Elementary Algebra Practice Questions — ACT — Page 13

Question 241Question →

An artist is designing a rectangular stained-glass window. The total area of the window, in square inches, is represented by the polynomial $8x^2 + 2x - 3$ . The area of the central blue glass section, in square inches, is represented by the product $(2x - 3)(x - 2)$ . The remaining portion of the window is made of clear glass. Which of the following expressions represents the area, in square inches, of the clear glass section?

6x^2 + 9x - 9

6x^2 - 5x + 3

6x^2 + 9x + 3

8x^2 + 7x - 9

6x^2 + 2x - 9

Show answer & explanation

Answer: $6x^2 + 9x - 9$

Answer

The expression representing the area of the clear glass section is

6x^2 + 9x - 9

.

The correct expression is

6x^2 + 9x - 9

. The area of the clear glass is found by expanding

(2x - 3)(x - 2)

to get

2x^2 - 7x + 6

and then subtracting this from the total area:

(8x^2 + 2x - 3) - (2x^2 - 7x + 6) = 8x^2 + 2x - 3 - 2x^2 + 7x - 6 = 6x^2 + 9x - 9

.

Step-by-Step Solution

1

Expand the product representing the area of the blue glass section using binomial multiplication.

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

Converting the factored area of the blue glass section to standard form is necessary before subtraction.

2

Set up the subtraction of the blue glass area from the total area.

(8x^2 + 2x - 3) - (2x^2 - 7x + 6)

The clear glass area is found by subtracting the blue glass area from the total window area.

3

Distribute the negative sign to each term of the second polynomial and combine like terms.

8x^2 + 2x - 3 - 2x^2 + 7x - 6 = (8x^2 - 2x^2) + (2x + 7x) + (-3 - 6) = 6x^2 + 9x - 9

Distributing the subtraction sign changes the signs of all terms inside the second set of parentheses, allowing for correct simplification.

Key Concept

Operations on Polynomials

Estimated Time:1m 30s

Question 242Question →

A commercial building has two water reservoirs. Reservoir $X$ contains $1,200$ gallons of water and is draining at a constant rate of $18$ gallons per minute. Reservoir $Y$ contains $360$ gallons of water and is being filled at a constant rate of $22$ gallons per minute. After how many minutes will both reservoirs contain the exact same amount of water?

Show answer & explanation

Answer: 21

Answer

The two reservoirs will contain the same amount of water after 21 minutes.

The correct answer is 21 minutes. By setting the expressions for the volume of both reservoirs equal (

1,200 - 18t = 360 + 22t

) and isolating the variable, we find

40t = 840

, which simplifies to

t = 21

.

Step-by-Step Solution

1

Translate the physical scenario for Reservoir

X

into an algebraic expression.

1,200 - 18t

Reservoir

X

starts with

1,200

gallons and loses

18

gallons per minute over

t

minutes.

2

Translate the physical scenario for Reservoir

Y

into an algebraic expression.

360 + 22t

Reservoir

Y

starts with

360

gallons and gains

22

gallons per minute over

t

minutes.

3

Set the two expressions equal to each other and solve for

t

.

1,200 - 18t = 360 + 22t \implies 840 = 40t \implies t = 21

Equating the two volume expressions allows us to find the time

t

at which the volumes are equal.

Key Concept

Translating and Solving Algebraic Word Problems

Alternative Method

Instead of solving algebraically, one can check the rates of change relative to each other. The distance between the initial volumes is

1,200 - 360 = 840

gallons. Since they are moving toward each other (one draining, one filling), their relative rate of convergence is

18 + 22 = 40

gallons per minute. Dividing the total volume difference by the rate of convergence gives

840 / 40 = 21

minutes.

Estimated Time:1m 30s

Question 243Question →

A local coffee shop sells two types of coffee blends: House Blend and organic Reserve Blend. A bag of Reserve Blend costs $4.50$ dollars less than three times the cost of a bag of House Blend. A customer purchases $3$ bags of House Blend and $2$ bags of Reserve Blend for a total of $54.00$ dollars, excluding tax. What is the cost, in dollars, of a bag of Reserve Blend?

7.00

10.50

15.00

16.50

19.50

Show answer & explanation

Answer: 16.50

Answer

The cost of a bag of Reserve Blend is 16.50 dollars.

The cost of a bag of Reserve Blend is 16.50 dollars. Let

x

represent the cost of a bag of House Blend. A bag of Reserve Blend costs

3x - 4.50

. Since the total cost for

3

bags of House Blend and

2

bags of Reserve Blend is

54.00

, we write the linear equation

3x + 2(3x - 4.50) = 54

. Expanding the terms gives

3x + 6x - 9 = 54

, which simplifies to

9x - 9 = 54

. Adding

9

to both sides yields

9x = 63

. Dividing by

9

gives

x = 7.00

. Substituting

7.00

back into the Reserve Blend expression yields

3(7.00) - 4.50 = 16.50

dollars.

Step-by-Step Solution

1

Define variables for the costs of each blend.

Let

x

represent the cost of a bag of House Blend in dollars. The cost of a bag of Reserve Blend is represented by the expression

3x - 4.50

.

Establishing algebraic expressions for the unknowns based on the problem description allows us to set up a linear equation.

2

Set up the linear equation based on the total cost of the purchase.

The total cost of

3

bags of House Blend and

2

bags of Reserve Blend is

54.00

dollars:

3(x) + 2(3x - 4.50) = 54

.

The sum of the individual total costs of the two blends must equal the overall purchase total.

3

Distribute and combine like terms to simplify the equation.

3x + 6x - 9 = 54

simplifies to

9x - 9 = 54

.

Applying the distributive property removes the parentheses, allowing like terms to be combined.

4

Isolate the variable term by adding

9

to both sides.

9x = 63

.

To solve for

x

, we must first isolate the term containing the variable by performing the inverse operation.

5

Solve for

x

by dividing both sides by

9

.

x = 7.00

.

Dividing isolates

x

, giving the cost of a bag of House Blend.

6

Calculate the cost of a bag of Reserve Blend using the expression from Step 1.

3(7.00) - 4.50 = 21.00 - 4.50 = 16.50

dollars.

The question asks for the cost of a bag of Reserve Blend, not the House Blend, so we evaluate the expression

3x - 4.50

at

x = 7.00

.

Key Concept

Setting up and solving a single-variable linear equation to solve a real-world word problem with multiple unknown quantities.

Estimated Time:1m 30s

Question 244Question →

A shipping company charges a rate based on the weight of a package. The total cost $C$ , in dollars, to ship a package of weight $w$ pounds is given by the formula $C = \frac{5}{8}(w - 2) + 6.50$ for packages weighing more than $2$ pounds. If the shipping cost for a certain package is $\$14.00$ , what is the weight of the package, in pounds?

Show answer & explanation

Answer: 14

Answer

The weight of the package is 14 pounds.

Substituting

C = 14.00

into the formula gives

14.00 = \frac{5}{8}(w - 2) + 6.50

. Subtracting

6.50

from both sides yields

7.50 = \frac{5}{8}(w - 2)

. Multiplying both sides by the reciprocal

\frac{8}{5}

yields

12 = w - 2

. Finally, adding

2

to both sides gives the weight

w = 14

pounds.

Step-by-Step Solution

1

Substitute the total shipping cost into the formula.

14.00 = \frac{5}{8}(w - 2) + 6.50

Since the shipping cost

C

is given as

14.00

, we substitute this value into the formula to solve for the unknown weight

w

.

2

Subtract

6.50

from both sides of the equation.

7.50 = \frac{5}{8}(w - 2)

Subtracting

6.50

isolates the term containing the variable

w

on the right side of the equation.

3

Multiply both sides of the equation by the reciprocal of the fraction.

12 = w - 2

Multiplying by

\frac{8}{5}

eliminates the fractional coefficient of

\frac{5}{8}

on the right side.

4

Add

2

to both sides of the equation to solve for

w

.

w = 14

Adding

2

isolates the variable

w

, giving the final weight of the package.

Key Concept

Solving linear equations with fractional and decimal terms

Estimated Time:1m 30s

Question 245Question →

A shipping company charges a flat fee of $15$ dollars plus $2.50$ dollars per pound for the first $10$ pounds of a package's weight. For any weight exceeding $10$ pounds, the rate is $4.00$ dollars per pound. If the total shipping charge for a package was $59.00$ dollars, what was the total weight of the package, in pounds?

11.0

11.6

14.75

17.6

18.5

Show answer & explanation

Answer: $14.75$

Answer

The package's total weight was 14.75 pounds.

The correct answer of 14.75 pounds is found by setting up the equation representing the total shipping cost. First, calculate the cost for the first 10 pounds, which is the flat fee of 15.00 dollars plus 2.50 dollars per pound for 10 pounds: 15 + 2.50(10) = 40.00 dollars. Since the total charge of 59.00 dollars is greater than 40.00 dollars, the package must weigh more than 10 pounds. Let w be the total weight of the package. The remaining weight exceeding 10 pounds is w - 10, which is charged at 4.00 dollars per pound. Setting up the equation: 40 + 4(w - 10) = 59. Solving for w gives: 4(w - 10) = 19, which simplifies to w - 10 = 4.75, so w = 14.75.

Step-by-Step Solution

1

Calculate the cost for a package weighing exactly 10 pounds.

The cost is 15 + 2.50 * 10 = 40.00 dollars.

To determine whether the package exceeds 10 pounds by comparing it to the total charge of 59.00 dollars.

2

Set up an equation for the total cost where the weight w exceeds 10 pounds.

40 + 4.00 * (w - 10) = 59.00

To express the cost of the first 10 pounds plus the cost of the excess weight at 4.00 dollars per pound.

3

Solve the equation for the total weight w.

4 * (w - 10) = 19 -> w - 10 = 4.75 -> w = 14.75

To isolate and find the value of the variable representing the package weight.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:1m 30s

Question 246Question →

A craft cider company produces two specialty blends. The production cost, in dollars per gallon, of the premium blend is represented by the expression $\frac{3}{4}(d - 8)$ , where $d$ is the wholesale cost, in dollars, of a bushel of apples. The production cost, in dollars per gallon, of the dry blend is represented by the expression $0.2(2d + 5)$ . If the production cost per gallon is the same for both blends, what is this production cost, in dollars per gallon?

1.2

1.8

9

20

45

Show answer & explanation

Answer: 9

Answer

The production cost is 9 dollars per gallon.

The correct answer is the value obtained by setting the two cost expressions equal, solving for

d = 20

, and then substituting

d = 20

back into either expression to find the cost of

9

dollars per gallon.

Step-by-Step Solution

1

Set the two production cost expressions equal to each other.

\frac{3}{4}(d - 8) = 0.2(2d + 5)

The problem states that the production cost per gallon is the same for both blends, so their algebraic representations must be equal.

2

Express the decimal as a fraction and solve for

d

.

15(d - 8) = 4(2d + 5) \implies 15d - 120 = 8d + 20 \implies 7d = 140 \implies d = 20

Converting

0.2

to

\frac{1}{5}

allows us to clear fractions by multiplying both sides of the equation by the least common multiple of the denominators, which is

20

. We then distribute, isolate the variable terms, and solve for

d

.

3

Substitute the value of

d

back into either of the original cost expressions to find the production cost.

\text{Cost} = \frac{3}{4}(20 - 8) = \frac{3}{4}(12) = 9

The question asks for the production cost per gallon, not the value of the variable

d

. Substituting

d = 20

into the premium blend cost expression gives the final cost.

Key Concept

Solving linear equations involving fractions and decimals, and evaluating expressions using the solved variable value.

Alternative Method

Convert all numbers to decimals. The premium blend cost is

0.75(d - 8) = 0.75d - 6

. The dry blend cost is

0.2(2d + 5) = 0.4d + 1

. Equating them gives

0.75d - 6 = 0.4d + 1 \implies 0.35d = 7 \implies d = 20

. Then find the cost:

0.75(20) - 6 = 15 - 6 = 9

dollars.

Estimated Time:1m 30s

Question 247Question →

A landscape architect is designing a park layout. The total area of the park is represented by the polynomial $A(t) = (2t - 3)(3t^2 + 4t - 1)$ square meters, while the area allocated for a parking lot is represented by $P(t) = 2t^2(3t - 5)$ square meters, where $t$ represents a scaling factor. The remaining green space in the park is modeled by the polynomial $g(t) = A(t) - P(t)$ . When $g(t)$ is written in the standard form $at^3 + bt^2 + ct + d$ , where $a$ , $b$ , $c$ , and $d$ are constants, what is the value of $b$ ?

Show answer & explanation

Answer: 9

Answer

The value of

b

, the coefficient of the

t^2

term in the standard form of

g(t)

, is

9

.

Expanding the polynomials gives

A(t) = 6t^3 - t^2 - 14t + 3

and

P(t) = 6t^3 - 10t^2

. Subtracting

P(t)

from

A(t)

yields

g(t) = (6t^3 - t^2 - 14t + 3) - (6t^3 - 10t^2) = 9t^2 - 14t + 3

. In the standard form

at^3 + bt^2 + ct + d

, the coefficient

b

of the

t^2

term is

9

.

Step-by-Step Solution

1

Expand the polynomial

A(t) = (2t - 3)(3t^2 + 4t - 1)

6t^3 - t^2 - 14t + 3

To represent the total area as a single polynomial in standard form before subtraction.

2

Expand the polynomial

P(t) = 2t^2(3t - 5)

6t^3 - 10t^2

To represent the parking lot area as a simplified polynomial in standard form.

3

Subtract

P(t)

from

A(t)

to find the green space polynomial

g(t)

9t^2 - 14t + 3

Subtracting

P(t)

from

A(t)

requires distributing the negative sign to both terms, yielding

-t^2 - (-10t^2) = 9t^2

.

4

Identify the coefficient

b

of the

t^2

term in

at^3 + bt^2 + ct + d

9

Comparing

g(t) = 9t^2 - 14t + 3

to the standard cubic form reveals that

a = 0

and

b = 9

.

Key Concept

Polynomial operations including expansion of products and subtraction with negative sign distribution

Question 248Question →

A marathon runner plans to run a total of $d$ miles this week. The runner schedules $\frac{2}{3}(d - 6)$ miles of the run on soft dirt trails and $0.4(d + 15)$ miles on asphalt roads. If the runner completes only these two segments for a total of $34$ miles, what is the value of $d$ ?

Show answer & explanation

Answer: 30

Answer

The value of

d

is 30.

The value of

d

that satisfies the equation is

30

because substituting

30

back into the original equation yields a true statement:

\frac{2}{3}(30 - 6) + 0.4(30 + 15) = 16 + 18 = 34

.

Step-by-Step Solution

1

Set up the equation based on the given information: the trail running distance plus the road running distance equals the total distance.

\frac{2}{3}(d - 6) + 0.4(d + 15) = 34

This relates the individual segment distances to the total planned mileage of

34

miles.

2

Distribute the coefficients to eliminate the parentheses.

\frac{2}{3}d - 4 + 0.4d + 6 = 34

Distributing

\frac{2}{3}

to

(d-6)

gives

\frac{2}{3}d - 4

, and distributing

0.4

to

(d+15)

gives

0.4d + 6

.

3

Combine the constant terms on the left side of the equation.

\frac{2}{3}d + 0.4d + 2 = 34

Combining the constants

-4

and

6

yields

2

.

4

Subtract 2 from both sides of the equation to isolate the variable terms.

\frac{2}{3}d + 0.4d = 32

Subtracting

2

from both sides simplifies the equation to have variable terms on one side and constant terms on the other.

5

Convert the decimal

0.4

to a fraction to combine the coefficients of

d

.

\frac{2}{3}d + \frac{2}{5}d = 32

Converting

0.4

to

\frac{4}{10} = \frac{2}{5}

allows us to work with a common denominator.

6

Find a common denominator to add the fractions.

\frac{10}{15}d + \frac{6}{15}d = 32 \Rightarrow \frac{16}{15}d = 32

The least common multiple of

3

and

5

is

15

.

7

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

d

to solve for

d

.

d = 32 \times \frac{15}{16} \Rightarrow d = 2 \times 15 \Rightarrow d = 30

Multiplying by

\frac{15}{16}

isolates

d

on the left side of the equation.

Key Concept

Solving linear equations involving fractions and decimals by distributing, combining like terms, and isolating the variable.

Alternative Method

Instead of converting the decimal

0.4

to a fraction, you can multiply the entire equation by a common multiple like

15

to eliminate both the fraction and the decimal:

15 \left[ \frac{2}{3}(d - 6) + 0.4(d + 15) \right] = 15(34) \Rightarrow 10(d - 6) + 6(d + 15) = 510 \Rightarrow 10d - 60 + 6d + 90 = 510 \Rightarrow 16d + 30 = 510 \Rightarrow 16d = 480 \Rightarrow d = 30

. This method avoids working with fractional coefficients.

Estimated Time:1m 30s

Question 249Question →

The volume of a water tank, in cubic feet, is given by the polynomial $V(t) = 3t(2t - 1)(t + 4)$ , where $t$ represents the time in hours since the pump was turned on. When the polynomial is written in standard form, what is the coefficient of the $t^2$ term?

Show answer & explanation

Answer: 21

Answer

21

To find the coefficient of the

t^2

term, we perform polynomial multiplication on the expression

V(t) = 3t(2t - 1)(t + 4)

. First, multiplying the binomials

(2t - 1)(t + 4)

yields

2t^2 + 7t - 4

. Next, distributing the

3t

term to the trinomial yields

6t^3 + 21t^2 - 12t

. The coefficient of the

t^2

term in this standard form polynomial is 21.

Step-by-Step Solution

1

Expand the product of the two binomials

(2t - 1)(t + 4)

.

2t^2 + 7t - 4

Use the distributive property (FOIL) to multiply the binomials:

2t(t) + 2t(4) - 1(t) - 1(4) = 2t^2 + 8t - t - 4 = 2t^2 + 7t - 4

.

2

Multiply the resulting trinomial by the monomial

3t

.

6t^3 + 21t^2 - 12t

Distribute

3t

to each term:

3t(2t^2) + 3t(7t) + 3t(-4) = 6t^3 + 21t^2 - 12t

.

3

Identify the coefficient of the

t^2

term.

21

The coefficient is the numerical factor of the

t^2

term, which is 21.

Key Concept

Operations on Polynomials

Question 250Question →

A computer hardware engineer is designing a rectangular silicon microchip. The total surface area of the microchip, in square millimeters, is represented by the expression $2y(3y^2 - 4y + 5)$ . A square sensor is embedded in the center of the chip, occupying an area represented by the expression $(2y - 3)^2$ square millimeters. Which of the following expressions represents the remaining surface area of the microchip, in square millimeters, that is not occupied by the sensor?

6y^3 - 12y^2 + 10y - 9

6y^3 - 12y^2 - 2y + 9

6y^3 - 12y^2 + 22y - 9

-6y^2 + 22y - 9

6y^3 - 12y^2 + 22y + 9

Show answer & explanation

Answer: $6y^3 - 12y^2 + 22y - 9$

Answer

The correct expression is

6y^3 - 12y^2 + 22y - 9

.

To find the remaining surface area of the microchip, subtract the area of the sensor from the total area. First, expand the expression for the total area:

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

. Next, expand the expression for the sensor area:

(2y - 3)^2 = 4y^2 - 12y + 9

. Subtract the sensor area from the total area, distributing the negative sign to all terms:

(6y^3 - 8y^2 + 10y) - (4y^2 - 12y + 9) = 6y^3 - 8y^2 + 10y - 4y^2 + 12y - 9

. Combining like terms yields the expression

6y^3 - 12y^2 + 22y - 9

.

Step-by-Step Solution

1

Expand the total area of the microchip by distributing the monomial

2y

to each term of the trinomial

3y^2 - 4y + 5

.

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

This determines the polynomial representing the total area before any components are subtracted.

2

Expand the area of the square sensor by squaring the binomial

(2y - 3)

.

(2y - 3)^2 = 4y^2 - 12y + 9

This determines the polynomial representing the area occupied by the sensor, ensuring the middle term

-12y

is correctly calculated.

3

Subtract the sensor's area from the total area, ensuring the negative sign is distributed to every term in the sensor's polynomial.

(6y^3 - 8y^2 + 10y) - (4y^2 - 12y + 9) = 6y^3 - 8y^2 + 10y - 4y^2 + 12y - 9

This sets up the subtraction of the two polynomials while distributing the negative sign to avoid common sign errors.

4

Combine like terms to find the final simplified expression in standard descending order.

6y^3 - 12y^2 + 22y - 9

This combines the cubic, quadratic, linear, and constant terms to find the remaining area.

Key Concept

Polynomial operations including monomial multiplication, binomial expansion, and subtraction with sign distribution.

Estimated Time:1m 30s

Question 251Question →

A botanist models the growth of a rare seedling. The number of weeks $w$ that the seedling has been growing satisfies the linear equation:

\frac{1}{4}(3w - 8) + 0.6 = \frac{1}{5}(2w + 7)

If the seedling's growth continues to follow this model, what is the value of 3 less than 5 times the number of weeks the seedling has been growing?

-37

-17

22

25

37

Show answer & explanation

Answer: 37

Answer

37

Solving the given equation for

w

yields

w = 8

. The question asks for the value of 3 less than 5 times the number of weeks, which translates to the expression

5w - 3

. Substituting

w = 8

into the expression results in

5(8) - 3 = 37

.

Step-by-Step Solution

1

Multiply both sides of the equation by the least common multiple of the denominators, which is 20.

5(3w - 8) + 12 = 4(2w + 7)

This clears the fractions and simplifies the equation to integer coefficients.

2

Distribute the coefficients across the terms inside the parentheses.

15w - 40 + 12 = 8w + 28

This removes the parentheses so terms can be grouped.

3

Combine the constant terms on the left side of the equation.

15w - 28 = 8w + 28

Combining

-40

and

12

simplifies the expression on the left.

4

Isolate the variable

w

by moving the variable terms to the left side and constant terms to the right side.

7w = 56

Subtracting

8w

and adding

28

to both sides groups like terms together.

5

Solve for

w

by dividing both sides by 7.

w = 8

This isolates the variable

w

to find the number of weeks.

6

Translate '3 less than 5 times the number of weeks' into an algebraic expression and evaluate it for

w = 8

.

5w - 3 = 5(8) - 3 = 37

This translates the verbal question into mathematical terms and calculates the final value.

Key Concept

Solving multi-step linear equations containing fractions and decimals, and translating verbal expressions into algebraic terms.

Alternative Method

Instead of clearing the fractions first, one could convert the fractions to decimals:

0.25(3w - 8) + 0.6 = 0.2(2w + 7)

. Distribute to get

0.75w - 2 + 0.6 = 0.4w + 1.4

, which simplifies to

0.75w - 1.4 = 0.4w + 1.4

. Subtracting

0.4w

and adding

1.4

to both sides yields

0.35w = 2.8

, which gives

w = \frac{2.8}{0.35} = 8

. Then, compute

5(8) - 3 = 37

.

Estimated Time:1m 30s

Question 252Question →

What is the greatest integer value of $x$ that satisfies the inequality $3 - 2x > 10$ ?

Show answer & explanation

Answer: -4

Answer

The correct answer is

-4

.

Subtracting 3 from both sides of

3 - 2x > 10

gives

-2x > 7

. When dividing both sides by

-2

, the inequality sign must be flipped, yielding

x < -3.5

. The greatest integer less than

-3.5

is

-4

.

Step-by-Step Solution

1

Subtract 3 from both sides of the inequality to isolate the variable term.

-2x > 7

Subtracting 3 from both sides keeps the inequality balanced while moving the constant term to the right side.

2

Divide both sides by

-2

and reverse the inequality sign.

x < -3.5

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

3

Identify the greatest integer that satisfies the inequality.

-4

The integers that are strictly less than

-3.5

are

-4, -5, -6, \dots

. The largest (greatest) of these integers is

-4

.

Key Concept

Solving linear inequalities by applying the sign-reversal rule when dividing by a negative number and identifying integer boundary values.

Question 253Question →

Which inequality represents all real values of $x$ for which the inequality $\frac{5 - 2x}{3} \geq \frac{x - 6}{2}$ is true?

x \geq 4

x \leq \frac{11}{3}

x \leq 4

x \geq \frac{11}{3}

x \leq -\frac{8}{7}

Show answer & explanation

Answer: $x \leq 4$

Answer

The correct inequality is

x \leq 4

.

The correct inequality is

x \leq 4

. Multiplying both sides by the least common multiple,

6

, yields the inequality

2(5 - 2x) \geq 3(x - 6)

. Distributing the coefficients results in

10 - 4x \geq 3x - 18

. Gathering the variable terms by subtracting

3x

gives

10 - 7x \geq -18

. Subtracting

10

from both sides results in

-7x \geq -28

. Dividing both sides by

-7

and reversing the inequality sign results in the final solution

x \leq 4

.

Step-by-Step Solution

1

Multiply both sides of the inequality by

6

(the least common multiple of

2

and

3

) to eliminate the fractions.

2(5 - 2x) \geq 3(x - 6)

Eliminating denominators simplifies the linear inequality for solving.

2

Distribute the constants on both sides.

10 - 4x \geq 3x - 18

Expanding the terms allows combining like terms next.

3

Subtract

3x

from both sides of the inequality.

10 - 7x \geq -18

Grouping all variable terms on one side of the inequality.

4

Subtract

10

from both sides of the inequality.

-7x \geq -28

Isolating the variable term on the left side.

5

Divide both sides by

-7

and reverse the inequality sign.

x \leq 4

Dividing or multiplying both sides of an inequality by a negative number requires reversing the inequality sign direction.

Key Concept

Solving linear inequalities by clearing denominators and applying the sign-reversal rule when dividing by a negative number.

Estimated Time:1m 15s

Question 254Question →

A rectangular garden has a length of $3x^2$ meters and a width of $2x^3 - 5$ meters. A square storage shed built inside the garden has a side length of $x^2 - 3$ meters. Which of the following polynomials represents the area of the remaining garden space, in square meters, that is not covered by the shed?

6x^5 - x^4 - 15x^2 - 9

6x^6 - x^4 - 9x^2 - 9

6x^5 - x^4 - 21x^2 + 9

6x^5 - x^4 - 9x^2 - 9

6x^5 - x^4 - 9x^2 + 9

Show answer & explanation

Answer: $6x^5 - x^4 - 9x^2 - 9$

Answer

The polynomial

6x^5 - x^4 - 9x^2 - 9

represents the remaining area.

To find the remaining area, subtract the area of the square shed from the area of the rectangular garden. The area of the garden is

3x^2(2x^3 - 5) = 6x^5 - 15x^2

. The area of the shed is

(x^2 - 3)^2 = x^4 - 6x^2 + 9

. Subtracting the two gives

(6x^5 - 15x^2) - (x^4 - 6x^2 + 9) = 6x^5 - 15x^2 - x^4 + 6x^2 - 9 = 6x^5 - x^4 - 9x^2 - 9

, which represents the correct remaining garden space.

Step-by-Step Solution

1

Calculate the area of the rectangular garden.

\text{Area}_{\text{garden}} = 3x^2(2x^3 - 5) = 6x^5 - 15x^2

The area of a rectangle is the product of its length and width. Applying the distributive property and the exponent rule for multiplication (

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

), we get

3x^2 \cdot 2x^3 = 6x^5

and

3x^2 \cdot (-5) = -15x^2

.

2

Calculate the area of the square storage shed.

\text{Area}_{\text{shed}} = (x^2 - 3)^2 = x^4 - 6x^2 + 9

The area of a square is the square of its side length. We expand

(x^2 - 3)^2

using the binomial square pattern

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

, which gives

(x^2)^2 - 2(x^2)(3) + (-3)^2 = x^4 - 6x^2 + 9

.

3

Subtract the area of the shed from the area of the garden and simplify.

\text{Remaining Area} = 6x^5 - x^4 - 9x^2 - 9

We subtract the shed's area from the garden's area:

(6x^5 - 15x^2) - (x^4 - 6x^2 + 9)

. Distributing the negative sign gives

6x^5 - 15x^2 - x^4 + 6x^2 - 9

. Combining like terms and writing in descending order yields

6x^5 - x^4 - 9x^2 - 9

.

Key Concept

Operations on Polynomials

Question 255Question →

What is the smallest integer value of $y$ that satisfies the inequality $5(2 - y) < 3(y - 6)$ ?

Show answer & explanation

Answer: 4

Answer

The smallest integer value of

y

that satisfies the inequality is

4

.

Evaluating the inequality leads to

y > 3.5

. The smallest integer greater than

3.5

is

4

. Substituting

y = 4

into the original inequality gives

5(2 - 4) < 3(4 - 6) \implies -10 < -6

, which is true. Substituting the next smallest integer,

3

, gives

-5 < -9

, which is false.

Step-by-Step Solution

1

Distribute the coefficients to the terms inside the parentheses on both sides.

10 - 5y < 3y - 18

To clear the parentheses and simplify the terms.

2

Subtract

3y

and

10

from both sides of the inequality to group the variable terms on one side and constants on the other.

-8y < -28

To isolate the variable term.

3

Divide both sides by

-8

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

y > 3.5

To solve the inequality for

y

.

4

Identify the smallest integer that satisfies the inequality

y > 3.5

.

4

To find the smallest integer value greater than

3.5

.

Key Concept

Solving multi-step linear inequalities involving distribution and division by a negative number.

Alternative Method

Instead of subtracting variables to the left, we can add

5y

to both sides to keep the variable coefficient positive:

10 - 5y < 3y - 18 \implies 10 < 8y - 18 \implies 28 < 8y \implies y > 3.5

. This avoids the need to divide by a negative number and flip the sign, reducing the risk of a sign-flip error.

Estimated Time:1m 0s

Question 256Question →

A rectangular swimming pool has a length of $3x - 2$ meters and a width of $2x - 5$ meters. A concrete walkway with a uniform width of $2$ meters surrounds the pool. Which of the following expressions represents the area, in square meters, of the walkway?

20x - 12

-18x + 8

20x + 8

12x^2 - 18x + 8

12x^2 - 12

Show answer & explanation

Answer: $20x - 12$

Answer

The expression representing the area of the walkway is

20x - 12

.

The correct answer is the expression

20x - 12

. The total area including the walkway is

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

. The pool's area is

(3x - 2)(2x - 5) = 6x^2 - 19x + 10

. Subtracting the pool's area from the total area and distributing the negative sign to all terms gives:

(6x^2 + x - 2) - (6x^2 - 19x + 10) = 6x^2 + x - 2 - 6x^2 + 19x - 10 = 20x - 12

.

Step-by-Step Solution

1

Determine the dimensions of the entire area including the pool and the walkway.

The total length is

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

meters, and the total width is

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

meters.

Since the walkway surrounds the pool on all sides, we add twice the width of the walkway to both the length and width of the pool.

2

Calculate the total area including the walkway.

Total Area =

(3x + 2)(2x - 1) = 6x^2 - 3x + 4x - 2 = 6x^2 + x - 2

square meters.

The area is found by multiplying the total length and the total width using binomial multiplication.

3

Calculate the area of the pool itself.

Pool Area =

(3x - 2)(2x - 5) = 6x^2 - 15x - 4x + 10 = 6x^2 - 19x + 10

square meters.

The area of the pool is the product of its length and width.

4

Subtract the pool area from the total area to find the area of the walkway.

Walkway Area =

(6x^2 + x - 2) - (6x^2 - 19x + 10) = 6x^2 + x - 2 - 6x^2 + 19x - 10 = 20x - 12

square meters.

Subtracting the inner pool area from the total area yields the remaining area of the walkway surrounding it.

Key Concept

Calculating the difference between two binomial products to find the area of a border.

Question 257Question →

The position of a particle moving along a straight line is described by the polynomial $s(t) = (2t^2 - 3t + 4)(t + 3)$ , where $t$ represents the time in seconds and $s(t)$ represents the displacement in meters. When the polynomial is expanded and written in the standard form $at^3 + bt^2 + ct + d$ , what is the value of $b + c$ ?

Show answer & explanation

Answer: -2

Answer

The value of

b + c

is

-2

.

Expanding the polynomial

s(t) = (2t^2 - 3t + 4)(t + 3)

yields

2t^3 + 3t^2 - 5t + 12

. Comparing this to the standard form

at^3 + bt^2 + ct + d

, we find

b = 3

and

c = -5

. Summing these coefficients gives

b + c = 3 + (-5) = -2

.

Step-by-Step Solution

1

Multiply each term of the trinomial

2t^2 - 3t + 4

by each term of the binomial

t + 3

.

2t^3 + 6t^2 - 3t^2 - 9t + 4t + 12

Apply the distributive property to expand the product of the polynomials.

2

Combine the like terms (the

t^2

terms and the

t

terms).

2t^3 + 3t^2 - 5t + 12

Group and add the coefficients of like terms to simplify the polynomial.

3

Identify the coefficients

b

(for the

t^2

term) and

c

(for the

t

term) and find their sum.

b = 3

and

c = -5

, so

b + c = 3 + (-5) = -2

Compare the simplified polynomial to the standard form

at^3 + bt^2 + ct + d

to find the coefficients, then calculate the requested sum.

Key Concept

Operations on Polynomials

Alternative Method

Instead of expanding the whole expression, you can find the individual coefficients directly. The

t^2

term is formed by

(2t^2)(3) + (-3t)(t) = 6t^2 - 3t^2 = 3t^2

, so

b = 3

. The

t

term is formed by

(-3t)(3) + (4)(t) = -9t + 4t = -5t

, so

c = -5

. Thus,

b + c = 3 - 5 = -2

.

Estimated Time:1m 30s

Question 258Question →

A student translates a word problem into an inequality. The problem states: "One-fourth of the difference of $x$ and $3$ is greater than the sum of $1$ and one-third of the quantity $2x$ minus $1$ ." Which of the following inequalities represents the correct set of all real values of $x$ that satisfy this condition?

x > -\frac{17}{5}

x < -\frac{44}{5}

x < -3

x < -\frac{17}{5}

x < -\frac{8}{3}

Show answer & explanation

Answer: $x < -\frac{17}{5}$

Answer

The set of all real values of

x

satisfying the condition is

x < -\frac{17}{5}

.

The correct inequality is obtained by translating the word problem statement as

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

. Multiplying the entire inequality by

12

yields

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

. Distributing the coefficients results in

3x - 9 > 12 + 8x - 4

, which simplifies to

3x - 9 > 8x + 8

. Moving all terms containing

x

to the left and constants to the right gives

-5x > 17

. Dividing both sides by

-5

and reversing the inequality sign results in the solution

x < -\frac{17}{5}

.

Step-by-Step Solution

1

Translate the verbal description into an algebraic inequality.

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

The phrase "one-fourth of the difference of

x

and

3

" translates to

\frac{x - 3}{4}

, and "the sum of

1

and one-third of the quantity

2x

minus

1

" translates to

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

.

2

Multiply all terms by the least common multiple of the denominators, which is

12

, to clear the fractions.

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

Multiplying both sides of an inequality by a positive number maintains the direction of the inequality sign while eliminating fractional coefficients.

3

Expand both sides of the inequality by distributing the coefficients.

3x - 9 > 12 + 8x - 4

Distribution allows the variable and constant terms to be separated and simplified.

4

Simplify the constants on the right side and move all variable terms to one side and constants to the other.

-5x > 17

Subtracting

8x

from both sides and adding

9

to both sides isolates the variable term on the left side of the inequality.

5

Divide both sides by

-5

and reverse the inequality sign.

x < -\frac{17}{5}

Dividing or multiplying an inequality by a negative number reverses the direction of the inequality sign.

Key Concept

Solving linear inequalities by translating word problems and applying the sign-flip rule when dividing by a negative number.

Estimated Time:1m 30s

Question 259Question →

What is the greatest integer value of $k$ that satisfies the inequality $8 - 3(2k - 5) \geq 4(k + 6)$ ?

Show answer & explanation

Answer: -1

Answer

The greatest integer value of

k

that satisfies the inequality is

-1

.

Solving the inequality step-by-step yields

k \leq -0.1

. The greatest integer less than or equal to

-0.1

is

-1

.

Step-by-Step Solution

1

Distribute the coefficients to remove parentheses

8 - 6k + 15 \geq 4k + 24

Expanding the terms makes it possible to combine like terms on each side of the inequality.

2

Combine like terms on the left side

23 - 6k \geq 4k + 24

Simplifying the constant values on the left side (

8 + 15 = 23

) simplifies the expression.

3

Subtract

4k

from both sides

23 - 10k \geq 24

This groups all the variable terms on the left-hand side.

4

Subtract

23

from both sides

-10k \geq 1

This isolates the variable term on the left-hand side.

5

Divide by

-10

and flip the inequality sign

k \leq -0.1

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

6

Identify the greatest integer satisfying the inequality

k = -1

The largest integer that is less than or equal to

-0.1

is

-1

.

Key Concept

Solving linear inequalities and applying the sign-flip rule when dividing by a negative number.

Estimated Time:1m 30s

Question 260Question →

A business analyst models a company's weekly net profit, $N(x)$ , in dollars, as the difference between its weekly revenue, $R(x) = (2x + 5)(30 - x)$ , and its weekly production cost, $C(x) = (x - 4)^2 + 150$ , where $x$ represents the number of items sold. When $N(x)$ is simplified and written in standard form as $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are integers, what is the value of the coefficient $b$ ?

Show answer & explanation

Answer: 63

Answer

The coefficient

b

is

63

.

Expanding the revenue polynomial gives

R(x) = -2x^2 + 55x + 150

and the cost polynomial gives

C(x) = x^2 - 8x + 166

. Subtracting them gives

N(x) = (-2x^2 + 55x + 150) - (x^2 - 8x + 166) = -3x^2 + 63x - 16

. The coefficient of the

x

term, which corresponds to

b

, is

63

.

Step-by-Step Solution

1

Expand the revenue expression

R(x) = (2x + 5)(30 - x)

using polynomial multiplication.

R(x) = -2x^2 + 55x + 150

To find the polynomial representing revenue in standard form.

2

Expand the cost expression

C(x) = (x - 4)^2 + 150

using binomial squaring.

C(x) = x^2 - 8x + 166

To find the polynomial representing cost in standard form.

3

Subtract the cost polynomial from the revenue polynomial:

N(x) = R(x) - C(x)

, distributing the negative sign through all terms of the cost polynomial.

N(x) = -3x^2 + 63x - 16

To determine the net profit polynomial

N(x)

in standard form.

4

Identify the coefficient

b

of the

x

term in the standard form

ax^2 + bx + c

.

b = 63

To answer the specific question asking for the coefficient of the middle term.

Key Concept

Polynomial subtraction and expansion of algebraic expressions.

Elementary Algebra

Answer

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