Algebra Practice Questions — SAT — Page 20

For which values of $x$ is the inequality $-4(2x - 5) \geq 3x + 9$ true?

x \geq 1

x \leq -1

x \leq 1

x \leq -\frac{29}{11}

Show answer & explanation

Answer: $x \leq 1$

Answer

The inequality is true for all values of

x

such that

x

is less than or equal to

1

.

The correct inequality representing the solution set is the one that shows the variable is less than or equal to one.

Step-by-Step Solution

1

Distribute the factor of

-4

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

-8x + 20 \geq 3x + 9

Applying the distributive property gives

-4 \times 2x = -8x

and

-4 \times -5 = 20

.

2

Subtract

3x

from both sides of the inequality to collect all terms with the variable

x

on the left side.

-11x + 20 \geq 9

Grouping the variable terms helps isolate the variable.

3

Subtract

20

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

-11x \geq -11

Subtracting twenty from both sides moves the constant terms to the right side of the inequality.

4

Divide both sides by

-11

and reverse the direction of the inequality sign.

x \leq 1

Dividing or multiplying an inequality by a negative number requires flipping the inequality symbol.

Key Concept

Linear Inequalities in One Variable

Question 382Question →

An electric vehicle is connected to a charging station. The energy stored in the vehicle's battery pack, $E$ , in kilowatt-hours (kWh), can be modeled by the equation $E = 0.75m + 24.0$ , where $m$ is the number of minutes the vehicle has been charging. Which of the following is the best interpretation of the number $0.75$ in this context?

The initial energy stored in the battery pack, in kilowatt-hours, before the vehicle began charging

The increase in the charging time, in minutes, for each additional kilowatt-hour of energy stored in the battery pack

The increase in the energy stored in the battery pack, in kilowatt-hours, for each additional minute the vehicle is charged

The total energy stored in the battery pack, in kilowatt-hours, after the vehicle has been charging for

1

minute

Show answer & explanation

Answer: The increase in the energy stored in the battery pack, in kilowatt-hours, for each additional minute the vehicle is charged

Answer

The increase in the energy stored in the battery pack, in kilowatt-hours, for each additional minute the vehicle is charged

The coefficient of

m

in the linear equation

E = 0.75m + 24.0

is the slope of the line. In context, the slope represents the rate of change of the dependent variable (stored energy,

E

) per unit increase of the independent variable (time,

m

). Thus,

0.75

represents an increase of

0.75

kilowatt-hours in stored energy for each additional minute of charging.

Step-by-Step Solution

1

Identify the form and components of the given equation.

The equation

E = 0.75m + 24.0

is in the slope-intercept form

y = mx + b

, where

m = 0.75

is the slope and

b = 24.0

is the y-intercept.

To relate the numerical values in the equation to their mathematical interpretations.

2

Interpret the meaning of the slope in context.

The slope,

0.75

, represents the unit rate of change of the dependent variable (energy

E

in kWh) per unit change of the independent variable (time

m

in minutes).

To explain what the coefficient of

m

means in real-world terms.

3

Combine the units and direction of change.

Since

0.75

is positive, the energy increases by

0.75

kWh for every additional

1

minute of charging time.

To formulate the final contextual interpretation of the slope.

Key Concept

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

Question 383Question →

In the $xy$ -plane, the graph of the linear equation $y = mx + b$ , where $m$ and $b$ are constants, passes through the points $(-2, 5)$ and $(4, 7)$ . What is the value of $m + b$ ?

6

-2

4

\frac{14}{3}

Show answer & explanation

Answer: $6$

Answer

The value of

m + b

is

6

.

The correct answer is

6

. By finding the slope

m

using the change in

y

divided by the change in

x

, we get

m = \frac{1}{3}

. Using the slope-intercept form with the point

(4, 7)

allows us to solve for

b = \frac{17}{3}

. Adding the two values together yields

m + b = \frac{1}{3} + \frac{17}{3} = 6

.

Step-by-Step Solution

1

Calculate the slope

m

of the line using the two points

(-2, 5)

and

(4, 7)

.

m = \frac{7 - 5}{4 - (-2)} = \frac{2}{6} = \frac{1}{3}

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

m = \frac{y_2 - y_1}{x_2 - x_1}

.

2

Substitute the slope

m = \frac{1}{3}

and the coordinates of one point, such as

(4, 7)

, into the slope-intercept equation

y = mx + b

to solve for

b

.

7 = \frac{1}{3}(4) + b \Rightarrow 7 = \frac{4}{3} + b \Rightarrow b = \frac{17}{3}

Substituting a known point and the slope into the slope-intercept form allows us to isolate and solve for the constant

y

-intercept.

3

Add the calculated values of

m

and

b

to find

m + b

.

m + b = \frac{1}{3} + \frac{17}{3} = \frac{18}{3} = 6

The question asks for the sum of the slope

m

and the

y

-intercept

b

.

Key Concept

Linear Equations in Two Variables

Question 384Question →

In the system of equations below, $x$ and $y$ are variables.

2x - y = 5

3x - 2y = -17

What is the value of $x$ ?

7

25

27

49

Show answer & explanation

Answer: 27

Answer

The value of

x

is 27.

Isolating

y

from the first equation gives

y = 2x - 5

. Substituting this expression into the second equation yields

3x - 2(2x - 5) = -17

. Distributing the

-2

gives

3x - 4x + 10 = -17

, which simplifies to

-x + 10 = -17

. Subtracting

10

from both sides results in

-x = -27

, meaning

x = 27

. Alternatively, multiplying the first equation by

2

gives

4x - 2y = 10

, and subtracting the second equation from it eliminates

y

directly to give

x = 10 - (-17) = 27

.

Step-by-Step Solution

1

Isolate the variable

y

in the first equation,

2x - y = 5

.

y = 2x - 5

This allows for substitution of the expression for

y

into the second equation.

2

Substitute the expression for

y

into the second equation,

3x - 2y = -17

.

3x - 2(2x - 5) = -17

To create a single-variable equation in terms of

x

.

3

Distribute the coefficient

-2

to the terms inside the parentheses and simplify the equation.

3x - 4x + 10 = -17

, which simplifies to

-x + 10 = -17

Applying the distributive property correctly, noting that

-2 \times -5 = 10

.

4

Subtract

10

from both sides of the equation to isolate the term with

x

.

-x = -27

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

5

Multiply both sides of the equation by

-1

to solve for

x

.

x = 27

To find the positive value of the variable

x

.

Key Concept

Solving systems of linear equations using the substitution method.

Estimated Time:1m 30s

Question 385Question →

A specialized cooling system is used to lower the temperature of a chemical solution in a laboratory. The temperature of the solution, $T$ , in degrees Celsius, can be modeled by the linear equation $T = 85.4 - 1.25m$ , where $m$ is the number of minutes since the cooling process began. According to the model, how many minutes does it take for the temperature of the solution to decrease by $15$ degrees Celsius?

Show answer & explanation

Answer: 12

Answer

12 minutes

The correct answer is 12. In the equation

T = 85.4 - 1.25m

, the coefficient of

m

is

-1.25

, which indicates that the temperature decreases by

1.25

degrees Celsius for each minute that passes. To find how many minutes it takes for the temperature to decrease by

15

degrees Celsius, divide the total decrease by the rate of decrease:

\frac{15}{1.25} = 12

.

Step-by-Step Solution

1

Identify the rate of change from the linear equation.

The rate of decrease is

1.25

degrees Celsius per minute.

The slope of the linear equation

T = 85.4 - 1.25m

is

-1.25

, representing the change in temperature per minute.

2

Divide the target temperature change by the rate of change to find the time.

12 minutes

To find the number of minutes for a

15

-degree decrease at a rate of

1.25

degrees per minute, calculate

\frac{15}{1.25}

.

Key Concept

Interpreting the slope of a linear relationship in context

Question 386Question →

A rideshare driver has a daily goal of earning at least $150. So far today, the driver has earned$ 45. The driver earns $12 per ride plus an average tip of$ 3 per ride. If the driver must pay a daily vehicle fee of $15, what is the minimum number of additional rides the driver must complete today to meet or exceed the daily earnings goal?

Show answer & explanation

Answer: 8

Answer

The minimum number of additional rides the driver must complete today is 8.

The driver earns

12 plus a

3 tip per ride, which is

15 per ride. Starting with

45 and subtracting the

15 fee leaves the driver with

30 before completing any new rides. To reach at least

150, the driver needs to earn at least

120 more. Dividing

120 by the

15 rate per ride gives a minimum of 8 rides.

Step-by-Step Solution

1

Define the variable

x

for the number of additional rides and write an inequality representing the total net earnings.

45 + 12x + 3x - 15 \geq 150

To represent the condition that the driver's total earnings, including initial earnings and new rides, minus the fee, must be at least $150.

2

Simplify the left side of the inequality by combining the constants and the

x

terms.

30 + 15x \geq 150

To group like terms and simplify the expression before solving.

3

Subtract 30 from both sides of the inequality to isolate the variable term.

15x \geq 120

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

4

Divide both sides of the inequality by 15 to solve for

x

.

x \geq 8

To find the minimum value of

x

that satisfies the inequality.

Key Concept

Solving linear inequalities in one variable to find a minimum threshold value in a real-world scenario.

Question 387Question →

In the $xy$ -plane, a line is represented by the equation $ay - bx = 24$ , where $a$ and $b$ are constants. If the line has a $y$ -intercept of $(0, -3)$ and passes through the point $(4, 5)$ , what is the slope of the line?

Show answer & explanation

Answer: 2

Answer

The slope of the line is 2.

Substituting the y-intercept

(0, -3)

into the equation

ay - bx = 24

yields

-3a = 24

, which gives

a = -8

. Substituting the point

(4, 5)

and

a = -8

into the equation yields

-8(5) - 4b = 24

, which simplifies to

-40 - 4b = 24

, giving

b = -16

. Re-assembling the equation gives

-8y + 16x = 24

. Solving for

y

in terms of

x

yields

y = 2x - 3

, where the coefficient of

x

is the slope, 2.

Step-by-Step Solution

1

Substitute the y-intercept

(0, -3)

into the given equation

ay - bx = 24

to find the value of

a

.

a = -8

Since the y-intercept lies on the line, its coordinates satisfy the line's equation.

2

Substitute the point

(4, 5)

and the value of

a = -8

into the equation to find the value of

b

.

b = -16

Since the point

(4, 5)

lies on the line, its coordinates must satisfy the equation.

3

Write the resulting equation

-8y + 16x = 24

in slope-intercept form (

y = mx + c

) to identify the slope.

y = 2x - 3

, which gives a slope of

2

.

The slope of a line in the form

y = mx + c

is represented by the coefficient

m

of

x

.

Key Concept

Finding the slope of a line from a given equation by determining its constant coefficients using known points.

Estimated Time:1m 30s

Question 388Question →

A digital archiving team is scanning a large collection of historical documents. The number of unscanned documents remaining in the collection, $D$ , can be modeled by the equation $D = 14{}200 - 350t$ , where $t$ represents the number of hours the team has been scanning. Which of the following is the best interpretation of $350$ in this context?

The number of hours it takes the team to scan one document

The initial number of unscanned documents at the start of the shift

The number of documents the team scans each hour

The number of unscanned documents remaining after one hour of the shift

Show answer & explanation

Answer: The number of documents the team scans each hour

Answer

The number of documents the team scans each hour

In the linear model

D = 14{}200 - 350t

, the coefficient

-350

is the slope of the line, which indicates the rate of change of the dependent variable. Since the remaining number of unscanned documents decreases by

350

for each hour that passes, the value

350

represents the number of documents scanned per hour by the team.

Step-by-Step Solution

1

Identify the role of each component in the linear equation

D = 14{}200 - 350t

.

The equation is in the form

y = b + mx

, where the constant

b = 14{}200

is the y-intercept (the starting value of

D

when

t = 0

) and the coefficient

m = -350

is the slope (the constant rate of change of

D

per unit increase in

t

).

Understanding the structure of the linear model helps distinguish between initial values and rates of change.

2

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

The slope of

-350

indicates that for each unit increase in the independent variable

t

(hours), the dependent variable

D

(unscanned documents remaining) decreases by

350

.

The slope represents the constant rate of change of the remaining documents over time.

3

Select the option that matches this rate of change.

A rate of

350

documents per hour means the archiving team scans

350

documents every hour.

This directly matches the definition of the slope in this context.

Key Concept

Interpreting Linear Relationships in Context

Estimated Time:1m 15s

Question 389Question →

A catering company charges a flat setup fee of $500 plus$ 35 per guest. A company has budgeted at most $2,400 for a banquet. If the company also wants to purchase a celebration cake for$ 150, what is the maximum number of guests they can invite to the banquet without exceeding their budget?

Show answer & explanation

Answer: 50

Answer

The maximum number of guests the company can invite without exceeding their budget is 50.

The total cost of the banquet consists of a flat setup fee (

500), a celebration cake (

150), and a variable cost of

35 per guest (

35g

, where

g

represents the number of guests). Since the company has budgeted at most

2,400, the sum of these expenses must be less than or equal to

2,400. This is represented by the inequality

35g + 500 + 150 \le 2400

. Combining the constants yields

35g + 650 \le 2400

. Subtracting 650 from both sides gives

35g \le 1750

. Dividing both sides by 35 results in

g \le 50$. Therefore, the maximum number of guests they can invite without exceeding their budget is 50.

Step-by-Step Solution

1

Set up the inequality representing the total expenses.

35g + 500 + 150 \le 2400

The sum of the fixed setup fee, the cake cost, and the variable per-guest cost must not exceed the maximum budget of $2,400.

2

Simplify the inequality by combining the constants.

35g + 650 \le 2400

Combining the fixed expenses (

500 setup fee and

150 cake) simplifies the calculation.

3

Isolate the variable term by subtracting the fixed cost from both sides.

35g \le 1750

This determines the portion of the budget that can be allocated specifically to guests.

4

Solve for the variable by dividing both sides by the cost per guest.

g \le 50

Dividing the remaining guest budget by the rate per guest gives the maximum number of guests allowed.

Key Concept

Formulating and solving multi-step linear inequalities in one variable to determine a maximum value in context.

Question 390Question →

Which of the following represents all possible values of $x$ that satisfy the inequality $-3(2x - 5) - 4x \geq 35$ ?

x \geq -2

x \leq -2

x \leq -5

x \leq 2

Show answer & explanation

Answer: $x \leq -2$

Answer

The inequality is satisfied by all values of

x

such that

x \leq -2

.

To solve the inequality, distribute

-3

across the terms in the parentheses to get

-6x + 15 - 4x \geq 35

. Grouping the

x

terms gives

-10x + 15 \geq 35

. Subtracting 15 from both sides yields

-10x \geq 20

. Finally, dividing both sides by

-10

and reversing the inequality symbol gives

x \leq -2

.

Step-by-Step Solution

1

Distribute the negative coefficient

-3

to both terms inside the parentheses.

-6x + 15 - 4x \geq 35

Applying the distributive property

a(b + c) = ab + ac

allows the removal of parentheses to simplify the expression.

2

Combine the variable terms

-6x

and

-4x

on the left side of the inequality.

-10x + 15 \geq 35

Combining like terms simplifies the left side of the inequality to prepare for isolating the variable.

3

Isolate the variable term by subtracting 15 from both sides of the inequality.

-10x \geq 20

Using the subtraction property of inequality maintains the balance of the inequality while isolating the term with the variable.

4

Divide both sides of the inequality by

-10

and reverse the direction of the inequality symbol.

x \leq -2

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

Key Concept

Solving multi-step linear inequalities in one variable, including distributing terms and reversing the inequality direction when dividing by a negative number.

Estimated Time:1m 30s

Question 391Question →

A commercial bakery uses an automated flour silo. The mass of the flour in the silo, $F$ , in kilograms, is modeled as a linear function of the time $t$ , in hours, after the bakery opens. The table below shows the mass of the flour remaining in the silo at two different times during the day:

Time (hours), $t$	Mass of flour (kilograms), $F$
$3$	$1,850$
$7$	$1,490$

If the mass of the flour in the silo decreases at a constant rate of $r$ kilograms per hour, what is the value of $r$ ?

Show answer & explanation

Answer: 90

Answer

90

The rate of decrease of the flour is represented by the magnitude of the slope of the linear relationship. Using the points

(3, 1850)

and

(7, 1490)

from the table, the slope is calculated as

\frac{1490 - 1850}{7 - 3} = \frac{-360}{4} = -90

. This indicates that the mass of the flour decreases by

90

kilograms per hour. Therefore, the value of

r

is

90

.

Step-by-Step Solution

1

Identify the data points representing time and mass from the table.

(t_1, F_1) = (3, 1850)

and

(t_2, F_2) = (7, 1490)

We need two coordinates to find the slope of the linear relationship.

2

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

Slope =

\frac{1490 - 1850}{7 - 3} = -90

The slope formula

\frac{y_2 - y_1}{x_2 - x_1}

gives the rate of change of the mass of the flour per hour.

3

Determine the value of

r

based on the rate of decrease.

r = 90

The rate of decrease is the positive magnitude of the rate of change.

Key Concept

Interpreting rate of change (slope) from tabular data in a linear context

Estimated Time:1m 30s

Question 392Question →

An online retail company determines that the relationship between the selling price of a product, $x$ dollars, and the daily number of units sold, $y$ , can be modeled by a linear equation. When the selling price is $12$ dollars, the company sells $80$ units per day. For every $3$ dollars increase in the selling price, the number of units sold daily decreases by $15$ . Which of the following equations represents this relationship?

y = -5x + 140

y = -5x + 80

y = -0.2x + 82.4

y = -5x + 20

Show answer & explanation

Answer: $y = -5x + 140$

Answer

The equation

y = -5x + 140

The relationship between price

x

and units sold

y

is linear. The slope

m

represents the rate of change:

m = \frac{\Delta y}{\Delta x} = \frac{-15}{3} = -5

. Using the point-slope form with the known point

(12, 80)

, we get

y - 80 = -5(x - 12)

. Distributing the

-5

gives

y - 80 = -5x + 60

. Adding

80

to both sides results in the equation

y = -5x + 140

.

Step-by-Step Solution

1

Calculate the slope (

m

) of the linear relation.

m = \frac{\text{change in } y}{\text{change in } x} = \frac{-15}{3} = -5

The slope is the rate of change, which represents the decrease of

15

units for every

3

dollars increase in price.

2

Set up the equation using point-slope form with the point

(12, 80)

and the slope

m = -5

.

y - 80 = -5(x - 12)

Point-slope form allows us to write the equation of a line given its slope and a point it passes through.

3

Simplify the equation into slope-intercept form.

y - 80 = -5x + 60 \Rightarrow y = -5x + 140

Distributing the slope

-5

to

-12

yields

+60

, and adding

80

to both sides isolates the variable

y

.

Key Concept

Linear Equations in Two Variables

Alternative Method

Substitute the point

(12, 80)

and slope

m = -5

into the slope-intercept equation

y = mx + b

to find

b

:

80 = -5(12) + b \Rightarrow 80 = -60 + b \Rightarrow b = 140

. Thus,

y = -5x + 140

.

Estimated Time:1m 30s

Question 393Question →

A coffee roasting company uses a commercial roasting machine. The temperature of the roasting drum, $T$ , in degrees Fahrenheit ( $^\circ\text{F}$ ), is modeled as a linear function of the roasting time, $t$ , in minutes, where $0 \leq t \leq 10$ . After $2$ minutes of roasting, the temperature of the drum is $280^\circ\text{F}$ . After $5$ minutes of roasting, the temperature of the drum is $385^\circ\text{F}$ . What is the rate of temperature increase, in degrees Fahrenheit per minute, of the roasting drum?

Show answer & explanation

Answer: 35

Answer

The rate of temperature increase of the roasting drum is

35

degrees Fahrenheit per minute.

The temperature,

T

, is modeled as a linear function of time,

t

. The constant rate of temperature increase corresponds to the slope of this linear function. Given two points,

(2, 280)

and

(5, 385)

, the slope is calculated as the change in temperature divided by the change in time:

\frac{385 - 280}{5 - 2} = \frac{105}{3} = 35

degrees Fahrenheit per minute.

Step-by-Step Solution

1

Identify the coordinates representing the relationship between time and temperature.

The two points are

(2, 280)

and

(5, 385)

, where the first coordinate is the time,

t

, in minutes, and the second coordinate is the temperature,

T

, in degrees Fahrenheit.

To find the rate of change of a linear relationship, we first need to determine two points

(t_1, T_1)

and

(t_2, T_2)

from the given context.

2

Calculate the rate of temperature increase as the slope of the line passing through these two points.

The slope

m

is given by

m = \frac{385 - 280}{5 - 2} = \frac{105}{3} = 35

.

The rate of temperature increase per minute is the constant slope of the linear relationship.

Key Concept

Slope of a linear relationship in context

Question 394Question →

The graph of a linear equation in the $xy$ -plane has an $x$ -intercept of $(k, 0)$ and a $y$ -intercept of $(0, 3k)$ , where $k$ is a positive constant. If the line passes through the point $(2, 12)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 6

Answer

The value of

k

is

6

.

The correct answer is

6

. The slope of the line can be found using the two intercepts

(k, 0)

and

(0, 3k)

:

m = \frac{3k - 0}{0 - k} = -3

. The equation of the line in slope-intercept form is

y = -3x + 3k

. Since the line passes through the point

(2, 12)

, we substitute

x = 2

and

y = 12

into the equation to get

12 = -3(2) + 3k

. Simplifying the equation yields

12 = -6 + 3k

, which gives

18 = 3k

. Dividing both sides by

3

results in

k = 6

.

Step-by-Step Solution

1

Find the slope of the line using the

x

-intercept

(k, 0)

and the

y

-intercept

(0, 3k)

.

The slope of the line is

-3

.

The slope formula is

m = \frac{y_2 - y_1}{x_2 - x_1}

. Substituting the points

(k, 0)

and

(0, 3k)

gives

m = \frac{3k - 0}{0 - k} = \frac{3k}{-k} = -3

.

2

Write the equation of the line in slope-intercept form.

The equation of the line is

y = -3x + 3k

.

Using the slope-intercept form

y = mx + b

, we substitute the slope

m = -3

and the

y

-intercept value

b = 3k

from the point

(0, 3k)

.

3

Substitute the given point

(2, 12)

into the equation and solve for

k

.

The value of

k

is

6

.

Substituting

x = 2

and

y = 12

into

y = -3x + 3k

yields

12 = -3(2) + 3k

. This simplifies to

12 = -6 + 3k

. Adding

6

to both sides gives

18 = 3k

, and dividing by

3

gives

k = 6

.

Key Concept

Linear equations in two variables, finding equations from intercepts, and constant determination via point substitution.

Question 395Question →

A researcher monitored the mass of a cooling block of metal over time as it underwent sublimation in a vacuum chamber. The table below shows the mass of the metal block, $M$ , in grams, for several values of time, $t$ , in hours after the sublimation process began.

Time, $t$ (hours)	Mass, $M$ (grams)
$2$	$112.4$
$5$	$104.6$
$8$	$96.8$

If the relationship between $M$ and $t$ is linear, which of the following is the best interpretation of the slope of the graph of this relationship in the $tM$ -plane?

The mass of the metal block decreases by

2.6

grams each hour.

The initial mass of the metal block before sublimation began was

117.6

grams.

The mass of the metal block decreases by

7.8

grams each hour.

The mass of the metal block after

1

hour of sublimation is

115.0

grams.

Show answer & explanation

Answer: The mass of the metal block decreases by $2.6$ grams each hour.

Answer

The mass of the metal block decreases by

2.6

grams each hour.

The correct answer is the option stating that the mass of the metal block decreases by

2.6

grams each hour. The slope of a linear model is the rate of change, which is computed as the change in the dependent variable divided by the change in the independent variable. Here,

\frac{104.6 - 112.4}{5 - 2} = -2.6

, representing a change of

-2.6

grams per hour.

Step-by-Step Solution

1

Calculate the slope of the linear relationship using two points from the table.

Using the points

(2, 112.4)

and

(5, 104.6)

, the slope is

\frac{104.6 - 112.4}{5 - 2} = \frac{-7.8}{3} = -2.6

.

The slope represents the constant rate of change of the dependent variable (

M

) per unit change in the independent variable (

t

).

2

Interpret the calculated slope value in the context of the problem.

A slope of

-2.6

means that the mass

M

decreases by

2.6

grams for every

1

hour increase in time

t

.

The negative sign indicates a decrease, and the rate is expressed in units of the dependent variable per unit of the independent variable (grams per hour).

Key Concept

Interpreting the slope of a linear relationship in context.

Estimated Time:1m 30s

Question 396Question →

An online retailer offers free shipping on orders of $75 or more. A customer has placed a shirt costing$ 22.50 and a pair of pants costing $34.80 in their shopping cart. They want to buy some pairs of socks that cost$ 4.50 per pair to qualify for free shipping. What is the minimum number of pairs of socks the customer must add to their cart to qualify for free shipping?

Show answer & explanation

Answer: 4

Answer

The minimum number of pairs of socks the customer must add to the cart is 4.

The customer needs to spend at least

75 to qualify for free shipping. The current subtotal of the cart is

22.50 +

34.80 =

57.30. The remaining amount needed to reach

75 is

75.00 -

57.30 =

17.70. Let

s

represent the number of pairs of socks purchased at

4.50 per pair. This can be represented by the inequality

4.50s \geq 17.70

. Dividing both sides of the inequality by

4.50

gives

s \geq 3.93$. Since the customer cannot purchase a fraction of a pair of socks, the minimum number of pairs of socks they must purchase is 4.

Step-by-Step Solution

1

Calculate the total cost of the items already in the shopping cart.

22.50 +

34.80 = $57.30

To determine how much has already been spent before adding socks.

2

Set up an inequality to represent the total cost of the order including

s

pairs of socks meeting the free shipping threshold of $75.

57.30 + 4.50s \geq 75

The total cost of the shirt, pants, and socks must be greater than or equal to $75.

3

Subtract $57.30 from both sides of the inequality to find the minimum amount that needs to be spent on socks.

4.50s \geq 17.70

To isolate the term representing the cost of the socks.

4

Divide both sides of the inequality by $4.50 to find the minimum number of pairs of socks.

s \geq 3.93

To isolate the variable representing the number of pairs of socks.

5

Determine the smallest integer value of

s

that satisfies the inequality.

4

The number of pairs of socks must be a whole number, and 4 is the smallest integer greater than or equal to 3.93.

Key Concept

Solving linear inequalities in one variable to find the minimum integer solution in a real-world context.

Estimated Time:1m 30s

Question 397Question →

A municipal swimming pool is being filled with water at a constant rate. After $2$ hours of filling, the pool contains $14,500$ gallons of water. After $5$ hours of filling, the pool contains $20,200$ gallons of water. If the relationship between the time the pool has been filling, $t$ , in hours, and the volume of water in the pool, $V$ , in gallons, is linear, which of the following equations represents this relationship?

V = 1,900t + 14,500

V = 5,700t + 3,100

V = 1,900t + 10,700

V = 1,900t + 18,300

Show answer & explanation

Answer: $V = 1,900t + 10,700$

Answer

The equation representing the relationship is

V = 1,900t + 10,700

.

The correct equation is

V = 1,900t + 10,700

. The rate of change of the water volume is the change in volume divided by the change in time:

\frac{20,200 - 14,500}{5 - 2} = \frac{5,700}{3} = 1,900

gallons per hour. Using the point-slope form with the coordinate point

(2, 14,500)

yields

V - 14,500 = 1,900(t - 2)

. Simplifying this expression gives

V - 14,500 = 1,900t - 3,800

, which results in

V = 1,900t + 10,700

.

Step-by-Step Solution

1

Identify two data points from the problem context.

The two coordinate pairs representing

(t, V)

are

(2, 14,500)

and

(5, 20,200)

.

These points will allow us to calculate the slope and the y-intercept of the linear equation.

2

Calculate the slope (

m

) using the slope formula

m = \frac{V_2 - V_1}{t_2 - t_1}

.

m = \frac{20,200 - 14,500}{5 - 2} = \frac{5,700}{3} = 1,900

.

The slope represents the constant rate, in gallons per hour, at which the pool is being filled.

3

Substitute the slope

m = 1,900

and the point

(2, 14,500)

into the point-slope form equation

V - V_1 = m(t - t_1)

to solve for

V

.

V - 14,500 = 1,900(t - 2) \implies V - 14,500 = 1,900t - 3,800 \implies V = 1,900t + 10,700

.

This yields the equation representing the volume of water

V

in the pool at any time

t

.

Key Concept

Determining a linear equation in two variables given two points from a word problem context.

Alternative Method

Instead of solving the linear equation algebraically, you can test the coordinates of the two given points

(2, 14,500)

and

(5, 20,200)

in the answer choices. Substituting

t = 2

and

t = 5

into the correct equation

V = 1,900t + 10,700

satisfies both conditions:

1,900(2) + 10,700 = 14,500

and

1,900(5) + 10,700 = 20,200

. None of the other options satisfy both points.

Estimated Time:1m 30s

Question 398Question →

A residential solar power system stores electricity in a battery. The total energy stored in the battery, $E$ , in kilowatt-hours (kWh), is modeled by a linear function of the time $t$ , in hours, since sunrise. After $4$ hours of sunlight, the energy stored is $18$ kWh. After $7$ hours of sunlight, the energy stored is $27$ kWh. Which of the following is the best interpretation of the slope of the graph of this function in the $tE$ -plane?

The energy stored in the battery increases by

3

kilowatt-hours per hour of sunlight.

The initial energy stored in the battery at sunrise is

6

kilowatt-hours.

It takes

3

hours of sunlight to increase the energy stored in the battery by

1

kilowatt-hour.

The energy stored in the battery increases by

9

kilowatt-hours per hour of sunlight.

Show answer & explanation

Answer: The energy stored in the battery increases by $3$ kilowatt-hours per hour of sunlight.

Answer

The energy stored in the battery increases by

3

kilowatt-hours per hour of sunlight.

To find the slope of the linear relationship, we use the formula for the rate of change:

\text{slope} = \frac{\Delta E}{\Delta t}

. Using the two points

(4, 18)

and

(7, 27)

, the slope is

\frac{27 - 18}{7 - 4} = \frac{9}{3} = 3

kilowatt-hours per hour. In the context of the problem, the slope represents the rate at which the stored energy increases per hour of sunlight. Therefore, the energy stored in the battery increases by

3

kilowatt-hours per hour of sunlight.

Step-by-Step Solution

1

Identify the coordinates

(t, E)

from the given values.

The two data points are

(4, 18)

and

(7, 27)

.

These points represent the time in hours and the corresponding energy stored in the battery.

2

Calculate the slope of the linear relationship using the slope formula.

The slope is

\frac{27 - 18}{7 - 4} = \frac{9}{3} = 3

.

The slope of a linear relationship represents the constant rate of change of the dependent variable (

E

) per unit change of the independent variable (

t

).

3

Interpret the meaning of the calculated slope in the context of the problem.

A slope of

3

means that the energy stored in the battery,

E

, increases by

3

kilowatt-hours (kWh) for every

1

hour of sunlight,

t

.

The units of the slope are units of

E

divided by units of

t

, which is kilowatt-hours per hour.

Key Concept

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

Estimated Time:1m 30s

Question 399Question →

What is the complete set of solutions for the inequality $7 - 3(2x - 5) \leq -4x + 8$ ?

x \leq 7

x \geq -8

x \geq 7

x \geq -7

Show answer & explanation

Answer: $x \geq 7$

Answer

The complete set of solutions is represented by the inequality

x \geq 7

.

To solve the inequality

7 - 3(2x - 5) \leq -4x + 8

, we first distribute

-3

to get

7 - 6x + 15 \leq -4x + 8

. Combining constant terms on the left side gives

22 - 6x \leq -4x + 8

. Adding

4x

and subtracting

22

from both sides isolates the variable, resulting in

-2x \leq -14

. Dividing both sides by

-2

and flipping the inequality symbol yields the solution

x \geq 7

.

Step-by-Step Solution

1

Distribute the

-3

to both terms inside the parentheses.

7 - 6x + 15 \leq -4x + 8

Applying the distributive property,

-3(2x) = -6x

and

-3(-5) = 15

.

2

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

22 - 6x \leq -4x + 8

Adding

7

and

15

yields

22

.

3

Isolate the variable term on one side and the constant term on the other side.

-2x \leq -14

Adding

4x

to both sides yields

-2x

, and subtracting

22

from both sides yields

-14

.

4

Divide both sides by

-2

and flip the inequality sign.

x \geq 7

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

Key Concept

Solving multi-step linear inequalities in one variable including negative coefficient division and the distributive property.

Question 400Question →

A technician is monitoring the pressure of a gas inside a container during an experiment. The pressure $P$ , in kilopascals ( $\text{kPa}$ ), and the time elapsed $t$ , in minutes, are related by a linear equation. At $t = 4$ minutes, the pressure is $112\text{ kPa}$ . At $t = 12$ minutes, the pressure is $136\text{ kPa}$ . If the pressure continues to increase at this constant rate, what will the pressure be, in $\text{kPa}$ , at $t = 15$ minutes?

Show answer & explanation

Answer: 145

Answer

145

The relationship between pressure

P

and time

t

is linear, which can be modeled by the equation

P = mt + b

, where

m

is the rate of change (slope) and

b

is the pressure at

t = 0

. Using the points

(4, 112)

and

(12, 136)

, the slope is calculated as

m = \frac{136 - 112}{12 - 4} = \frac{24}{8} = 3

. Substituting the point

(4, 112)

and

m = 3

into the equation

P = mt + b

yields

112 = 3(4) + b

, which simplifies to

112 = 12 + b

, so

b = 100

. The linear equation is

P = 3t + 100

. Substituting

t = 15

into this equation gives

P = 3(15) + 100 = 45 + 100 = 145

.

Step-by-Step Solution

1

Calculate the rate of change (slope,

m

) using the two given coordinate points

(4, 112)

and

(12, 136)

.

m = \frac{136 - 112}{12 - 4} = \frac{24}{8} = 3

To find the constant rate at which the pressure is increasing per minute.

2

Set up the linear equation using the point-slope form

P - P_1 = m(t - t_1)

with the point

(4, 112)

.

P - 112 = 3(t - 4) \Rightarrow P = 3t + 100

To establish the linear relationship between pressure and time.

3

Substitute

t = 15

into the linear equation to find the pressure at

15

minutes.

P = 3(15) + 100 = 145

To determine the pressure at the requested time of

15

minutes.

Key Concept

Finding and applying a linear equation in two variables from two points.

Algebra

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer