = -4p^2q + 3q^3

Simplifying the combined coefficients yields the final, simplified expression.

Key Concept

Simplifying Expressions and Combining Like Terms

Question 143Question →

If the polynomial $2x^2 + kx - 12$ can be factored as the product of two binomials with integer coefficients, what is the greatest possible integer value of $k$ ?

Show answer & explanation

Answer: 23

Answer

The greatest possible integer value of

k

is 23.

The greatest possible value of

k

is 23. This is found by setting the factored form of the expression as

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

. We find that

ab = -12

and

k = 2b + a

. The possible integer pairs for

(a, b)

that multiply to

-12

are tested to find the maximum sum of

2b + a

. The maximum is achieved when

b = 12

and

a = -1

, yielding

2(12) - 1 = 23

.

Step-by-Step Solution

1

Set up the algebraic representation for the binomial factors with integer coefficients.

The polynomial must factor into the form

(2x + a)(x + b)

where

a

and

b

are integers.

Since the leading coefficient is 2 (a prime number), the coefficients of the linear terms in the binomial factors must be 2 and 1 to obtain a product of

2x^2

.

2

Expand the product of the binomials to relate the parameters

a

and

b

to the original expression.

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

, which implies

ab = -12

and

k = 2b + a

.

By equating the coefficients of corresponding terms in the expanded expression and the original polynomial, we establish relationships for the constant term and the linear coefficient.

3

Analyze the factors of

-12

to find the integer values of

a

and

b

that maximize the linear term coefficient.

Choosing

b = 12

and

a = -1

gives

ab = -12

and results in

k = 2(12) + (-1) = 23

.

To maximize

2b + a

, we choose the largest possible positive factor of

-12

for the term multiplied by 2, which is 12, paired with the corresponding negative factor

-1

for

a

.

Key Concept

Factoring quadratic trinomials of the form

ax^2 + bx + c

with

a > 1

Question 144Question →

A company offers two monthly data storage plans. Plan A costs $32.00 plus$ 0.08 per gigabyte of data stored. Plan B costs $15.00 plus$ 0.14 per gigabyte for the first 150 gigabytes of data stored, and $0.12 per gigabyte for all data stored beyond the first 150 gigabytes. For how many gigabytes of data stored in a single month would the monthly cost under both plans be exactly the same?

Show answer & explanation

Answer: 350

Answer

The monthly costs under both plans are exactly the same when the data stored is 350 gigabytes.

Equating the two cost functions yields

32 + 0.08g = 15 + 0.14(150) + 0.12(g - 150)

, which simplifies to

32 + 0.08g = 18 + 0.12g

. Solving for

g

gives

0.04g = 14

, which results in

g = 350

.

Step-by-Step Solution

1

Determine the cost function for Plan A.

C_A(g) = 32 + 0.08g

Plan A charges a flat fee of

32.00 and

0.08 per gigabyte.

2

Determine the cost function for Plan B when the usage exceeds 150 gigabytes.

C_B(g) = 15 + 0.14(150) + 0.12(g - 150) = 18 + 0.12g

Plan B charges

15.00 flat,

0.14 per gigabyte for the first 150 gigabytes, and $0.12 per gigabyte for any additional usage.

3

Equate the two cost functions and solve for

g

.

32 + 0.08g = 18 + 0.12g \implies 14 = 0.04g \implies g = 350

To find the usage where both plans cost the same, set their cost functions equal and isolate the variable

g

.

Key Concept

Solving linear equations in real-life contexts involving piecewise rates

Question 145Question →

If $a = -27$ , $b = -\frac{1}{4}$ , and $c = -2$ , what is the value of the algebraic expression $a^{-4/3} - b^{-2} \cdot c^{-5}$ ?

-\frac{79}{162}

\frac{79}{162}

\frac{83}{162}

\frac{163}{2}

\frac{1295}{2592}

Show answer & explanation

Answer: $\frac{83}{162}$

Answer

The correct value of the algebraic expression is

\frac{83}{162}

.

Evaluating each term of the expression with the given values yields:

a^{-4/3} = \frac{1}{81}

,

b^{-2} = 16

, and

c^{-5} = -\frac{1}{32}

. Applying the order of operations, we multiply

16

by

-\frac{1}{32}

first to obtain

-\frac{1}{2}

. We then subtract this product from

\frac{1}{81}

, which simplifies to

\frac{1}{81} + \frac{1}{2} = \frac{83}{162}

.

Step-by-Step Solution

1

Substitute the given values of

a

,

b

, and

c

into the algebraic expression.

The expression is written as

(-27)^{-4/3} - (-\frac{1}{4})^{-2} \cdot (-2)^{-5}

.

This establishes the numerical expression to be evaluated.

2

Evaluate the first term,

(-27)^{-4/3}

.

(-27)^{-4/3} = \frac{1}{(-27)^{4/3}} = \frac{1}{((-27)^{1/3})^4} = \frac{1}{(-3)^4} = \frac{1}{81}

.

A negative exponent indicates a reciprocal, and a fractional exponent of

4/3

indicates taking the cube root and then raising to the fourth power.

3

Evaluate the second term,

(-\frac{1}{4})^{-2}

.

(-\frac{1}{4})^{-2} = (-4)^2 = 16

.

Raising a fraction to a negative integer power is equivalent to raising its reciprocal to the corresponding positive integer power.

4

Evaluate the third term,

(-2)^{-5}

.

(-2)^{-5} = \frac{1}{(-2)^5} = -\frac{1}{32}

.

Evaluating a negative base raised to an odd negative power results in a negative unit fraction.

5

Substitute the evaluated terms back into the original expression and apply the order of operations.

E = \frac{1}{81} - 16 \cdot \left(-\frac{1}{32}\right) = \frac{1}{81} - \left(-\frac{16}{32}\right) = \frac{1}{81} + \frac{1}{2} = \frac{2 + 81}{162} = \frac{83}{162}

.

Multiplication must be performed before subtraction according to standard mathematical order of operations.

Key Concept

Evaluating expressions containing multiple variables with fractional exponents, negative bases, and standard order of operations.

Estimated Time:2m 0s

Question 146Question →

A large rectangle has a length of $4x + 3$ inches and a width of $3x - 2$ inches. A smaller rectangular region is cut out from the center. The cut-out region has a length of $2x - 1$ inches and a width of $x - 5$ inches. The area of the remaining region can be written as the polynomial $Ax^2 + Bx + C$ , where $A$ , $B$ , and $C$ are integers. What is the value of the coefficient $B$ ?

Show answer & explanation

Answer: 12

Answer

The correct answer is 12, which is the coefficient of the linear term in the simplified remaining area polynomial.

The remaining area is found by subtracting the area of the smaller rectangle from the area of the larger rectangle. The area of the larger rectangle is

(4x + 3)(3x - 2) = 12x^2 + x - 6

. The area of the smaller rectangle is

(2x - 1)(x - 5) = 2x^2 - 11x + 5

. Subtracting the two yields

(12x^2 + x - 6) - (2x^2 - 11x + 5) = 10x^2 + 12x - 11

. Matching this to

Ax^2 + Bx + C

, the coefficient

B

of the linear term is

12

.

Step-by-Step Solution

1

Find the area of the outer rectangle.

Area =

12x^2 + x - 6

Multiply the length

(4x + 3)

and the width

(3x - 2)

using the distributive property.

2

Find the area of the inner rectangle.

Area =

2x^2 - 11x + 5

Multiply the length

(2x - 1)

and the width

(x - 5)

using the distributive property.

3

Subtract the inner area from the outer area to find the remaining area.

Remaining Area =

10x^2 + 12x - 11

Subtract the polynomial

(2x^2 - 11x + 5)

from

(12x^2 + x - 6)

by distributing the negative sign and combining like terms.

4

Identify the coefficient

B

of the

x

term.

B = 12

Match the simplified polynomial

10x^2 + 12x - 11

with the form

Ax^2 + Bx + C

.

Key Concept

Operations on Polynomials (multiplication and subtraction of polynomials)

Question 147Question →

Match each of the unsimplified algebraic expressions on the left with its equivalent simplified form on the right. (Assume all variables represent real numbers.)

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches pair the first expression with

6r^2 - 5rs + 6s^2

, the second expression with

2r^2 - 2rs + s^2

, the third expression with

3s^2 - 4rs

, and the fourth expression with

6r^2 - 2r^2s + 5s^2

.

Each expression is correctly simplified by performing distribution first (carefully tracking negative signs and binomial expansion rules) and then combining terms that share the exact same variable powers.

Step-by-Step Solution

1

Simplify the expression

3r(2r - s) - 2s(r - 3s)

.

6r^2 - 5rs + 6s^2

Distribute

3r

to get

6r^2 - 3rs

. Then distribute

-2s

to get

-2rs + 6s^2

(noting that a negative times a negative is positive). Combine the like terms

-3rs

and

-2rs

to get

-5rs

.

2

Simplify the expression

(2r - s)^2 - 2(r^2 - rs)

.

2r^2 - 2rs + s^2

Square the binomial

(2r - s)^2

using the formula

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

to get

4r^2 - 4rs + s^2

. Distribute

-2

to get

-2r^2 + 2rs

. Combine

4r^2 - 2r^2

to get

2r^2

, and

-4rs + 2rs

to get

-2rs

.

3

Simplify the expression

\frac{1}{2}(4r^2 - 6rs) - (r - s)(2r + 3s)

.

3s^2 - 4rs

Distribute

\frac{1}{2}

to get

2r^2 - 3rs

. Expand the binomial product to get

2r^2 + 3rs - 2rs - 3s^2 = 2r^2 + rs - 3s^2

. Subtract this entire expression:

(2r^2 - 3rs) - (2r^2 + rs - 3s^2) = 2r^2 - 3rs - 2r^2 - rs + 3s^2

. Combine like terms to get

-4rs + 3s^2

.

4

Simplify the expression

r^2(6 - s) - s(r^2 - 5s)

.

6r^2 - 2r^2s + 5s^2

Distribute

r^2

to get

6r^2 - r^2s

. Distribute

-s

to get

-sr^2 + 5s^2

. Combine the like terms

-r^2s

and

-sr^2

(since multiplication is commutative) to get

-2r^2s

.

Key Concept

Simplifying expressions by applying the distributive property, expanding products of binomials, and combining like terms.

Question 148Question →

Which of the following represents the completely factored form of the expression $4x^4 - 16x^2$ ?

4x^2(x - 2)^2

4x^3(x - 4)

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

(2x^2 - 4x)^2

16x^2(x^2 - 1)

Show answer & explanation

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

Answer

The completely factored form of the expression is

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

.

The correct answer is found by first identifying the greatest common factor of the terms in the polynomial

4x^4 - 16x^2

, which is

4x^2

. Factoring this out results in

4x^2(x^2 - 4)

. Next, the binomial

x^2 - 4

is recognized as a difference of squares and is factored into

(x - 2)(x + 2)

. Putting these together gives the completely factored form

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

.

Step-by-Step Solution

1

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

4x^4 - 16x^2

.

The GCF of

4x^4

and

16x^2

is

4x^2

. Factoring it out yields

4x^2(x^2 - 4)

.

Factoring out the greatest common factor simplifies the polynomial and reveals remaining factorable patterns.

2

Factor the remaining binomial expression inside the parentheses,

x^2 - 4

.

Since

x^2 - 4

is a difference of squares (

x^2 - 2^2

), it factors into

(x - 2)(x + 2)

.

A difference of squares of the form

a^2 - b^2

always factors into

(a - b)(a + b)

.

3

Combine the factored parts to write the completely factored expression.

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

This combines the GCF and the factored difference of squares to represent the original expression in its simplest factored parts.

Key Concept

Factoring a polynomial completely by first extracting the greatest common factor (GCF) and then applying the difference of squares formula.

Estimated Time:1m 0s

Question 149Question →

If $(x - 4)^2 = -4x + 28$ , what is the sum of the solutions to the equation?

-8

-4

4

8

28

Show answer & explanation

Answer: 4

Answer

The sum of the solutions to the equation is

4

.

The correct answer of

4

is found by first expanding

(x - 4)^2

to obtain

x^2 - 8x + 16 = -4x + 28

. Setting the equation to zero gives the quadratic equation

x^2 - 4x - 12 = 0

. Factoring the trinomial yields

(x - 6)(x + 2) = 0

, which results in the solutions

x = 6

and

x = -2

. Adding these solutions together gives

6 + (-2) = 4

.

Step-by-Step Solution

1

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

x^2 - 8x + 16 = -4x + 28

Applying the binomial expansion formula

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

to

(x - 4)^2

yields

x^2 - 8x + 16

.

2

Rearrange the equation into standard quadratic form by adding

4x

and subtracting

28

from both sides.

x^2 - 4x - 12 = 0

To solve a quadratic equation by factoring, all terms must be moved to one side so that the equation is set to zero.

3

Factor the quadratic expression

x^2 - 4x - 12

.

(x - 6)(x + 2) = 0

Find two integers that multiply to the constant term

-12

and add to the linear coefficient

-4

. These numbers are

-6

and

2

.

4

Set each binomial factor to zero and solve for

x

.

x = 6

or

x = -2

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

5

Find the sum of the solutions.

6 + (-2) = 4

Adding the two individual solutions gives the requested sum.

Key Concept

Solving quadratic equations by expanding binomials, rearranging terms into standard form, factoring the trinomial, and applying the zero-product property.

Alternative Method

Once the equation is rewritten in the standard form

x^2 - 4x - 12 = 0

, Vieta's formulas can be used to find the sum of the solutions directly. For any quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is given by

-\frac{b}{a}

. Substituting

a = 1

and

b = -4

into the formula gives

-\frac{-4}{1} = 4

.

Estimated Time:1m 0s

Question 150Question →

A landscaping company charges a flat equipment fee plus a fixed hourly rate for lawn maintenance. For a job that took $4$ hours, the company charged a total of $\$190$ . For a different job that took $7$ hours, the company charged a total of $\$295$ . The company also offers a package that includes a $15\%$ discount on the hourly rate, but the flat equipment fee remains the same. Under this discounted package, what is the total charge, in dollars, for a job that takes $8$ hours?

Show answer & explanation

Answer: 288

Answer

The total charge for an

8

-hour job under the discounted package is

288

dollars.

By translating the problem into the equations

F + 4H = 190

and

F + 7H = 295

, we find that the hourly rate

H

is

35

dollars and the flat fee

F

is

50

dollars. Applying a

15\%

discount to the hourly rate yields a new rate of

29.75

dollars per hour. The total cost for

8

hours is then

50 + 8(29.75) = 288

dollars.

Step-by-Step Solution

1

Define variables for the flat fee and hourly rate, and translate the given information into a system of linear equations.

F + 4H = 190

and

F + 7H = 295

, where

F

is the flat fee and

H

is the hourly rate.

To represent the cost structure algebraically.

2

Solve the system of equations by elimination or substitution.

H = 35

and

F = 50

.

To determine the individual cost components (flat fee and hourly rate).

3

Apply the

15\%

discount to the hourly rate.

H_{\text{discounted}} = 35 \times (1 - 0.15) = 29.75

.

To find the new hourly rate under the discounted package.

4

Calculate the total cost for

8

hours of work with the flat fee and the discounted hourly rate.

\text{Total Cost} = 50 + 8(29.75) = 288

.

To answer the question asking for the total charge of an

8

-hour job under the discounted package.

Key Concept

Translating verbal statements into a system of linear equations and solving them to evaluate a modified expression.

Question 151Question →

If $a$ and $b$ represent positive real numbers, which of the following is an equivalent form of the expression below?

\frac{(a^{-1} + b^{-1})^{-2}}{ab}

\frac{a^2 + b^2}{ab}

\frac{(a+b)^2}{4ab}

\frac{ab}{(a+b)^2}

\frac{(a+b)^2}{ab}

\frac{1}{a^2 b^2 (a+b)^2}

Show answer & explanation

Answer: $\frac{ab}{(a+b)^2}$

Answer

The correct answer is the fraction with

ab

in the numerator and the square of the sum

(a+b)

in the denominator, which is \frac{ab}{(a+b)^2}.

The correct answer is found by first rewriting

a^{-1} + b^{-1}

as

\frac{1}{a} + \frac{1}{b}

, which simplifies to

\frac{a+b}{ab}

using a common denominator. Raising this to the power of

-2

gives

\frac{a^2 b^2}{(a+b)^2}

. Finally, dividing this expression by

ab

reduces the exponents of

a

and

b

by 1, resulting in

\frac{ab}{(a+b)^2}

.

Step-by-Step Solution

1

Rewrite the negative exponents in the numerator as reciprocals.

The expression inside the parentheses becomes

\frac{1}{a} + \frac{1}{b}

.

By the definition of negative exponents,

x^{-1} = \frac{1}{x}

.

2

Find a common denominator to add the fractions inside the parentheses.

\frac{1}{a} + \frac{1}{b} = \frac{b}{ab} + \frac{a}{ab} = \frac{a+b}{ab}

.

To add fractions, they must share a common denominator, which is the product of

a

and

b

.

3

Apply the negative power of

-2

to the simplified fraction.

\left(\frac{a+b}{ab}\right)^{-2} = \left(\frac{ab}{a+b}\right)^2 = \frac{a^2 b^2}{(a+b)^2}

.

An expression raised to a negative exponent is equal to the reciprocal of the expression raised to the positive exponent.

4

Divide the result by the denominator

ab

.

\frac{a^2 b^2}{(a+b)^2} \cdot \frac{1}{ab} = \frac{ab}{(a+b)^2}

.

Dividing by a term is equivalent to multiplying by its reciprocal, and applying the exponent quotient rule simplifies

\frac{a^2 b^2}{ab}

to

ab

.

Key Concept

Properties of exponents including negative power rules, quotient rules, and algebraic fraction addition.

Alternative Method

An alternative method is to substitute small positive integer values for

a

and

b

. Let

a = 1

and

b = 2

. The expression evaluates to

\frac{(1^{-1} + 2^{-1})^{-2}}{1 \cdot 2} = \frac{(1 + 0.5)^{-2}}{2} = \frac{1.5^{-2}}{2} = \frac{(3/2)^{-2}}{2} = \frac{4/9}{2} = \frac{2}{9}

. Evaluating the correct expression with these values yields

\frac{1 \cdot 2}{(1+2)^2} = \frac{2}{9}

, which matches.

Estimated Time:2m 0s

Question 152Question →

A square has a side length of $x^3 - 3$ inches. Which of the following expressions represents the area, in square inches, of the square?

x^6 - 9

x^9 - 6x^3 + 9

x^6 - 6x^3 + 9

x^6 + 9

x^9 - 9

Show answer & explanation

Answer: $x^6 - 6x^3 + 9$

Answer

The expression

x^6 - 6x^3 + 9

To find the area of a square, square its side length:

(x^3 - 3)^2

. Expanding this gives

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

.

Step-by-Step Solution

1

Write the formula for the area of a square.

A = s^2

, where

s

is the side length.

The area of a square is the side length squared.

2

Substitute the given side length

s = x^3 - 3

into the area formula.

A = (x^3 - 3)^2

We replace

s

with the algebraic expression for the side length.

3

Expand the squared binomial

(x^3 - 3)^2

using the identity

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

.

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

Expanding the binomial yields the correct simplified polynomial in standard descending order.

Key Concept

Expanding a squared binomial and applying exponent rules for powers of powers.

Estimated Time:1m 0s

Question 153Question →

The weight of Box A is $1.5$ pounds more than $\frac{2}{3}$ of the weight of Box B. If the weight of Box A is $13.5$ pounds, what is the weight of Box B, in pounds?

8

10.5

13.5

18

22.5

Show answer & explanation

Answer: 18

Answer

The weight of Box B is 18 pounds.

The weight of 18 pounds is correct. Translating the word problem yields the linear equation

A = \frac{2}{3}B + 1.5

. Substituting the weight of Box A (

13.5

) gives

13.5 = \frac{2}{3}B + 1.5

. Subtracting

1.5

from both sides results in

12 = \frac{2}{3}B

. Multiplying both sides by the reciprocal

\frac{3}{2}

isolates

B

, giving

B = 12 \times \frac{3}{2} = 18

.

Step-by-Step Solution

1

Translate the verbal description into an algebraic equation.

Let

A

represent the weight of Box A and

B

represent the weight of Box B. The description translates to the equation:

A = \frac{2}{3}B + 1.5

.

This establishes the mathematical relationship between the weights of the two boxes.

2

Substitute the given value for Box A into the equation and isolate the term containing the variable B.

13.5 = \frac{2}{3}B + 1.5 \implies 12 = \frac{2}{3}B

.

Substituting

13.5

for

A

and subtracting

1.5

from both sides simplifies the equation to isolate the fraction term.

3

Solve for B by multiplying both sides of the equation by the reciprocal of the coefficient.

B = 12 \times \frac{3}{2} = 18

.

Multiplying by the reciprocal

\frac{3}{2}

isolates

B

to find its value.

Key Concept

Translating real-world descriptions into multi-step linear equations and solving them using inverse operations.

Estimated Time:1m 0s

Question 154Question →

A digital marketing firm allocates a total monthly advertising budget of $B$ dollars between search engine ads and social media ads. The amount allocated to social media ads is $\$800$ less than twice the amount allocated to search engine ads. The firm then increases its total monthly budget by $20\%$ and allocates this entire increase to search engine ads. As a result, the new amount allocated to search engine ads is equal to the amount originally allocated to social media ads. What was the firm's original total monthly advertising budget, $B$ ?

\$1,600

\$2,200

\$2,400

\$4,000

\$8,000

Show answer & explanation

Answer: $\$4,000$

Answer

The original total monthly advertising budget was

\$4,000

.

By defining the original search engine advertising budget as

S

, the original social media advertising budget as

M = 2S - 800

, and the original total budget as

B = 3S - 800

, we can set up the equation

S + 0.20B = M

using the condition that the new search engine budget equals the original social media budget. Solving this equation yields

S = 1600

. Substituting

S = 1600

into the expression for the total budget gives

B = 3(1600) - 800 = 4000

dollars.

Step-by-Step Solution

1

Define the variables representing each budget allocation.

Let

S

represent the original amount allocated to search engine ads, and let

M

represent the original amount allocated to social media ads. The original total budget is

B = S + M

.

Establishing clear variables is necessary to translate the verbal relationships into equations.

2

Translate the relationship between the two advertising categories into an equation.

M = 2S - 800

The problem states that the social media ads budget is

\$800

less than twice the search engine ads budget.

3

Express the original total budget

B

in terms of a single variable,

S

.

B = S + (2S - 800) = 3S - 800

Substituting the expression for

M

into the total budget equation simplifies the system to a single variable.

4

Set up the equation for the new budget allocation after the

20\%

increase.

S + 0.20B = M

An increase of

20\%

of the total budget is

0.20B

, which is added entirely to the search engine ads budget, making it equal to the original social media ads budget.

5

Substitute the expressions for

B

and

M

in terms of

S

into the new budget equation.

S + 0.20(3S - 800) = 2S - 800

Substituting the relationships found in steps 2 and 3 allows us to solve for

S

directly.

6

Solve the equation for

S

.

S + 0.60S - 160 = 2S - 800 \Rightarrow 1.60S - 160 = 2S - 800 \Rightarrow 640 = 0.40S \Rightarrow S = 1,600

Distributing, combining like terms, and isolating the variable yields the original search engine ads allocation.

7

Calculate the original total budget,

B

.

B = 3(1,600) - 800 = 4,800 - 800 = 4,000

The question asks for the original total budget

B

, so we must evaluate the expression for

B

using the value of

S

.

Key Concept

Translating verbal descriptions of algebraic relationships into systems of linear equations and solving them.

Question 155Question →

When the expression $(5x^2 - 8x + 2) - (2x^2 - 4x - 9)$ is written in the standard form $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are integers, what is the value of $b$ ?

Show answer & explanation

Answer: -4

Answer

The correct answer is

-4

, which is the coefficient of the

x

term after simplifying the expression.

To subtract polynomials, we distribute the negative sign to all terms of the polynomial being subtracted and then combine like terms. Simplifying

(5x^2 - 8x + 2) - (2x^2 - 4x - 9)

gives

5x^2 - 8x + 2 - 2x^2 + 4x + 9 = 3x^2 - 4x + 11

. Comparing this to

ax^2 + bx + c

shows that

b

, the coefficient of the

x

term, is

-4

.

Step-by-Step Solution

1

Distribute the subtraction sign to all terms inside the second set of parentheses.

-2x^2 + 4x + 9

Subtracting a polynomial is equivalent to adding its opposite, which means changing the sign of every term in that polynomial.

2

Group and combine like terms from both polynomials.

3x^2 - 4x + 11

Like terms (terms with the same variable raised to the same power) can be combined by adding or subtracting their coefficients:

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

,

(-8x + 4x) = -4x

, and

(2 + 9) = 11

.

3

Compare the simplified expression to the standard form

ax^2 + bx + c

to identify the value of

b

.

b = -4

In the standard form

ax^2 + bx + c

, the coefficient of the

x

term is represented by

b

. In the simplified expression

3x^2 - 4x + 11

, the coefficient of

x

is

-4

.

Key Concept

Polynomial Subtraction and Combining Like Terms

Question 156Question →

If the polynomial $x^4 - 13x^2 + 36$ is factored completely into the product of four linear binomials of the form $(x - r_1)(x - r_2)(x - r_3)(x - r_4)$ , where $r_1 < r_2 < r_3 < r_4$ , what is the value of the expression $r_1 + 2r_2 + 3r_3 + 4r_4$ ?

Show answer & explanation

Answer: 11

Answer

The value of the expression is 11.

Factoring the quartic polynomial

x^4 - 13x^2 + 36

as a quadratic in

x^2

yields

(x^2 - 9)(x^2 - 4)

. Applying the difference of squares identity to each factor gives

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

. Matching these to the template

(x - r_1)(x - r_2)(x - r_3)(x - r_4)

with the condition

r_1 < r_2 < r_3 < r_4

yields

r_1 = -3

,

r_2 = -2

,

r_3 = 2

, and

r_4 = 3

. Evaluating the linear combination

-3 + 2(-2) + 3(2) + 4(3)

results in

11

.

Step-by-Step Solution

1

Substitute

u = x^2

and factor the quadratic trinomial.

u^2 - 13u + 36 = (u - 9)(u - 4) \Rightarrow (x^2 - 9)(x^2 - 4)

To reduce the degree of the polynomial and make it easier to factor.

2

Apply the difference of squares formula to each binomial factor.

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

Both

x^2 - 9

and

x^2 - 4

are differences of squares.

3

Determine the roots

r_1, r_2, r_3, r_4

in ascending order.

r_1 = -3

,

r_2 = -2

,

r_3 = 2

,

r_4 = 3

We write each factor as

(x - r_i)

to find the roots, and then sort them from least to greatest according to the inequality constraint.

4

Calculate the value of the requested expression.

11

Substitute the sorted values into the linear combination:

(-3) + 2(-2) + 3(2) + 4(3) = -3 - 4 + 6 + 12 = 11

.

Key Concept

Factoring a quartic polynomial of quadratic form followed by factoring differences of squares.

Question 157Question →

If the equation $2x(x - 1) = 3 - x$ is solved for $x$ , what is the larger of the two solutions?

Show answer & explanation

Answer: 1.5

Answer

The larger of the two solutions is

1.5

.

Expanding the equation

2x(x - 1) = 3 - x

results in

2x^2 - 2x = 3 - x

. Setting this quadratic equation to zero yields

2x^2 - x - 3 = 0

. Factoring the expression gives

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

. Setting the first factor to zero,

2x - 3 = 0

, results in

x = 1.5

. Setting the second factor to zero,

x + 1 = 0

, results in

x = -1

. Comparing the two values,

1.5

is the larger solution.

Step-by-Step Solution

1

Expand and rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

.

2x^2 - x - 3 = 0

A quadratic equation must be set to zero before solving it by factoring.

2

Factor the quadratic trinomial by grouping.

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

Factoring allows us to apply the zero product property.

3

Set each factor to zero to solve for the roots.

x = 1.5

and

x = -1

If a product equals zero, at least one of its factors must be zero.

4

Compare the solutions to find the larger value.

1.5

Comparing

1.5

and

-1

shows that

1.5

is the greater value.

Key Concept

Solving quadratic equations by factoring when the leading coefficient is greater than 1

Question 158Question →

When the expression $3x^2(2x - y) - 2y(x^2 - 3xy) - (4x^3 - xy^2)$ is simplified to the form $Ax^3 + Bx^2y + Cxy^2$ , where $A$ , $B$ , and $C$ are integers, what is the value of $A + B + C$ ?

-8

-2

1

2

4

Show answer & explanation

Answer: 4

Answer

4

The correct sum of the coefficients is 4. By distributing all terms correctly:

3x^2(2x - y) = 6x^3 - 3x^2y

,

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

, and

-(4x^3 - xy^2) = -4x^3 + xy^2

. Combining the terms yields

2x^3 - 5x^2y + 7xy^2

, which gives

A = 2

,

B = -5

, and

C = 7

. Summing these values gives

2 - 5 + 7 = 4

.

Step-by-Step Solution

1

Distribute

3x^2

to the first parenthetical expression:

3x^2(2x - y)

6x^3 - 3x^2y

Applying the distributive property multiplies

3x^2

by both

2x

and

-y

.

2

Distribute

-2y

to the second parenthetical expression:

-2y(x^2 - 3xy)

-2x^2y + 6xy^2

Applying the distributive property multiplies

-2y

by both

x^2

and

-3xy

, noting that a negative times a negative is a positive.

3

Distribute the negative sign to the third parenthetical expression:

-(4x^3 - xy^2)

-4x^3 + xy^2

Distributing the negative sign changes the signs of both terms inside the parenthesis.

4

Combine the expanded parts:

(6x^3 - 3x^2y) + (-2x^2y + 6xy^2) + (-4x^3 + xy^2)

and group like terms

(6x^3 - 4x^3) + (-3x^2y - 2x^2y) + (6xy^2 + xy^2)

Grouping like terms together makes it easier to combine their coefficients.

5

Combine the coefficients of the like terms

2x^3 - 5x^2y + 7xy^2

Combining the coefficients gives

A = 2

,

B = -5

, and

C = 7

.

6

Calculate the sum

A + B + C

2 + (-5) + 7 = 4

Adding the coefficients together yields the final numerical value.

Key Concept

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

Question 159Question →

A square metal sheet has a side length of $x + 4$ inches. A rectangular region with a width of $x$ inches and a length of $x + 2$ inches is removed from the sheet. Which of the following expressions represents the area, in square inches, of the remaining portion of the sheet?

-2x + 16

8x + 14

10x + 16

6x + 16

6x + 8

Show answer & explanation

Answer: 6x + 16

Answer

The expression that represents the remaining area is

6x + 16

.

To find the remaining area, subtract the area of the removed rectangle from the area of the original square sheet. The area of the square is

(x + 4)^2 = x^2 + 8x + 16

and the area of the rectangle is

x(x + 2) = x^2 + 2x

. Subtracting these yields

(x^2 + 8x + 16) - (x^2 + 2x) = x^2 + 8x + 16 - x^2 - 2x = 6x + 16

.

Step-by-Step Solution

1

Find the area of the original square metal sheet.

Area of the square is

(x + 4)^2 = x^2 + 8x + 16

square inches.

The area of a square is calculated by squaring its side length.

2

Find the area of the removed rectangular region.

Area of the rectangle is

x(x + 2) = x^2 + 2x

square inches.

The area of a rectangle is calculated by multiplying its width by its length.

3

Subtract the area of the rectangular region from the area of the square sheet.

(x^2 + 8x + 16) - (x^2 + 2x) = x^2 + 8x + 16 - x^2 - 2x = 6x + 16

square inches.

Subtracting the removed area from the total area gives the remaining area, distributing the negative sign to all terms inside the parentheses.

Key Concept

Operations on Polynomials

Estimated Time:45s

Question 160Question →

A school club sold 20 rolls of wrapping paper and 25 boxes of greeting cards for a fundraiser. The price of a roll of wrapping paper was $\frac{3}{4}$ of the price of a box of greeting cards. If the club raised a total of $800 from these sales, what was the price of a single roll of wrapping paper?

$5.00

$20.00

$15.00

$20.65

$26.67

Show answer & explanation

Answer: $15.00

Answer

The price of a single roll of wrapping paper was $15.00.

The correct answer is

15.00. By representing the price of a roll of wrapping paper as

W

and the price of a box of greeting cards as

G

, the problem states that

W = \frac{3}{4}G

, which means

G = \frac{4}{3}W

. The total revenue equation is

20W + 25G = 800

. Substituting

G

gives

20W + 25\left(\frac{4}{3}W\right) = 800

, which simplifies to

20W + \frac{100}{3}W = 800

. Multiplying the entire equation by 3 to clear the fraction results in

60W + 100W = 2400

. Combining like terms gives

160W = 2400

, which yields

W = 15$.

Step-by-Step Solution

1

Define variables for the unknowns and translate the price relationship into an algebraic equation.

Let

W

be the price of a roll of wrapping paper and

G

be the price of a box of greeting cards. The relationship is given by

W = \frac{3}{4}G

, which can be rearranged to express

G

in terms of

W

:

G = \frac{4}{3}W

.

Expressing one variable in terms of another allows us to set up a single-variable linear equation.

2

Write the linear equation representing the total revenue from the fundraiser sales.

The total revenue from selling 20 rolls of wrapping paper and 25 boxes of greeting cards is

20W + 25G = 800

. Substituting

G = \frac{4}{3}W

gives:

20W + 25\left(\frac{4}{3}W\right) = 800

, which simplifies to

20W + \frac{100}{3}W = 800

.

This sets up the equation that we need to solve to find the value of

W

.

3

Clear the fraction by multiplying all terms by 3 and solve for

W

.

Multiplying the entire equation by 3 yields:

3(20W) + 3\left(\frac{100}{3}W\right) = 3(800) \implies 60W + 100W = 2400

. Combining like terms gives

160W = 2400

. Dividing by 160 yields

W = 15

.

Clearing the denominator simplifies the equation to a standard linear form that can be solved directly.

Key Concept

Solving linear equations derived from real-world contexts, particularly those involving fractional relationships and multi-step isolation.

Alternative Method

Instead of expressing

G

in terms of

W

first, solve for

G

directly by substituting

W = \frac{3}{4}G

into the revenue equation. This gives

20\left(\frac{3}{4}G\right) + 25G = 800 \implies 15G + 25G = 800 \implies 40G = 800 \implies G = 20

. Then, calculate

W = \frac{3}{4}(20) = 15

.

Estimated Time:2m 30s

Elementary Algebra

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