Elementary Algebra Practice Questions — ACT — Page 10

For all non-zero real numbers

a

,

b

, and

c

, the expression below is simplified:

\frac{(2a^2 b^{-1} c^3)^3}{4a^5 (b^2 c^{-2})^{-2}}

Which of the following is equivalent to this expression?

\frac{2ac^5}{b^7}

\frac{abc^5}{2}

2abc^5

4abc^5

2b^2c^{10}

Show answer & explanation

Answer: $2abc^5$

Answer

The expression is equivalent to

2abc^5

.

The correct answer is obtained by first simplifying the numerator and denominator using the power of a product and power of a power rules, then dividing the coefficients and subtracting the exponents of like bases. This yields the simplified expression

2abc^5

.

Step-by-Step Solution

1

Apply the power of a product rule to the numerator.

(2a^2 b^{-1} c^3)^3 = 2^3 \cdot (a^2)^3 \cdot (b^{-1})^3 \cdot (c^3)^3 = 8 a^6 b^{-3} c^9

Each factor inside the parentheses must be raised to the power of 3, multiplying the exponents of the variables.

2

Apply the power of a product rule to the denominator's parentheses.

4a^5 (b^2 c^{-2})^{-2} = 4a^5 \cdot (b^2)^{-2} \cdot (c^{-2})^{-2} = 4a^5 b^{-4} c^4

The terms inside the parentheses are raised to the power of -2, multiplying their exponents.

3

Divide the simplified numerator by the simplified denominator.

2a^1 b^1 c^5

, which is

2abc^5

Divide the coefficients (8 / 4 = 2) and subtract the exponents of the same bases:

a^{6-5} = a^1

,

b^{-3 - (-4)} = b^1

, and

c^{9-4} = c^5

.

Key Concept

Properties of Exponents in Algebraic Expressions

Alternative Method

Alternatively, you can rewrite the expression by eliminating negative exponents first. Convert

b^{-1}

to

\frac{1}{b}

,

b^2

to itself, and

c^{-2}

to

\frac{1}{c^2}

within the parentheses, apply the outer exponents, and then simplify the resulting complex fraction.

Estimated Time:1m 30s

Question 182Question →

Two polynomial expressions are defined as $P(x) = (2x - 4)(x^2 - 5x + c)$ and $Q(x) = (x^2 + 3x - 4)(2x + a)$ , where $a$ and $c$ are constants. When $P(x)$ is expanded and simplified, it has no $x$ term. If the coefficient of the $x^2$ term in the expanded and simplified form of $Q(x)$ is equal to the coefficient of the $x^2$ term in the expanded and simplified form of $P(x)$ , what is the value of $a$ ?

Show answer & explanation

Answer: -20

Answer

The value of

a

is

-20

.

Expanding

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

yields

2x^3 - 14x^2 + (2c + 20)x - 4c

. Since there is no

x

term,

2c + 20 = 0

, which means

c = -10

. This leaves the coefficient of the

x^2

term in

P(x)

as

-14

. Expanding

Q(x) = (x^2 + 3x - 4)(2x + a)

yields

2x^3 + (a + 6)x^2 + (3a - 8)x - 4a

, so the coefficient of its

x^2

term is

a + 6

. Setting

a + 6 = -14

and solving for

a

gives

a = -20

.

Step-by-Step Solution

1

Expand the polynomial

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

using the distributive property.

P(x) = 2x^3 - 14x^2 + (2c + 20)x - 4c

To identify the coefficients of each term in

P(x)

.

2

Set the coefficient of the

x

term,

2c + 20

, to

0

and solve for

c

.

c = -10

The problem states that

P(x)

has no

x

term when simplified, which means its coefficient must be zero.

3

Determine the coefficient of the

x^2

term in

P(x)

.

The coefficient of

x^2

is

-14

.

This coefficient will be equated to the

x^2

coefficient of

Q(x)

as per the problem constraints.

4

Expand the polynomial

Q(x) = (x^2 + 3x - 4)(2x + a)

using the distributive property.

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

To identify the coefficient of the

x^2

term in

Q(x)

.

5

Set the coefficient of the

x^2

term in

Q(x)

, which is

a + 6

, equal to the coefficient of the

x^2

term in

P(x)

, which is

-14

, and solve for

a

.

a = -20

The problem states that these two coefficients are equal.

Key Concept

Operations on Polynomials

Estimated Time:2m 30s

Question 183Question →

A rectangle has a length of $y + 7$ inches and a width of $y - 3$ inches. Which of the following expressions represents the area, in square inches, of the rectangle?

y^2 - 21

y^2 + 4y + 21

y^2 + 4y - 21

y^2 - 4y - 21

5y - 21

Show answer & explanation

Answer: y^2 + 4y - 21

Answer

The expression

y^2 + 4y - 21

The area of a rectangle is found by multiplying its length by its width. The product of the dimensions is represented by the expression

(y + 7)(y - 3)

. Applying the distributive property gives

y^2 - 3y + 7y - 21

. Combining the like terms

-3y

and

7y

results in

+4y

. Therefore, the simplified expression for the area is

y^2 + 4y - 21

.

Step-by-Step Solution

1

Set up the area formula for the rectangle by multiplying its length and width.

Area

= (y + 7)(y - 3)

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

2

Use the distributive property to expand the product of the two binomials.

Area

= y(y) + y(-3) + 7(y) + 7(-3) = y^2 - 3y + 7y - 21

Each term in the first binomial must be multiplied by each term in the second binomial.

3

Combine the like terms to simplify the expression into standard form.

Area

= y^2 + 4y - 21

Combining

-3y

and

+7y

yields

+4y

.

Key Concept

Finding the area of a rectangle by multiplying binomial expressions using the distributive property.

Estimated Time:45s

Question 184Question →

If $x$ is a real number such that $\frac{(3^x)^4 \cdot 9^{2-x}}{27^{x-1}} = 81$ , what is the value of $x$ ?

Show answer & explanation

Answer: 3

Answer

The value of

x

is 3.

Rewriting the bases in terms of 3, the expression becomes

\frac{3^{4x} \cdot 3^{4-2x}}{3^{3x-3}} = 3^4

. Combining the numerator using the product rule gives

3^{2x+4}

in the numerator. Dividing by the denominator using the quotient rule gives

3^{(2x+4)-(3x-3)} = 3^{-x+7}

. Equating this to

3^4

gives

-x + 7 = 4

, which yields

x = 3

.

Step-by-Step Solution

1

Express each base in terms of the common base 3

9 = 3^2

,

27 = 3^3

, and

81 = 3^4

Converting all terms to a common base allows the use of exponent rules to combine them.

2

Apply the power of a power rule

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

to rewrite each exponent

(3^x)^4 = 3^{4x}

,

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

, and

(3^3)^{x-1} = 3^{3x-3}

This simplifies individual terms by multiplying their exponents.

3

Apply the product rule of exponents

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

to combine the terms in the numerator

3^{4x} \cdot 3^{4-2x} = 3^{4x + 4 - 2x} = 3^{2x + 4}

Multiplying exponential terms with the same base is simplified by adding their exponents.

4

Apply the quotient rule of exponents

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

to simplify the fraction

3^{(2x+4) - (3x-3)} = 3^{-x+7}

Dividing exponential terms with the same base is simplified by subtracting the exponent in the denominator from the exponent in the numerator.

5

Equate the exponents of the simplified base 3 expression and base 3 representation of 81

-x + 7 = 4

, which solves to

x = 3

Since the bases are equal, the powers must be equal for the equation to hold true.

Key Concept

Properties of Exponents in Algebraic Expressions

Question 185Question →

What is the greatest integer value of $x$ that satisfies the inequality $\frac{2 - 5x}{3} - \frac{x - 4}{2} \geq 3$ ?

Show answer & explanation

Answer: -1

Answer

The greatest integer value of

x

that satisfies the inequality is -1.

The correct answer is -1 because solving the inequality leads to

x \leq -\frac{2}{13}

. Since

-\frac{2}{13}

is approximately

-0.154

, the set of integers satisfying the inequality is

\{-1, -2, -3, \dots\}

. The greatest integer in this set is -1.

Step-by-Step Solution

1

Multiply the entire inequality by the least common multiple of the denominators (6) to eliminate the fractions.

2(2 - 5x) - 3(x - 4) \geq 18

Multiplying by a positive number clears the fractions without changing the direction of the inequality.

2

Distribute the coefficients and combine like terms on the left side of the inequality.

16 - 13x \geq 18

Simplifying the expressions on each side makes it easier to isolate the variable.

3

Subtract 16 from both sides to isolate the term with the variable

x

.

-13x \geq 2

Moving the constant terms to one side prepares the inequality for division.

4

Divide both sides by -13 and reverse the direction of the inequality sign.

x \leq -\frac{2}{13}

Dividing by a negative number requires flipping the inequality sign to maintain a true statement.

5

Determine the greatest integer that is less than or equal to

-\frac{2}{13}

.

-1

Since

-\frac{2}{13} \approx -0.154

, the largest integer that is less than or equal to this value is -1.

Key Concept

Solving multi-step linear inequalities, including clearing fractional coefficients and reversing the inequality sign when multiplying or dividing by a negative number.

Question 186Question →

What is the maximum integer value of $k$ that satisfies the inequality $8 - 5k > 28$ ?

Show answer & explanation

Answer: -5

Answer

The maximum integer value that satisfies the inequality is

-5

.

Subtracting 8 from both sides of the inequality

8 - 5k > 28

yields

-5k > 20

. Dividing both sides of the inequality by

-5

and reversing the inequality sign results in

k < -4

. The largest integer strictly less than

-4

is

-5

.

Step-by-Step Solution

1

Subtract 8 from both sides of the inequality.

-5k > 20

To isolate the term with the variable on the left side of the inequality.

2

Divide both sides by

-5

and reverse the inequality sign.

k < -4

Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality symbol.

3

Determine the largest integer strictly less than

-4

.

-5

Because the inequality is strict (

<

), the value of

k

cannot be equal to

-4

. The greatest integer less than

-4

is

-5

.

Key Concept

Solving linear inequalities and reversing the inequality sign when dividing by a negative number.

Question 187Question →

For the opening night of a school play, a total of $320$ tickets were sold, raising a total of $\$2{}140$ in ticket sales. Student tickets were sold for $\$5$ each, and adult tickets were sold for $\$8$ each. How many adult tickets were sold?

Show answer & explanation

Answer: 180

Answer

180

The correct answer is

180

. By setting up a single-variable linear equation where

a

represents the number of adult tickets, the problem translates to

8a + 5(320 - a) = 2140

. Simplifying this yields

3a + 1600 = 2140

, which simplifies to

3a = 540

and results in

a = 180

.

Step-by-Step Solution

1

Define variables for the unknown quantities based on the given total.

Let

a

represent the number of adult tickets sold. The number of student tickets sold is represented as

320 - a

.

Since the total number of tickets is

320

, subtracting the number of adult tickets from the total yields the number of student tickets.

2

Construct a linear equation using the ticket prices and the total revenue.

8a + 5(320 - a) = 2140

Multiplying the quantity of each ticket type by its respective price (

8

dollars for adult and

5

dollars for student) yields the total ticket sales revenue of

2,140

dollars.

3

Solve the equation for the variable

a

.

8a + 1600 - 5a = 2140 \implies 3a + 1600 = 2140 \implies 3a = 540 \implies a = 180

Apply the distributive property, group like terms, isolate the variable term, and divide to solve for the number of adult tickets.

Key Concept

Translating word problems into a single-variable linear equation and solving for the unknown quantity.

Question 188Question →

If $\frac{2}{3}(3x - 4) - \frac{1}{4}(x + 2) = \frac{22}{3}$ , what is the value of $3x + 2$ ?

Show answer & explanation

Answer: 20

Answer

The correct answer is 20, which is the value of the expression

3x + 2

when

x = 6

.

The value of the expression

3x + 2

is 20. Solving the linear equation

\frac{2}{3}(3x - 4) - \frac{1}{4}(x + 2) = \frac{22}{3}

yields

x = 6

. Substituting

x = 6

into

3x + 2

gives

3(6) + 2 = 20

.

Step-by-Step Solution

1

Multiply the entire equation by the least common multiple of the denominators (12).

8(3x - 4) - 3(x + 2) = 88

This clears the fractions to make solving the equation simpler.

2

Distribute the coefficients to remove parentheses.

24x - 32 - 3x - 6 = 88

This allows like terms to be grouped together.

3

Combine like terms on the left side of the equation.

21x - 38 = 88

Simplifies the equation to prepare for isolating the variable.

4

Add 38 to both sides of the equation.

21x = 126

Isolates the variable term on one side of the equation.

5

Divide both sides by 21 to solve for

x

.

x = 6

Finds the value of the variable

x

.

6

Substitute

x = 6

into the target expression

3x + 2

.

3(6) + 2 = 20

Calculates the final requested value.

Key Concept

Solving multi-step linear equations with fractions and evaluating algebraic expressions.

Question 189Question →

For a certain real number $x$ , the equation $\frac{3}{4}(x - 3) - \frac{1}{3}(2x + 5) = -2$ is true. What is the value of $2x - 5$ ?

Show answer & explanation

Answer: 41

Answer

The value of the expression

2x - 5

is 41.

Solving the linear equation by multiplying both sides by the least common denominator of 12 yields the simplified equation

9(x - 3) - 4(2x + 5) = -24

. Expanding the terms gives

9x - 27 - 8x - 20 = -24

, which simplifies to

x - 47 = -24

. Adding 47 to both sides gives

x = 23

. Finally, evaluating the expression

2x - 5

for

x = 23

results in

2(23) - 5 = 41

.

Step-by-Step Solution

1

Multiply both sides of the equation by the least common denominator of 12 to eliminate fractions.

9(x - 3) - 4(2x + 5) = -24

Multiplying by the LCD clears all fractional coefficients, making the equation easier to solve.

2

Distribute the constants and expand the terms on the left side of the equation.

9x - 27 - 8x - 20 = -24

Applying the distributive property removes the parentheses.

3

Combine like terms on the left side of the equation.

x - 47 = -24

Simplifying the equation makes it easier to isolate the variable

x

.

4

Isolate the variable

x

by adding 47 to both sides of the equation.

x = 23

This determines the value of the unknown variable

x

.

5

Substitute

x = 23

into the target expression

2x - 5

.

41

The question asks for the value of the expression

2x - 5

, not just the value of

x

.

Key Concept

Solving multi-step linear equations with fractional coefficients by clearing the denominators and then evaluating algebraic expressions.

Alternative Method

Instead of multiplying by the LCD first, distribute the fractions directly:

\frac{3}{4}x - \frac{9}{4} - \frac{2}{3}x - \frac{5}{3} = -2

. Combine the

x

terms:

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

. Combine the constant terms:

-\frac{9}{4} - \frac{5}{3} = -\frac{27}{12} - \frac{20}{12} = -\frac{47}{12}

. This gives the equation

\frac{1}{12}x - \frac{47}{12} = -2

. Add

\frac{47}{12}

to both sides:

\frac{1}{12}x = -2 + \frac{47}{12} = -\frac{24}{12} + \frac{47}{12} = \frac{23}{12}

. Multiply by 12 to get

x = 23

, then evaluate

2x - 5 = 41

.

Estimated Time:1m 30s

Question 190Question →

Which of the following inequalities represents the complete set of real values of $x$ that satisfy the inequality $15 \leq 3 - 4x$ ?

x \geq -3

x \leq -15

x \geq -15

x \leq -3

x \geq 3

Show answer & explanation

Answer: $x \leq -3$

Answer

The inequality

x \leq -3

Subtracting 3 from both sides of the inequality

15 \leq 3 - 4x

yields

12 \leq -4x

. Dividing both sides by

-4

and reversing the inequality sign results in

-3 \geq x

, which is equivalent to

x \leq -3

.

Step-by-Step Solution

1

Subtract 3 from both sides of the inequality.

12 \leq -4x

This isolates the variable term on the right side of the inequality.

2

Divide both sides of the inequality by

-4

and reverse the inequality sign.

-3 \geq x

Reversing the inequality sign is required whenever both sides of an inequality are multiplied or divided by a negative number.

3

Rewrite the inequality to place the variable on the left side.

x \leq -3

Reorganizing the inequality with

x

on the left side is the standard format for representing the solution set.

Key Concept

Solving linear inequalities by isolating the variable and reversing the inequality sign when dividing by a negative number.

Question 191Question →

A local library has two types of books: fiction and non-fiction. The number of fiction books is $120$ more than twice the number of non-fiction books. If the library has a total of $1\text{}860$ books, how many more fiction books than non-fiction books does the library have?

580

780

700

870

1\text{}280

Show answer & explanation

Answer: $700$

Answer

The library has

700

more fiction books than non-fiction books.

The correct answer is

700

because defining the non-fiction books as

n

gives

2n + 120

fiction books. Summing these to equal

1860

yields

3n + 120 = 1860

, which solves to

n = 580

. The number of fiction books is therefore

2(580) + 120 = 1280

. The difference between the two quantities is

1280 - 580 = 700

.

Step-by-Step Solution

1

Define variables for the two categories of books.

Let

n

be the number of non-fiction books. Then the number of fiction books,

f

, can be written as

f = 2n + 120

.

This translates the statement 'the number of fiction books is

120

more than twice the number of non-fiction books' into an algebraic expression.

2

Set up an equation representing the total number of books.

n + f = 1860 \implies n + (2n + 120) = 1860 \implies 3n + 120 = 1860

.

The sum of the fiction and non-fiction books must equal the total library inventory of

1860

books.

3

Solve the equation for the number of non-fiction books,

n

.

3n = 1740 \implies n = 580

.

Subtracting

120

from both sides and then dividing by

3

isolates the variable

n

.

4

Calculate the number of fiction books and find the difference.

Fiction books:

f = 2(580) + 120 = 1280

. Difference:

f - n = 1280 - 580 = 700

.

The question asks for the difference between the number of fiction and non-fiction books.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:1m 30s

Question 192Question →

A certain relationship between a number $x$ and other values is described as follows: one-fourth of the difference when $3x$ is subtracted from $2$ , decreased by one-third of the sum of $x$ and $3$ , is strictly greater than the difference when $x$ is subtracted from $1$ . Which of the following inequalities represents the complete set of all possible values of $x$ ?

x < 6

x > -18

x < -18

x > \frac{6}{25}

x < -\frac{2}{3}

Show answer & explanation

Answer: $x < -18$

Answer

x < -18

To find the correct solution set, translate the word problem into the inequality

\frac{2 - 3x}{4} - \frac{x + 3}{3} > 1 - x

. First, multiply all terms by the least common denominator,

12

, to clear the fractions, giving

3(2 - 3x) - 4(x + 3) > 12(1 - x)

. Distributing the constants yields

6 - 9x - 4x - 12 > 12 - 12x

. Combining like terms on the left side simplifies the expression to

-13x - 6 > 12 - 12x

. Adding

12x

to both sides results in

-x - 6 > 12

. Adding

6

to both sides gives

-x > 18

. Finally, dividing by

-1

and reversing the inequality sign results in the solution set

x < -18

.

Step-by-Step Solution

1

Translate the verbal description into an algebraic inequality.

\frac{2 - 3x}{4} - \frac{x + 3}{3} > 1 - x

To represent the relationships described in the word problem mathematically.

2

Multiply the entire inequality by the least common denominator, 12, to clear the fractions.

3(2 - 3x) - 4(x + 3) > 12(1 - x)

Clearing denominators makes it easier to combine like terms and solve for

x

.

3

Distribute the constants on both sides.

6 - 9x - 4x - 12 > 12 - 12x

To remove parentheses and separate individual terms for simplification.

4

Combine like terms on the left side of the inequality.

-13x - 6 > 12 - 12x

Simplifying the expressions on each side makes the inequality easier to solve.

5

Add

12x

and

6

to both sides of the inequality to isolate variables on one side.

-x > 18

To group variable terms on the left and constant terms on the right.

6

Divide both sides by

-1

and reverse the inequality sign.

x < -18

Dividing or multiplying an inequality by a negative number requires reversing the inequality sign to maintain a true statement.

Key Concept

Solving multi-step linear inequalities with rational terms and variable terms on both sides, including reversing the inequality sign when dividing by a negative number.

Alternative Method

Instead of clearing fractions first, you can distribute the division to each term (e.g.,

\frac{2}{4} - \frac{3x}{4} - \frac{x}{3} - \frac{3}{3} > 1 - x

), group the

x

terms on one side and constants on the other using fraction arithmetic, and then solve for

x

. However, clearing fractions with the LCD is generally faster and less prone to arithmetic errors.

Estimated Time:2m 0s

Question 193Question →

A manufacturing company produces sheet metal. The area of a large rectangular sheet is represented by the expression $2x^2(3x^3 - 4x + 5)$ square inches. A square piece with a side length of $3x - 4$ inches is cut out from the sheet. Which of the following expressions represents the area of the remaining metal sheet, in square inches, for all $x > 2$ ?

6x^6 - 8x^3 + x^2 + 24x - 16

6x^5 - 8x^3 + x^2 - 16

6x^5 - 8x^3 + x^2 - 24x + 16

6x^5 - 8x^3 + x^2 + 24x - 16

6x^5 - 8x^3 + x^2 + 24x + 16

Show answer & explanation

Answer: $6x^5 - 8x^3 + x^2 + 24x - 16$

Answer

The remaining area of the metal sheet is represented by

6x^5 - 8x^3 + x^2 + 24x - 16

.

To find the remaining area, the area of the square piece must be subtracted from the area of the large rectangular sheet. Expanding the large rectangular sheet's area yields

2x^2(3x^3 - 4x + 5) = 6x^5 - 8x^3 + 10x^2

. Expanding the area of the square piece yields

(3x - 4)^2 = 9x^2 - 24x + 16

. Subtracting the second polynomial from the first requires distributing the negative sign to all terms:

(6x^5 - 8x^3 + 10x^2) - (9x^2 - 24x + 16) = 6x^5 - 8x^3 + 10x^2 - 9x^2 + 24x - 16

. Combining the quadratic terms

10x^2 - 9x^2 = x^2

yields the correct simplified expression

6x^5 - 8x^3 + x^2 + 24x - 16

.

Step-by-Step Solution

1

Expand the expression for the area of the large rectangular sheet.

2x^2(3x^3 - 4x + 5) = 6x^5 - 8x^3 + 10x^2

Distribute the term

2x^2

to each term in the trinomial, using the exponent rule

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

to combine the variable factors.

2

Expand the expression for the area of the square piece.

(3x - 4)^2 = 9x^2 - 24x + 16

Apply the binomial squaring formula

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

, where

a = 3x

and

b = 4

.

3

Subtract the area of the square piece from the area of the large rectangular sheet.

(6x^5 - 8x^3 + 10x^2) - (9x^2 - 24x + 16) = 6x^5 - 8x^3 + 10x^2 - 9x^2 + 24x - 16

Distribute the negative sign to each term of the subtracted polynomial, changing the sign of each term inside the second parenthesis.

4

Combine like terms and write the final expression in descending order.

6x^5 - 8x^3 + x^2 + 24x - 16

Combine the quadratic terms:

10x^2 - 9x^2 = x^2

. Arrange the remaining terms in descending order of their exponents.

Key Concept

Polynomial operations involving monomial distribution with exponent rules, binomial squaring, and polynomial subtraction with sign distribution.

Estimated Time:2m 0s

Question 194Question →

A smart home heating system has two operating modes: Eco mode and Comfort mode. In a certain week, the system was active in one of these two modes for a total of $168$ hours. The number of hours the system was in Eco mode was $24$ hours less than three times the number of hours it was in Comfort mode. For how many hours was the heating system in Comfort mode during that week?

Show answer & explanation

Answer: 48

Answer

The heating system was in Comfort mode for 48 hours.

The correct answer is obtained by representing the hours in Comfort mode as

c

and Eco mode as

3c - 24

. Since their sum is

168

, we write the equation

c + 3c - 24 = 168

. Solving this linear equation gives

4c = 192

, which yields

c = 48

.

Step-by-Step Solution

1

Define variables for the Comfort and Eco modes.

Let

c

represent the hours in Comfort mode and

e

represent the hours in Eco mode.

To represent the unknown quantities algebraically.

2

Translate the given information into a system of equations.

c + e = 168

and

e = 3c - 24

The sum of the hours in both modes is

168

, and the relationship between Eco and Comfort hours is described.

3

Substitute the expression for

e

into the total hours equation.

c + (3c - 24) = 168

To create a single linear equation with one variable.

4

Solve for the variable

c

.

4c - 24 = 168 \Rightarrow 4c = 192 \Rightarrow c = 48

Simplifying the equation gives the value of

c

, which is the hours spent in Comfort mode.

Key Concept

Translating and Solving Algebraic Word Problems

Question 195Question →

In a chemistry laboratory, a beaker contains a mixture of water and acid. The volume of water in the beaker is $3$ liters more than twice the volume of acid. After $5$ liters of water are added to the beaker, the ratio of the volume of water to the volume of acid is $5$ to $2$ . If no acid was added or removed, what was the initial volume of water, in liters, in the beaker?

5

16

19

35

50

Show answer & explanation

Answer: 35

Answer

35

The correct answer of 35 liters is found by setting up the linear equation where the initial volume of acid is

a

and the initial volume of water is

2a + 3

. Adding 5 liters of water gives

2a + 8

liters of water. Setting the ratio of water to acid to

\frac{5}{2}

and solving gives

a = 16

. Finally, substituting

a = 16

into the expression for the initial volume of water (

2a + 3

) yields 35 liters.

Step-by-Step Solution

1

Define variables for the initial volumes of acid and water based on the given relationships.

Let

a

be the initial volume of acid in liters. The initial volume of water is

2a + 3

liters.

This translates the statement 'the volume of water is 3 liters more than twice the volume of acid' into algebraic expressions.

2

Set up a linear equation representing the ratio after adding 5 liters of water.

The new volume of water is

(2a + 3) + 5 = 2a + 8

liters. The ratio of water to acid is

\frac{2a + 8}{a} = \frac{5}{2}

.

This uses the new state of the mixture to form an equation that can be solved for

a

.

3

Solve the linear equation for

a

by cross-multiplying and isolating the variable.

2(2a + 8) = 5a \implies 4a + 16 = 5a \implies a = 16

.

Cross-multiplication removes the fractions and allows for standard term isolation.

4

Calculate the initial volume of water.

Initial water volume =

2a + 3 = 2(16) + 3 = 35

liters.

The question asks for the initial volume of water, which is represented by

2a + 3

, not the volume of acid

a

.

Key Concept

Solving linear equations in one variable derived from word problems.

Alternative Method

Instead of solving algebraically, one could test the answer choices. For example, testing 35 liters of water means the initial acid is

(35 - 3) / 2 = 16

liters. Adding 5 liters of water gives 40 liters of water. The ratio of water to acid is

40 : 16

, which simplifies to

5 : 2

. This confirms 35 is correct.

Estimated Time:1m 30s

Question 196Question →

A manufacturing company produces two types of metal alloys. The production cost, in dollars per ton, of Alloy A is modeled by the expression $800 - 15(x - 5)$ , where $x$ is the amount of stabilizer added in kilograms. The production cost of Alloy B, in dollars per ton, is modeled by the expression $520 + 5(x + 3)$ . If the company requires the production cost of Alloy A to be strictly less than the production cost of Alloy B, what is the minimum integer amount of stabilizer $x$ , in kilograms, that must be added?

Show answer & explanation

Answer: 18

Answer

The minimum integer amount of stabilizer that must be added is

18

kg.

The inequality representing the condition is

800 - 15(x - 5) < 520 + 5(x + 3)

. Expanding both sides gives

875 - 15x < 535 + 5x

. Isolating

x

yields

-20x < -340

. Dividing by

-20

and flipping the inequality sign results in

x > 17

. The minimum integer value that is strictly greater than

17

is

18

.

Step-by-Step Solution

1

Set up the linear inequality using the cost expressions for Alloy A and Alloy B.

800 - 15(x - 5) < 520 + 5(x + 3)

The cost of Alloy A must be strictly less than the cost of Alloy B.

2

Distribute and combine like terms to simplify both sides of the inequality.

875 - 15x < 535 + 5x

Simplifying the expressions makes it easier to isolate the variable.

3

Isolate the variable term on one side of the inequality.

-20x < -340

Subtracting

5x

and

875

from both sides moves all variable terms to the left and constant terms to the right.

4

Divide both sides by the negative coefficient

-20

and reverse the inequality sign.

x > 17

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

5

Identify the smallest integer that satisfies the inequality.

18

The solution requires a strict inequality

x > 17

, so the smallest integer value that is strictly greater than

17

is

18

.

Key Concept

Solving multi-step linear inequalities with variables on both sides, including reversing the inequality sign when dividing by a negative number.

Question 197Question →

If the equation $\frac{2}{5}(x - 4) + \frac{1}{3}(2x + k) = 7$ is true when $x = 9$ , what is the value of $k$ ?

-17

1

-3

3

8

Show answer & explanation

Answer: $-3$

Answer

-3

Substituting

x = 9

into the equation gives

\frac{2}{5}(9 - 4) + \frac{1}{3}(2(9) + k) = 7

. Simplifying the terms yields

2 + \frac{18 + k}{3} = 7

. Subtracting 2 from both sides results in

\frac{18 + k}{3} = 5

. Multiplying by 3 gives

18 + k = 15

. Subtracting 18 from both sides gives

k = -3

.

Step-by-Step Solution

1

Substitute

x = 9

into the given equation.

\frac{2}{5}(9 - 4) + \frac{1}{3}(2(9) + k) = 7

We are given that the equation is true when

x = 9

.

2

Simplify the operations inside the parentheses.

2 + \frac{18 + k}{3} = 7

Simplifying

\frac{2}{5}(5)

gives 2, and

2(9)

gives 18.

3

Subtract 2 from both sides of the equation.

\frac{18 + k}{3} = 5

Isolating the fraction term simplifies the equation.

4

Multiply both sides by 3.

18 + k = 15

Clearing the denominator allows us to isolate the variable

k

.

5

Subtract 18 from both sides of the equation.

k = -3

This isolates

k

to find its value.

Key Concept

Solving linear equations in one variable by substitution and simplification

Question 198Question →

Let $P(x) = (3x^2 - 2x + 4)(2x - 5) - (4x^3 - 7x^2 + x - 3)$ . When $P(x)$ is simplified and written in standard form, what is the coefficient of the $x^2$ term?

Show answer & explanation

Answer: -12

Answer

The coefficient of the

x^2

term is -12.

Expanding

(3x^2 - 2x + 4)(2x - 5)

yields

6x^3 - 19x^2 + 18x - 20

. Distributing the negative sign across the second polynomial yields

-4x^3 + 7x^2 - x + 3

. Combining the

x^2

terms gives

-19x^2 + 7x^2 = -12x^2

, so the coefficient of the

x^2

term is -12.

Step-by-Step Solution

1

Expand the product of the trinomial and the binomial:

(3x^2 - 2x + 4)(2x - 5)

6x^3 - 19x^2 + 18x - 20

To find the expanded form of the first polynomial component before subtraction

2

Distribute the negative sign across the second polynomial:

-(4x^3 - 7x^2 + x - 3)

-4x^3 + 7x^2 - x + 3

To prepare the second polynomial for combination of like terms

3

Combine the like terms from the two expanded components

2x^3 - 12x^2 + 17x - 17

To write the entire polynomial in standard form and identify the coefficient of

x^2

Key Concept

Operations on Polynomials

Question 199Question →

For all non-zero real numbers

p

and

q

, which of the following expressions is equivalent to

\frac{(p^2 + q^{-1})^2 - p^4}{q^{-2}}?

1

2p^2 q^2 + 1

2p^2 q + 1

2p^2 q^{-1} + 1

2p^2 + 1

Show answer & explanation

Answer: $2p^2 q + 1$

Answer

2p^2 q + 1

Expanding the binomial in the numerator yields

p^4 + 2p^2 q^{-1} + q^{-2}

. After subtracting

p^4

, the numerator is left as

2p^2 q^{-1} + q^{-2}

. Dividing this numerator term-by-term by the denominator

q^{-2}

gives

\frac{2p^2 q^{-1}}{q^{-2}} + \frac{q^{-2}}{q^{-2}}

. Applying the quotient property of exponents to each term yields

2p^2 q^{(-1) - (-2)} + 1 = 2p^2 q^1 + 1 = 2p^2 q + 1

.

Step-by-Step Solution

1

Expand the binomial in the numerator using the perfect square identity:

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

.

(p^2 + q^{-1})^2 = (p^2)^2 + 2(p^2)(q^{-1}) + (q^{-1})^2 = p^4 + 2p^2 q^{-1} + q^{-2}

To begin simplifying the numerator, we must resolve the exponent outside the parentheses.

2

Subtract

p^4

from the expanded numerator expression.

(p^4 + 2p^2 q^{-1} + q^{-2}) - p^4 = 2p^2 q^{-1} + q^{-2}

This simplifies the numerator by combining the

p^4

terms.

3

Divide each term in the simplified numerator by the denominator

q^{-2}

.

\frac{2p^2 q^{-1}}{q^{-2}} + \frac{q^{-2}}{q^{-2}} = 2p^2 q^{-1 - (-2)} + q^{-2 - (-2)}

To divide a polynomial by a monomial, we distribute the division to each term of the polynomial.

4

Apply the quotient rule of exponents,

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

, and simplify the terms.

2p^2 q^1 + q^0 = 2p^2 q + 1

Subtracting the exponents of

q

in each term simplifies the division to its final form.

Key Concept

Properties of Exponents in Algebraic Expressions

Question 200Question →

A commercial printing press charges a business a setup fee of $100$ plus $0.15$ per brochure printed. To encourage green practices, the press offers a recycling discount equal to $30$ less than $0.05$ per brochure printed. If the business has a budget of at most $380$ for their brochure order, what is the maximum number of brochures they can print?

1,250

1,550

2,500

3,100

3,500

Show answer & explanation

Answer: 2,500

Answer

2,500

The correct answer is 2,500 brochures. The total cost is the setup fee of

100 plus the printing cost of

0.15 per brochure, minus the discount of

30 less than

0.05 per brochure. Setting

b

as the number of brochures, the discount is

0.05b - 30

. The total cost equation is

100 + 0.15b - (0.05b - 30) = 130 + 0.10b

. Setting this less than or equal to

380

yields

130 + 0.10b \leq 380 \Rightarrow 0.10b \leq 250 \Rightarrow b \leq 2,500

.

Step-by-Step Solution

1

Define the variable and write the algebraic expressions for the cost components.

Let

b

be the number of brochures printed. The total cost before the discount is

100 + 0.15b

. The recycling discount is represented as

0.05b - 30

.

Establishing clear variables and translating the textual relationships into mathematical expressions is the first step in solving word problems.

2

Set up the inequality representing the budget constraint.

100 + 0.15b - (0.05b - 30) \leq 380

The total cost (original cost minus the discount) must be at most the budget of

380

.

3

Simplify the left side of the inequality.

100 + 0.15b - 0.05b + 30 \leq 380 \Rightarrow 130 + 0.10b \leq 380

Distributing the subtraction sign across the discount expression and combining like terms simplifies the inequality.

4

Solve for the variable

b

.

0.10b \leq 250 \Rightarrow b \leq 2500

Isolating

b

gives the maximum number of brochures that can be printed.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:1m 30s

Elementary Algebra

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Key Concept

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Answer

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Key Concept

Common Mistakes

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Key Concept

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Answer

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Key Concept

Common Mistakes

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Answer

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Key Concept

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Key Concept

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Key Concept

Common Mistakes

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Answer

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Answer

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Key Concept

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Answer

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Key Concept