Algebra Practice Questions — SAT — Page 21

Question 401Question →

A forestry service uses a remote weather station powered by a solar battery system. During a period of heavy cloud cover, the remaining charge in the battery, $C$ , in watt-hours, can be modeled by the linear equation $C = 360 - 4.5h$ , where $h$ represents the number of hours since the cloud cover began. Based on the model, what is the best interpretation of the $h$ -intercept of the relationship?

The number of hours, which is 80, after the cloud cover began when the battery is completely discharged.

The initial charge of the battery, which is 360 watt-hours, when the cloud cover began.

The rate, which is 4.5 watt-hours per hour, at which the battery charge decreases.

The number of hours, which is 90, after the cloud cover began when the battery is completely discharged.

Show answer & explanation

Answer: The number of hours, which is 80, after the cloud cover began when the battery is completely discharged.

Answer

The number of hours, which is 80, after the cloud cover began when the battery is completely discharged.

The

h

-intercept of the relationship is the value of

h

when

C = 0

. Substituting

0

for

C

in the equation

C = 360 - 4.5h

gives

0 = 360 - 4.5h

. Solving for

h

yields

4.5h = 360

, or

h = 80

. In context, this represents the number of hours after the cloud cover began when the battery is completely discharged.

Step-by-Step Solution

1

Identify the meaning of the

h

-intercept in the context of the linear equation.

The

h

-intercept occurs when the dependent variable,

C

(remaining charge), is equal to 0.

By definition, the horizontal intercept of a function occurs where the vertical coordinate is zero.

2

Substitute

C = 0

into the equation and solve for

h

.

0 = 360 - 4.5h \implies 4.5h = 360 \implies h = 80

.

This algebraic isolation finds the specific value of

h

when the charge is completely depleted.

3

Interpret the resulting value of

h

in the context of the problem.

At

h = 80

hours, the remaining battery charge is 0 watt-hours, meaning the battery is fully discharged.

Connecting the mathematical coordinate

(80, 0)

back to the units of hours and watt-hours provides the physical interpretation.

Key Concept

Interpreting Linear Relationships in Context

Question 402Question →

To calculate the remaining budget, $B$ , in dollars, for a community project, a coordinator uses the formula $B = 1,200 - 3(2x + 50)$ , where $x$ is the number of volunteer shifts scheduled. If the coordinator wants the remaining budget to be at most $450, which of the following inequalities represents all possible values of$ x$?

x \le 100

x \ge 150

x \ge 10

x \ge 100

Show answer & explanation

Answer: $x \ge 100$

Answer

The correct answer is

x \ge 100

.

The correct answer is the inequality stating that

x

is greater than or equal to

100

. Distributing

-3

to both terms inside the parentheses gives

1,200 - 6x - 150 \le 450

. Combining the constant terms on the left side results in

1,050 - 6x \le 450

. Subtracting

1,050

from both sides gives

-6x \le -600

. Dividing both sides by

-6

and reversing the inequality sign because of the division by a negative number yields

x \ge 100

.

Step-by-Step Solution

1

Substitute the formula for the remaining budget

B

into the inequality

B \le 450

.

1,200 - 3(2x + 50) \le 450

To find the possible values of

x

when the budget is at most

450

.

2

Distribute

-3

to both terms inside the parentheses.

1,200 - 6x - 150 \le 450

To simplify the expression by removing the parentheses.

3

Combine the constant terms

1,200

and

-150

on the left side.

1,050 - 6x \le 450

To simplify the inequality further before isolating the variable.

4

Subtract

1,050

from both sides of the inequality.

-6x \le -600

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

5

Divide both sides of the inequality by

-6

and reverse the inequality sign.

x \ge 100

Dividing by a negative number requires reversing the direction of the inequality sign to maintain the truth value.

Key Concept

Solving linear inequalities in one variable, including the distributive property and sign reversal when dividing by a negative number.

Question 403Question →

A commercial printing press has a reservoir of yellow ink. The volume of yellow ink in the reservoir, $V$ , in milliliters, after printing $p$ pages of a color brochure is modeled by the equation $V = 1,200 - 0.15p$ . According to the model, what is the decrease, in milliliters, in the volume of yellow ink in the reservoir for every 100 pages printed?

Show answer & explanation

Answer: 15

Answer

The volume of yellow ink in the reservoir decreases by 15 milliliters for every 100 pages printed.

In the equation

V = 1,200 - 0.15p

, the coefficient of

p

is

-0.15

. This represents the rate of change of the volume of yellow ink in the reservoir with respect to the number of pages printed. Specifically, it means the volume decreases by

0.15

milliliters for each additional page printed. To find the decrease in volume for every 100 pages printed, multiply the rate per page by 100:

0.15 \times 100 = 15

milliliters.

Step-by-Step Solution

1

Identify the rate of change per page from the linear equation.

The rate of change is 0.15 milliliters per page.

In the linear equation

V = 1,200 - 0.15p

, the coefficient of the independent variable

p

represents the change in the dependent variable

V

for each unit increase in

p

.

2

Calculate the decrease in ink volume for 100 pages.

15 milliliters

Since the volume decreases by 0.15 milliliters for each page printed, printing 100 pages results in a total decrease of

0.15 \times 100 = 15

milliliters.

Key Concept

Interpreting the slope (rate of change) of a linear equation in context.

Question 404Question →

If $-3(2x - 5) + 4x \geq -7$ , what is the maximum possible value of $x$ ?

Show answer & explanation

Answer: 11

Answer

The maximum possible value of

x

is

11

.

By applying the distributive property, combining like terms, and dividing by

-2

(while reversing the inequality sign), we find that the solution is

x \leq 11

. Thus, the maximum possible value of

x

is

11

.

Step-by-Step Solution

1

Apply the distributive property to simplify the left side of the inequality.

-6x + 15 + 4x \geq -7

Multiplying

-3

by each term inside the parentheses

(2x - 5)

yields

-6x

and

+15

.

2

Combine the variable terms on the left side.

-2x + 15 \geq -7

Combining

-6x

and

4x

gives

-2x

.

3

Subtract

15

from both sides of the inequality.

-2x \geq -22

To isolate the variable term

-2x

on the left side.

4

Divide both sides by

-2

and flip the inequality sign.

x \leq 11

Dividing both sides of an inequality by a negative number reverses the direction of the inequality sign.

5

Determine the maximum value from the solution set.

11

The solution set consists of all values less than or equal to

11

, so the greatest value is

11

.

Key Concept

Solving multi-step linear inequalities, including applying the distributive property and reversing the inequality sign when dividing by a negative number.

Question 405Question →

The total daily cost $C$ , in dollars, for a custom apparel company to manufacture $x$ shirts is modeled by a linear equation. The table below shows the total daily cost for two different numbers of shirts manufactured:

Number of shirts, $x$	Total daily cost, $C$ (dollars)
12	260
20	380

Based on the model, what is the total daily cost, in dollars, to manufacture 35 shirts?

262

536

605

785

Show answer & explanation

Answer: 605

Answer

605

To find the total daily cost to manufacture 35 shirts, we first determine the linear relationship

C = mx + b

between the number of shirts manufactured,

x

, and the total cost,

C

. The slope

m

is the constant rate of change, calculated as

m = \frac{380 - 260}{20 - 12} = \frac{120}{8} = 15

dollars per shirt. Using the point-slope form with the point

(12, 260)

, we get

C - 260 = 15(x - 12)

, which simplifies to

C = 15x + 80

. Substituting

35

for

x

in this equation yields

C = 15(35) + 80 = 525 + 80 = 605

. Therefore, the total daily cost to manufacture 35 shirts is 605 dollars.

Step-by-Step Solution

1

Calculate the slope (rate of change) of the linear relationship using the two points from the table,

(12, 260)

and

(20, 380)

.

Slope

m = \frac{380 - 260}{20 - 12} = \frac{120}{8} = 15

dollars per shirt.

Since the relationship is linear, the rate of change is constant.

2

Find the equation of the line using the point-slope form

C - C_1 = m(x - x_1)

with the point

(12, 260)

and slope

m = 15

.

C - 260 = 15(x - 12) \implies C = 15x - 180 + 260 \implies C = 15x + 80

.

This establishes the linear equation relating the number of shirts

x

and the total daily cost

C

.

3

Substitute

x = 35

into the linear equation to find the total daily cost to manufacture 35 shirts.

C = 15(35) + 80 = 525 + 80 = 605

dollars.

This evaluates the linear model at the desired quantity.

Key Concept

Determining a linear equation in two variables from a table of values and using it to predict a value.

Question 406Question →

A vertical farm uses an automated nutrient delivery system for a crop of lettuce. The total volume of nutrient solution, $V$ , in liters, remaining in the system's reservoir is modeled as a linear function of the time $t$ , in hours, since the system started running for the day. The table below shows several values of $t$ and the corresponding values of $V$ .

Time, $t$ (hours)	Volume, $V$ (liters)
3	415
5	385
8	340
12	280

According to the model, what was the initial volume of nutrient solution, in liters, in the reservoir when the system started running?

Show answer & explanation

Answer: 460

Answer

460

The correct answer is 460. The remaining volume of nutrient solution,

V

, is a linear function of time,

t

, which can be written in the form

V = mt + b

, where

m

is the rate of change (slope) and

b

is the initial volume (y-intercept). The rate of change can be found using any two points from the table, for example,

(3, 415)

and

(5, 385)

:

m = \frac{385 - 415}{5 - 3} = \frac{-30}{2} = -15

liters per hour. Using the point

(3, 415)

and substituting

m = -15

,

t = 3

, and

V = 415

into the equation

V = mt + b

yields

415 = -15(3) + b

. Simplifying this expression gives

415 = -45 + b

, and adding 45 to both sides gives

b = 460

. Therefore, the initial volume of nutrient solution in the reservoir was 460 liters.

Step-by-Step Solution

1

Calculate the constant rate of consumption (slope) of the nutrient solution.

The rate of consumption is 15 liters per hour.

The relationship between volume and time is linear. The slope

m

can be calculated from two coordinates from the table,

(3, 415)

and

(5, 385)

, as

m = \frac{385 - 415}{5 - 3} = -15

liters per hour.

2

Determine the initial volume of nutrient solution (the y-intercept) using the rate and a data point.

The initial volume is 460 liters.

Substitute the slope

m = -15

, the time

t = 3

, and the remaining volume

V = 415

into the slope-intercept equation

V = mt + b

. Solving

415 = -15(3) + b

gives

b = 460

.

Key Concept

Interpreting the y-intercept of a linear function in context as the initial value of the dependent variable when the independent variable is 0.

Estimated Time:1m 30s

Question 407Question →

A hybrid car's fuel tank has a capacity of $12$ gallons. When driving on a highway, the amount of fuel in the tank decreases at a constant rate. After driving for $1.5$ hours, $9.6$ gallons of fuel remain in the tank. If the amount of fuel in the tank, $F$ , in gallons, after driving for $t$ hours is modeled by a linear equation, what is the value of $F$ when $t = 4$ ?

Show answer & explanation

Answer: 5.6

Answer

The amount of fuel remaining in the tank after driving for

4

hours is

5.6

gallons.

The amount of fuel in the tank,

F

, and the driving time,

t

, share a linear relationship. The initial amount of fuel at

t = 0

is

12

gallons, representing the vertical intercept. The fuel decreases at a constant rate, which is the slope of the line. Over a period of

1.5

hours, the amount of fuel decreases by

12 - 9.6 = 2.4

gallons. The rate of decrease is

\frac{2.4}{1.5} = 1.6

gallons per hour, so the slope is

-1.6

. The linear model is

F = -1.6t + 12

. Substituting

t = 4

into this equation yields

F = -1.6(4) + 12 = -6.4 + 12 = 5.6

gallons.

Step-by-Step Solution

1

Calculate the constant rate of fuel consumption (the slope of the linear relationship).

The rate of consumption is

1.6

gallons per hour.

Since the fuel decreases at a constant rate, the change in fuel divided by the change in time gives the rate of consumption. In

1.5

hours, the fuel decreases from

12

gallons to

9.6

gallons, which is a decrease of

12 - 9.6 = 2.4

gallons. Thus, the rate of consumption is

\frac{2.4\text{ gallons}}{1.5\text{ hours}} = 1.6

gallons per hour.

2

Write the linear equation modeling the fuel remaining in the tank,

F

, as a function of time,

t

.

F = -1.6t + 12

The initial amount of fuel when

t = 0

is

12

gallons, which represents the vertical intercept (

b = 12

). The fuel decreases at a constant rate of

1.6

gallons per hour, which represents a slope of

m = -1.6

.

3

Substitute

t = 4

into the linear equation to find the value of

F

.

F = 5.6

Evaluating the equation at

t = 4

yields

F = -1.6(4) + 12 = -6.4 + 12 = 5.6

.

Key Concept

Linear Equations in Two Variables

Question 408Question →

A city reservoir contains water that is released at a constant rate for river conservation. The volume of water, $W$ , in millions of gallons, remaining in the reservoir $d$ days after the release begins is modeled by the equation $W = 240 - 1.8d$ . What is the best interpretation of the number $1.8$ in this context?

The reservoir loses 1 million gallons of water every 1.8 days.

The initial volume of water in the reservoir before the release begins is 1.8 million gallons.

The volume of water in the reservoir decreases by 1.8 million gallons each day.

The total duration of the controlled release is 1.8 days.

Show answer & explanation

Answer: The volume of water in the reservoir decreases by 1.8 million gallons each day.

Answer

The volume of water in the reservoir decreases by 1.8 million gallons each day.

The equation

W = 240 - 1.8d

is in slope-intercept form, where the slope is

-1.8

and the

y

-intercept is

240

. The slope represents the rate of change of the remaining volume of water with respect to time. A slope of

-1.8

indicates that the volume of water,

W

, decreases by

1.8

million gallons for every

1

day increase in the number of days,

d

.

Step-by-Step Solution

1

Identify the variables and constants in the given linear model.

In

W = 240 - 1.8d

,

W

is the remaining volume in millions of gallons,

d

is the time in days, 240 is the y-intercept (initial volume), and -1.8 is the slope (rate of change).

Understanding the components of a linear equation

y = mx + b

helps isolate the meaning of each constant.

2

Interpret the meaning of the slope in the context of the variables and units.

The coefficient of

d

is -1.8, which represents a rate of change of -1.8 million gallons per day.

The slope is the change in the dependent variable (

W

, in millions of gallons) per unit change in the independent variable (

d

, in days).

3

Relate the negative sign to the physical context.

A negative slope of -1.8 means that the volume decreases by 1.8 million gallons for each elapsed day.

In context, a constant release of water leads to a decrease in the remaining volume.

Key Concept

Interpreting slope in a linear relationship context

Estimated Time:1m 30s

Question 409Question →

A shipping container initially holds 450 packages. A crew unloads the container at a rate of 15 packages per hour, while an automated sorting machine unloads packages at a rate of $x$ packages per hour. The number of packages remaining in the container after 8 hours must be at most 130. The inequality $450 - 8(15 + x) \leq 130$ models this scenario. What is the minimum possible value of $x$ ?

Show answer & explanation

Answer: 25

Answer

The minimum possible value of

x

is 25.

To find the minimum possible value of

x

, solve the inequality

450 - 8(15 + x) \leq 130

. First, distribute the

-8

to obtain

450 - 120 - 8x \leq 130

. Simplify the constant terms on the left side to get

330 - 8x \leq 130

. Subtract 330 from both sides, yielding

-8x \leq -200

. Finally, divide both sides by

-8

and reverse the inequality sign because of the division by a negative number, which results in

x \geq 25

. The minimum possible value of

x

is 25.

Step-by-Step Solution

1

Distribute the factor of

-8

to both terms inside the parentheses.

450 - 120 - 8x \leq 130

To remove the parentheses and prepare to combine like terms.

2

Subtract 120 from 450 to simplify the constants on the left side.

330 - 8x \leq 130

To simplify the left side of the inequality before isolating the variable.

3

Subtract 330 from both sides of the inequality.

-8x \leq -200

To isolate the variable term on the left-hand side.

4

Divide both sides by

-8

and reverse the direction of the inequality symbol.

x \geq 25

Dividing by a negative value reverses the inequality sign. Since

x

must be greater than or equal to 25, the minimum possible value is 25.

Key Concept

Solving multi-step linear inequalities in one variable, including distributing coefficients and reversing the inequality sign when multiplying or dividing by a negative number.

Question 410Question →

An electric delivery van's remaining battery capacity, $B$ , in kilowatt-hours (kWh), after driving $d$ miles is modeled by a linear equation. After driving $45$ miles, the remaining battery capacity is $68$ kWh. After driving a total of $120$ miles, the remaining battery capacity is $38$ kWh. According to this model, how many miles can the van drive on a full charge before the battery capacity reaches $0$ kWh?

Show answer & explanation

Answer: 215

Answer

215

To find the maximum distance the van can drive on a full charge, we model the relationship between the remaining battery capacity,

B

, and the distance driven,

d

, using the linear equation

B = md + B_0

. First, calculate the rate of consumption (slope,

m

) using the points

(45, 68)

and

(120, 38)

:

m = \frac{38 - 68}{120 - 45} = \frac{-30}{75} = -0.4

kWh per mile. Next, find the initial battery capacity (

B_0

) by substituting the values from one of the points:

68 = -0.4(45) + B_0

, which simplifies to

68 = -18 + B_0

, so

B_0 = 86

kWh. Finally, determine the distance

d

when the battery is fully depleted (

B = 0

):

0 = -0.4d + 86

, which yields

0.4d = 86

, and

d = 215

miles.

Step-by-Step Solution

1

Calculate the rate of change of the battery capacity per mile driven.

The rate of change (slope) is

-0.4

kWh per mile.

The slope

m

represents the energy consumption rate and is calculated using the two data points

(45, 68)

and

(120, 38)

with the formula

m = \frac{38 - 68}{120 - 45} = \frac{-30}{75} = -0.4

.

2

Determine the initial battery capacity when the distance driven is 0 miles.

The initial battery capacity is

86

kWh.

Using the slope-intercept form

B = md + B_0

, substitute

m = -0.4

and the point

(45, 68)

to get

68 = -0.4(45) + B_0

. This simplifies to

68 = -18 + B_0

, so

B_0 = 86

.

3

Solve for the distance driven

d

when the battery capacity

B

is

0

kWh.

The distance is

215

miles.

Set

B = 0

in the linear model

B = -0.4d + 86

to get

0 = -0.4d + 86

. Solving for

d

gives

0.4d = 86

, which results in

d = \frac{86}{0.4} = 215

.

Key Concept

Interpreting and calculating slope, y-intercept, and x-intercept of a linear function in a real-world context.

Question 411Question →

For the inequality $-2(3x - 4) + 5 > -3x - 12$ , which inequality represents all possible values of $x$ ?

x > \frac{25}{3}

x < \frac{25}{9}

x < 3

x < \frac{25}{3}

Show answer & explanation

Answer: $x < \frac{25}{3}$

Answer

x < \frac{25}{3}

The correct inequality is obtained by first distributing

-2

to the terms inside the parentheses to get

-6x + 8 + 5 > -3x - 12

. Combining the constants on the left yields

-6x + 13 > -3x - 12

. Adding

3x

to both sides results in

-3x + 13 > -12

, and subtracting 13 from both sides gives

-3x > -25

. Finally, dividing by

-3

and flipping the inequality sign results in the correct solution.

Step-by-Step Solution

1

Distribute

-2

to the terms inside the parentheses on the left side of the inequality.

-6x + 8 + 5 > -3x - 12

Applying the distributive property simplifies the expression.

2

Combine the constant terms on the left side.

-6x + 13 > -3x - 12

Simplifying the constant terms makes it easier to isolate the variable.

3

Add

3x

to both sides to move all variable terms to the left side.

-3x + 13 > -12

This isolates the variable terms on one side of the inequality.

4

Subtract 13 from both sides to isolate the variable term.

-3x > -25

This isolates the term containing

x

.

5

Divide both sides by

-3

and reverse the direction of the inequality sign.

x < \frac{25}{3}

Dividing by a negative number requires reversing the inequality sign.

Key Concept

Solving linear inequalities in one variable using the distributive property and sign reversal rules.

Question 412Question →

A science museum offers two ticketing options for groups. Option A is a flat group rate of $125 plus$ 9.50 per person. Option B is a flat group rate of $50 plus$ 12.50 per person. For a group of $p$ people, Option A is less expensive than Option B. What is the minimum number of people in the group for this to be true?

Show answer & explanation

Answer: 26

Answer

The minimum number of people in the group is 26.

To find when Option A is less expensive than Option B, we set up the inequality representing their respective costs:

125 + 9.5p < 50 + 12.5p

. Subtracting

9.5p

from both sides gives

125 < 50 + 3p

. Subtracting

50

from both sides yields

75 < 3p

. Dividing by

3

gives

p > 25

. Because the group must consist of a whole number of people, the minimum integer value of

p

that is strictly greater than

25

is

26

.

Step-by-Step Solution

1

Write the inequality representing the cost comparison between the two ticketing options.

125 + 9.5p < 50 + 12.5p

Option A's cost must be strictly less than Option B's cost for Option A to be less expensive.

2

Isolate the variable

p

by subtracting

9.5p

and

50

from both sides of the inequality.

p > 25

Subtracting

9.5p

yields

125 < 50 + 3p

. Subtracting

50

yields

75 < 3p

. Dividing by

3

yields

p > 25

.

3

Identify the minimum integer value of

p

that satisfies the inequality.

26

Since the number of people must be a positive integer, the smallest integer strictly greater than

25

is

26

.

Key Concept

Solving linear inequalities in one variable and interpreting the solution set within a discrete real-world context.

Estimated Time:1m 30s

Question 413Question →

A team of glaciologists monitors the thickness of an alpine glacier during the summer season. The thickness of the glacier, $T$ , in meters, can be modeled by the equation $T = 54.5 - 0.12d$ , where $d$ represents the number of days since the start of the summer season, for $0 \leq d \leq 150$ . Which of the following is the best interpretation of the number $0.12$ in this context?

The initial thickness of the glacier, in meters, at the start of the summer season

The number of days it takes for the thickness of the glacier to decrease by

1

meter

The decrease in the thickness of the glacier, in meters, each day during the summer season

The total number of days from the start of the summer season until the glacier melts completely

Show answer & explanation

Answer: The decrease in the thickness of the glacier, in meters, each day during the summer season

Answer

The correct answer is the option stating that 0.12 is the decrease in the thickness of the glacier, in meters, each day during the summer season.

The linear model is given by

T = 54.5 - 0.12d

, which is in the slope-intercept form

y = mx + b

, where

m

is the slope and

b

is the y-intercept. In this equation, the slope

m

is

-0.12

, which represents the rate of change of the glacier's thickness per day. Since the slope is negative, the thickness decreases by

0.12

meters each day. Therefore, the number

0.12

represents the decrease in the thickness of the glacier, in meters, each day during the summer season.

Step-by-Step Solution

1

Identify the components of the linear equation

T = 54.5 - 0.12d

.

The constant term is

54.5

, and the coefficient of the variable

d

is

-0.12

.

To interpret a linear relationship in context, we must distinguish between the initial value (y-intercept) and the rate of change (slope).

2

Determine the meaning of the slope in this context.

The slope is

-0.12

meters per day, meaning the glacier's thickness

T

decreases by

0.12

meters for each day

d

that passes.

The coefficient of the independent variable in a linear equation represents the rate of change of the dependent variable per unit of the independent variable.

3

Relate the number

0.12

to the rate of change.

The positive value

0.12

represents the magnitude of this rate of change, which is the amount of decrease in thickness per day.

Since the slope is negative, the change is a decrease, and the rate of this decrease is

0.12

meters per day.

Key Concept

Interpreting the slope of a linear relationship in a real-world context

Question 414Question →

What is the solution set for the inequality $5 - 2(3x - 1) \geq -4x + 11$ ?

x \leq -2

x \geq -2

x \leq -4

x \geq -4

Show answer & explanation

Answer: $x \leq -2$

Answer

The correct solution is

x

is less than or equal to

-2

.

To solve the inequality, first distribute the negative two to the terms inside the parentheses to get

5 - 6x + 2 \geq -4x + 11

. Combining the constants on the left side yields

7 - 6x \geq -4x + 11

. Adding four

x

to both sides results in

7 - 2x \geq 11

, and subtracting seven from both sides isolates the variable term, giving

-2x \geq 4

. Dividing both sides by negative two and reversing the inequality sign yields the correct solution,

x

is less than or equal to negative two.

Step-by-Step Solution

1

Distribute the

-2

to the terms inside the parentheses.

5 - 6x + 2 \geq -4x + 11

To simplify the expression by removing the parentheses.

2

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

7 - 6x \geq -4x + 11

To group like terms together before isolating the variable.

3

Add

4x

to both sides of the inequality.

7 - 2x \geq 11

To move all variable terms to one side of the inequality.

4

Subtract

7

from both sides of the inequality.

-2x \geq 4

To isolate the variable term.

5

Divide both sides by

-2

and reverse the inequality sign.

x \leq -2

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

Key Concept

Solving linear inequalities in one variable using the distributive property and applying the sign reversal rule when dividing by a negative number.

Estimated Time:1m 30s

Question 415Question →

If $-5(2x - 9) + 3 \geq 18$ , what is the maximum possible value of $x$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

Distributing the

-5

gives

-10x + 45 + 3 \geq 18

. Combining constants yields

-10x + 48 \geq 18

. Subtracting 48 from both sides gives

-10x \geq -30

. Finally, dividing both sides by

-10

and reversing the inequality sign gives

x \leq 3

. The maximum possible value is therefore 3.

Step-by-Step Solution

1

Distribute

-5

to the terms inside the parentheses.

-10x + 45 + 3 \geq 18

Simplify the expression by expanding the parentheses.

2

Combine the constant terms

45

and

3

on the left side.

-10x + 48 \geq 18

Group like terms together.

3

Subtract

48

from both sides of the inequality.

-10x \geq -30

Isolate the variable term.

4

Divide both sides by

-10

and reverse the inequality symbol.

x \leq 3

Dividing by a negative number reverses the direction of the inequality.

Key Concept

Solving multi-step linear inequalities in one variable, applying the distributive property, and reversing the inequality sign when multiplying or dividing by a negative number.

Question 416Question →

A commercial cargo ship is unloading shipping containers at a port. The total mass of the ship and its remaining cargo, $M$ , in kilotonnes (kt), is a linear function of the number of hours, $h$ , since the unloading process began. After $3$ hours of unloading, the total mass of the ship and its cargo is $116$ kt. After $8$ hours of unloading, the total mass is $98$ kt. According to this model, what is the mass, in kilotonnes, of the cargo that is unloaded each hour?

Show answer & explanation

Answer: 3.6

Answer

The mass of the cargo unloaded each hour is

3.6

kilotonnes (or the equivalent fraction

\frac{18}{5}

).

The correct answer is

3.6

. Since the relationship between the total mass of the ship and cargo,

M

, and the elapsed time,

h

, is linear, the rate at which cargo is unloaded corresponds to the magnitude of the slope of the linear function. Using the two data points

(3, 116)

and

(8, 98)

, the slope

m

can be calculated as

\frac{98 - 116}{8 - 3} = \frac{-18}{5} = -3.6

. The negative sign indicates that the mass is decreasing, meaning that

3.6

kilotonnes of cargo are unloaded each hour.

Step-by-Step Solution

1

Set up the linear relationship model.

M = mh + b

Since the relationship between total mass

M

and time

h

is linear, it can be modeled by a linear equation where

m

is the slope (unloading rate) and

b

is the y-intercept (initial mass).

2

Determine the two data points from the given information.

(3, 116)

and

(8, 98)

After

3

hours of unloading, the mass is

116

kt, and after

8

hours, the mass is

98

kt.

3

Calculate the slope (rate of change) of the linear relationship.

m = -3.6

The slope is calculated as

m = \frac{98 - 116}{8 - 3} = \frac{-18}{5} = -3.6

.

4

Interpret the slope's value in context to find the amount of cargo unloaded per hour.

3.6

The slope is

-3.6

kilotonnes per hour, which means the total mass decreases by

3.6

kilotonnes each hour. Therefore, the mass of the cargo unloaded each hour is

3.6

kilotonnes.

Key Concept

Interpreting the slope of a linear relationship in context

Estimated Time:1m 30s

Question 417Question →

What is the complete set of solutions to the inequality $-\frac{2}{3}(6x - 9) + 4 > 12$ ?

x > -\frac{1}{2}

x < -\frac{1}{2}

x < -\frac{7}{2}

x < -2

Show answer & explanation

Answer: $x < -\frac{1}{2}$

Answer

The inequality is satisfied for all values of

x

such that

x < -\frac{1}{2}

.

Distributing

-\frac{2}{3}

across

(6x - 9)

yields

-4x + 6

. Adding 4 gives

-4x + 10 > 12

. Subtracting 10 from both sides results in

-4x > 2

. Dividing by

-4

and reversing the inequality sign yields

x < -\frac{1}{2}

.

Step-by-Step Solution

1

Distribute

-\frac{2}{3}

to both terms inside the parentheses:

(6x - 9)

.

-4x + 6 + 4 > 12

, which simplifies to

-4x + 10 > 12

.

Applying the distributive property removes the parentheses.

2

Subtract 10 from both sides of the inequality to isolate the term with

x

.

-4x > 2

Subtracting 10 from both sides maintains the inequality while simplifying the constant terms.

3

Divide both sides by

-4

and reverse the inequality sign.

x < -\frac{1}{2}

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

Key Concept

Solving linear inequalities in one variable requires distributing coefficient terms, combining constants, and reversing the inequality sign when multiplying or dividing by a negative number.

Estimated Time:1m 15s

Question 418Question →

A commercial coffee roaster heats a large batch of coffee beans. The temperature $T$ , in degrees Celsius ( $^\circ\text{C}$ ), of the beans $m$ minutes after heating begins is modeled by the equation $T = 18m + 31$ , where $m \le 10$ . Which of the following is the best interpretation of $18$ in this context?

The increase in the temperature of the beans, in degrees Celsius, for each minute of heating.

The initial temperature of the beans, in degrees Celsius, when heating begins.

The number of minutes it takes for the temperature of the beans to increase by

1^\circ\text{C}

.

The temperature of the beans, in degrees Celsius, after

18

minutes of heating.

Show answer & explanation

Answer: The increase in the temperature of the beans, in degrees Celsius, for each minute of heating.

Answer

The increase in the temperature of the beans, in degrees Celsius, for each minute of heating.

The equation is in the slope-intercept form

T = 18m + 31

, where

18

is the slope and

31

is the y-intercept. In this context,

T

represents the temperature of the beans and

m

represents the time in minutes. The slope,

18

, represents the rate of change of the temperature with respect to time, which means the temperature increases by

18^\circ\text{C}

for each minute of heating.

Step-by-Step Solution

1

Identify the component of the linear equation being questioned.

The number

18

is the coefficient of the variable

m

in the linear equation

T = 18m + 31

.

We need to determine what role the coefficient of the independent variable plays in a linear model.

2

Determine the mathematical meaning of the coefficient in a linear equation of the form

y = mx + b

.

The coefficient of

m

, which is

18

, represents the slope or the rate of change of

T

with respect to

m

.

In a linear function, the slope represents the change in the dependent variable per unit change in the independent variable.

3

Interpret the rate of change in the context of the problem.

Since

T

is the temperature in degrees Celsius (

^\circ\text{C}

) and

m

is the time in minutes, a rate of change of

18

means that the temperature increases by

18^\circ\text{C}

for every

1

minute of heating.

This matches the definition of the slope in the given context.

Key Concept

Interpreting the slope of a linear relationship in context

Estimated Time:1m 0s

Question 419Question →

A scientist measures the density of a core sample of ice as a function of depth. The density $d$ , in grams per cubic centimeter ( $\text{g/cm}^3$ ), of the ice at a depth of $x$ meters below the glacier surface is modeled by the equation $d = 0.0004x + 0.917$ . According to the model, what is the depth, in meters, at which the ice density is $0.935\text{ g/cm}^3$ ?

Show answer & explanation

Answer: 45

Answer

The depth is 45 meters.

To find the depth at which the density is

0.935\text{ g/cm}^3

, substitute

0.935

for the density

d

in the model equation, yielding

0.935 = 0.0004x + 0.917

. Subtracting

0.917

from both sides gives

0.018 = 0.0004x

. Dividing both sides by

0.0004

results in

x = 45

. Therefore, the depth is 45 meters.

Step-by-Step Solution

1

Substitute the target density of

0.935\text{ g/cm}^3

for

d

in the given model equation.

0.935 = 0.0004x + 0.917

This sets up the linear equation to solve for the corresponding depth

x

.

2

Subtract

0.917

from both sides of the equation.

0.018 = 0.0004x

This isolates the term containing the variable

x

on one side of the equation.

3

Divide both sides of the equation by

0.0004

to find the value of

x

.

x = 45

Dividing

0.018

by

0.0004

solves for the depth

x

in meters.

Key Concept

Solving linear relationships in context for the independent variable given a value of the dependent variable.

Question 420Question →

A community library is moving its collection of books to a new building. The number of books, $B$ , remaining to be packed $h$ hours after the packing begins is modeled by the linear equation $B = 12,500 - 450h$ , where $0 \leq h \leq 25$ . Which of the following is the best interpretation of $450$ in this context?

The number of books packed each hour

The initial number of books that need to be packed

The number of hours it takes to pack each book, which is

\frac{1}{450}

hour

The total number of hours it takes to pack all the books

Show answer & explanation

Answer: The number of books packed each hour

Answer

The number of books packed each hour

In the linear equation

B = 12,500 - 450h

, the coefficient of

h

is

-450

. This value represents the rate of change of the dependent variable,

B

(remaining books), with respect to the independent variable,

h

(hours). A slope of

-450

means that for every hour that passes, the number of remaining books decreases by

450

, which means

450

books are packed each hour.

Step-by-Step Solution

1

Identify the variable and constants in the given linear equation

B = 12,500 - 450h

.

B

represents the number of remaining books,

h

represents the number of hours spent packing,

12,500

is the constant term (

y

-intercept), and

-450

is the coefficient of

h

(slope).

Understanding the components of the linear model is essential to interpreting their real-world meanings.

2

Analyze the rate of change (slope) of the equation.

The slope is

-450

, which indicates that the number of remaining books decreases by

450

for every

1

hour that passes.

The coefficient of the independent variable in a linear equation in context represents the rate of change of the dependent variable with respect to the independent variable.

3

Translate the rate of change of the remaining books into the rate of packing.

A decrease of

450

remaining books per hour is equivalent to packing

450

books per hour.

Since books are being packed to be moved, a reduction in the remaining unpacked books corresponds directly to the number of books packed.

Key Concept

Interpreting the slope (rate of change) of a linear equation in a real-world context.

Estimated Time:1m 30s

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