Algebra Practice Questions — SAT — Page 22

Question 421Question →

In the inequality $-3(2x - 5) + 8 \geq 2(kx - 3) - 1$ , $k$ is a constant. If the solution set for $x$ is $x \leq 2$ , what is the value of $k$ ?

Show answer & explanation

Answer: 4.5

Answer

4.5

To find the value of

k

, simplify both sides of the inequality:

-3(2x - 5) + 8 \geq 2(kx - 3) - 1

becomes

-6x + 23 \geq 2kx - 7

. Grouping the variable terms on the left and constants on the right gives

-(6 + 2k)x \geq -30

. Because the solution set is

x \leq 2

, dividing by the negative coefficient reverses the inequality sign to yield

x \leq \frac{30}{6 + 2k}

. Setting the boundary value

\frac{30}{6 + 2k}

equal to 2 gives the equation

30 = 2(6 + 2k) = 12 + 4k

. Subtracting 12 from both sides gives

18 = 4k

, which results in

k = 4.5

.

Step-by-Step Solution

1

Simplify both sides of the inequality

-6x + 23 \geq 2kx - 7

Distribute the constants on both sides:

-3(2x - 5) + 8 = -6x + 15 + 8 = -6x + 23

, and

2(kx - 3) - 1 = 2kx - 6 - 1 = 2kx - 7

.

2

Isolate the variable terms on the left side and constants on the right side

-(6 + 2k)x \geq -30

Subtract

2kx

and

23

from both sides to get

-6x - 2kx \geq -7 - 23

, then factor out

x

to get

-(6 + 2k)x \geq -30

.

3

Relate the inequality to the given solution boundary

x \leq \frac{30}{6 + 2k}

Since the solution is

x \leq 2

, dividing both sides by the negative coefficient

-(6 + 2k)

reverses the inequality sign, yielding

x \leq \frac{-30}{-(6 + 2k)} = \frac{30}{6 + 2k}

.

4

Solve for

k

using the boundary equation

k = 4.5

Set the boundary expression equal to 2:

\frac{30}{6 + 2k} = 2

. Multiply both sides by

6 + 2k

to get

30 = 12 + 4k

, which simplifies to

18 = 4k

, giving

k = 4.5

.

Key Concept

Solving linear inequalities in one variable involving variable coefficients and applying the inequality direction flip when dividing by a negative value.

Question 422Question →

Which of the following represents all possible values of $x$ that satisfy the inequality $-2(x - 4) + 9 > 2x + 5$ ?

x < 3

x > 3

x < -1

x < 2

Show answer & explanation

Answer: $x < 3$

Answer

x < 3

Expanding the inequality by distributing

-2

yields

-2x + 8 + 9 > 2x + 5

, which simplifies to

-2x + 17 > 2x + 5

. Subtracting

2x

and

17

from both sides isolates the variable terms, giving

-4x > -12

. Dividing both sides by the negative coefficient

-4

and reversing the inequality sign results in the solution set representing all values of

x

less than

3

.

Step-by-Step Solution

1

Distribute the coefficient

-2

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

-2x + 8 + 9 > 2x + 5

Applying the distributive property removes the parentheses so like terms can be combined.

2

Combine the constant terms on the left side.

-2x + 17 > 2x + 5

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

3

Subtract

2x

and

17

from both sides of the inequality.

-4x > -12

This groups all terms containing the variable on the left side and all constant terms on the right side.

4

Divide both sides by

-4

and reverse the inequality sign.

x < 3

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

Key Concept

To solve a multi-step linear inequality, apply the distributive property, combine like terms, isolate the variable, and reverse the inequality sign when multiplying or dividing both sides by a negative number.

Question 423Question →

A truck rental company charges a daily fee of $45.00 plus$ 0.75 per mile driven. A driver rents a truck for one day, has a budget of at most $150.00, and is required to drive at least 60 miles for a delivery. What is the maximum number of additional miles the driver can drive beyond the required 60 miles without exceeding the budget?

Show answer & explanation

Answer: 80

Answer

80

To find the maximum number of additional miles, we set up the inequality

45 + 0.75(60 + a) \leq 150

, where

a

represents the number of additional miles driven. Distributing

0.75

yields

45 + 45 + 0.75a \leq 150

, which simplifies to

90 + 0.75a \leq 150

. Subtracting

90

from both sides gives

0.75a \leq 60

. Finally, dividing by

0.75

gives

a \leq 80

. The maximum value of

a

is therefore 80.

Step-by-Step Solution

1

Set up the inequality representing the total budget constraint.

45 + 0.75(60 + a) \leq 150

, where

a

is the number of additional miles.

The daily fee is

45, the per-mile rate is

0.75, the driver must drive at least 60 miles plus

a

additional miles, and the total cost cannot exceed the $150 budget.

2

Simplify the expression by distributing

0.75

and combining constant terms.

90 + 0.75a \leq 150

0.75 \times 60 = 45

, and adding the daily fee of

45

gives

90

.

3

Isolate the variable term by subtracting

90

from both sides.

0.75a \leq 60

This determines the remaining budget available for the additional miles.

4

Solve for

a

by dividing both sides by

0.75

.

a \leq 80

Dividing

60

by

0.75

gives the maximum number of additional miles.

Key Concept

Solving multi-step linear inequalities in context

Question 424Question →

Which inequality is equivalent to $-3(3x - 5) - 4 < -2x + 18$ ?

x < -1

x > -\frac{37}{7}

x > -1

x > -\frac{29}{7}

Show answer & explanation

Answer: $x > -1$

Answer

x > -1

The correct inequality is found by first expanding

-3(3x - 5)

to get

-9x + 15

. The inequality becomes

-9x + 15 - 4 < -2x + 18

, which simplifies to

-9x + 11 < -2x + 18

. Adding

2x

to both sides gives

-7x + 11 < 18

. Subtracting

11

from both sides gives

-7x < 7

. Dividing both sides by the negative coefficient

-7

requires reversing the inequality sign, yielding

x > -1

.

Step-by-Step Solution

1

Distribute

-3

to both terms inside the parentheses:

-3(3x - 5)

.

-9x + 15 - 4 < -2x + 18

This expands the expression using the distributive property, making sure to multiply both

3x

and

-5

by

-3

, which changes the sign of the constant term to positive

15

.

2

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

-9x + 11 < -2x + 18

Simplifying the constant terms (

15 - 4 = 11

) makes it easier to isolate the variable.

3

Add

2x

to both sides of the inequality to group the variable terms on the left.

-7x + 11 < 18

This combines the

x

terms on one side of the inequality.

4

Subtract

11

from both sides of the inequality to group the constant terms on the right.

-7x < 7

This isolates the term with the variable on the left side.

5

Divide both sides of the inequality by

-7

and reverse the inequality sign.

x > -1

Since we are dividing by a negative number, the inequality sign must be reversed from less than (

<

) to greater than (

>

).

Key Concept

Solving linear inequalities in one variable, including the distributive property and division by negative numbers.

Question 425Question →

An environmentalist studies the sound level of a waterfall. The sound level, $S$ , in decibels ( $\text{dB}$ ), at a distance of $d$ meters from the base of the waterfall is modeled by the equation $S = 92 - 0.15d$ , where $0 \leq d \leq 200$ . What is the best interpretation of the $S$ -intercept of the graph of this equation in the $dS$ -plane?

The sound level of the waterfall decreases by

0.15\text{ decibels}

for each meter of distance from its base.

The sound level of the waterfall is

91.85\text{ decibels}

at its base.

The sound level of the waterfall is

92\text{ decibels}

at its base.

The distance from the base where the sound level is

0\text{ decibels}

is

92\text{ meters}

.

Show answer & explanation

Answer: The sound level of the waterfall is $92\text{ decibels}$ at its base.

Answer

The sound level of the waterfall is

92\text{ decibels}

at its base.

The

S

-intercept of the graph in the

dS

-plane occurs where the horizontal variable,

d

, is equal to

0

. Substituting

d = 0

into the equation yields

S = 92 - 0.15(0) = 92

. Since

d

represents the distance from the base of the waterfall in meters and

S

represents the sound level in decibels, this means that at a distance of

0

meters (the base of the waterfall), the sound level is

92\text{ decibels}

.

Step-by-Step Solution

1

Identify the definition of the

S

-intercept in the context of the

dS

-plane.

The

S

-intercept is the point on the graph where the independent variable,

d

(distance), is equal to

0

.

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

2

Substitute

d = 0

into the linear equation.

S = 92 - 0.15(0) = 92

.

Evaluating the equation at

d = 0

determines the value of the sound level at zero meters from the base.

3

Interpret the mathematical result in the real-world context.

At a distance of

0

meters (which is the base of the waterfall), the sound level is

92\text{ decibels}

.

This connects the coordinate point

(0, 92)

to the physical properties of the scenario.

Key Concept

Interpreting the vertical intercept of a linear relationship in context

Estimated Time:1m 30s

Question 426Question →

In a certain board game, a player receives $15$ points for each quest completed, but loses $4$ points for each penalty card drawn. A player completes $12$ quests and draws $c$ penalty cards. If the player's total score is greater than $120$ points, what is the maximum possible value of $c$ ?

Show answer & explanation

Answer: 14

Answer

14

The player earns a base score of

15 \times 12 = 180

points from quests, and loses

4

points for each of the

c

penalty cards, resulting in a total score of

180 - 4c

points. Because the score must be greater than

120

, we write the inequality

180 - 4c > 120

. Subtracting

180

from both sides yields

-4c > -60

. Dividing by

-4

and reversing the inequality sign gives

c < 15

. Since

c

must represent a whole number of cards, the maximum possible value is the largest integer less than

15

, which is

14

.

Step-by-Step Solution

1

Set up the inequality representing the score constraint.

180 - 4c > 120

The player earns

15

points for each of the

12

quests (

15 \times 12 = 180

) and loses

4

points for each of the

c

penalty cards (

4c

), and this total must exceed

120

.

2

Isolate the variable term by subtracting

180

from both sides.

-4c > -60

To solve for

c

, we first subtract the constant term

180

from both sides of the inequality.

3

Divide both sides by

-4

and reverse the inequality sign.

c < 15

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

4

Determine the maximum integer value for

c

.

14

Since the number of penalty cards

c

must be a whole number and

c

must be strictly less than

15

, the largest possible value is

14

.

Key Concept

Solving linear inequalities in one variable involving multiplication or division by a negative number and interpreting the solution set in a discrete context.

Question 427Question →

A botanist is studying the transpiration rate of a certain plant species under controlled conditions. The transpiration rate $T$ , in grams of water vapor per square meter of leaf area per hour ( $\text{g/m}^2/\text{h}$ ), is modeled by a linear function of the relative humidity $H$ , expressed as a decimal where $0.20 \leq H \leq 0.80$ . The model is represented by the equation $T = -4.8H + C$ , where $C$ is a constant. Based on this model, if the relative humidity increases by $0.15$ , what is the corresponding decrease in the transpiration rate, in $\text{g/m}^2/\text{h}$ ?

Show answer & explanation

Answer: 0.72

Answer

The correct answer is 0.72.

In the linear equation

T = -4.8H + C

, the coefficient of

H

(which is

-4.8

) represents the slope of the line. The slope indicates that for every increase of

1.0

in the relative humidity

H

, the transpiration rate

T

decreases by

4.8

grams of water vapor per square meter of leaf area per hour. To find the decrease in transpiration rate corresponding to an increase of

0.15

in relative humidity, multiply the rate of change by the change in relative humidity:

4.8 \times 0.15 = 0.72

. Thus, the transpiration rate decreases by

0.72

grams of water vapor per square meter of leaf area per hour.

Step-by-Step Solution

1

Identify the slope of the linear model.

The slope is

-4.8

.

The coefficient of

H

in the linear equation

T = -4.8H + C

represents the rate of change of the transpiration rate with respect to the relative humidity.

2

Calculate the change in the transpiration rate (

\Delta T

) for an increase of

0.15

in the relative humidity (

\Delta H = 0.15

).

\Delta T = -4.8 \times 0.15 = -0.72

The change in the dependent variable is equal to the slope multiplied by the change in the independent variable.

3

Determine the magnitude of the decrease.

The decrease in the transpiration rate is

0.72

.

The negative sign in the change

\Delta T = -0.72

represents a decrease, so the amount of decrease is

0.72

.

Key Concept

Interpreting the slope of a linear relationship in context as the rate of change of the dependent variable with respect to the independent variable.

Question 428Question →

A manufacturing plant uses a heating chamber for curing composite materials. The temperature $T$ , in degrees Celsius ( $^\circ\text{C}$ ), of the chamber $m$ minutes after the heating element is turned on is modeled by the equation $T = 3.5m + 22$ , where $0 \leq m \leq 60$ . The chamber must reach a target temperature of $180^\circ\text{C}$ for the curing process. Which of the following is the best interpretation of the value $\frac{180 - 22}{3.5}$ in this context?

The rate, in degrees Celsius per minute, at which the temperature of the chamber increases

The temperature, in degrees Celsius, of the heating chamber after

180

minutes

The number of minutes it takes for the heating chamber to reach the target temperature of

180^\circ\text{C}

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

180^\circ\text{C}

Show answer & explanation

Answer: The number of minutes it takes for the heating chamber to reach the target temperature of $180^\circ\text{C}$

Answer

The number of minutes it takes for the heating chamber to reach the target temperature of

180^\circ\text{C}

The correct answer represents the time needed to reach the target temperature. To find when the temperature

T

reaches

180^\circ\text{C}

, we substitute

180

for

T

in the equation

T = 3.5m + 22

, giving

180 = 3.5m + 22

. Solving for time

m

requires subtracting the initial temperature of

22^\circ\text{C}

from the target temperature of

180^\circ\text{C}

, and then dividing by the heating rate of

3.5^\circ\text{C}

per minute. This algebraic isolation results in

m = \frac{180 - 22}{3.5}

, representing the duration in minutes to reach that target.

Step-by-Step Solution

1

Identify the meaning of each term in the linear model.

In the equation

T = 3.5m + 22

,

T

represents the chamber's temperature,

m

represents time in minutes,

3.5

represents the rate of temperature increase per minute, and

22

represents the initial temperature of the chamber.

Understanding the components of the linear equation helps contextualize the mathematical operations performed in the expression.

2

Set up the equation using the target temperature.

The target temperature is

180^\circ\text{C}

. Setting

T = 180

gives the linear equation

180 = 3.5m + 22

.

This establishes the relationship between the target temperature and the time

m

required to reach it.

3

Solve for the variable

m

representing time.

Subtract

22

from both sides to get

180 - 22 = 3.5m

, then divide both sides by

3.5

to isolate

m

, yielding

m = \frac{180 - 22}{3.5}

.

Isolating

m

shows that the expression is mathematically equivalent to the time, in minutes, at which the temperature reaches

180^\circ\text{C}

.

Key Concept

Interpreting expressions derived from linear relationships in real-world contexts

Alternative Method

Use dimensional analysis to verify the units of the expression. The numerator

(180 - 22)

represents the difference between two temperatures in degrees Celsius (

^\circ\text{C}

), which yields a temperature change in

^\circ\text{C}

. The denominator,

3.5

, represents the rate of change in degrees Celsius per minute (

^\circ\text{C}/\text{min}

). Dividing

^\circ\text{C}

by

^\circ\text{C}/\text{min}

results in minutes:

\frac{^\circ\text{C}}{^\circ\text{C}/\text{min}} = \text{min}

. This confirms the expression represents a time duration.

Estimated Time:1m 30s

Question 429Question →

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

x \leq 2

x \geq -\frac{6}{5}

x \geq 2

x \geq \frac{10}{7}

Show answer & explanation

Answer: $x \geq 2$

Answer

x \geq 2

Distributing

-2

over

3x - 4

yields

-6x + 8

. Combining the constants on the left gives

-6x + 13 \leq -x + 3

. Subtracting

13

and adding

x

to both sides results in

-5x \leq -10

. Dividing both sides by

-5

and flipping the inequality symbol yields the solution set of all values greater than or equal to

2

.

Step-by-Step Solution

1

Distribute the

-2

coefficient on the left side of the inequality.

-6x + 8 + 5 \leq -x + 3

Applying the distributive property simplifies the expression inside the parentheses.

2

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

-6x + 13 \leq -x + 3

Combining like terms simplifies the left-hand side before isolating the variable.

3

Subtract

13

from both sides of the inequality.

-6x \leq -x - 10

This begins the process of isolating the variable terms on one side and constant terms on the other.

4

Add

x

to both sides of the inequality.

-5x \leq -10

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

5

Divide both sides of the inequality by

-5

and reverse the direction of the inequality sign.

x \geq 2

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

Key Concept

Solving linear inequalities in one variable using algebraic properties, specifically distributing a negative value and reversing the inequality direction when dividing by a negative number.

Question 430Question →

A shipping service charges a flat rate of $15.50 to ship a box, plus$ 0.50 per ounce of the total weight of the box. A customer wants to ship a package and wants the total shipping cost to be at most $40.00. If the empty box weighs 14 ounces, what is the maximum weight, in ounces, of the contents that can be placed inside the box?

Show answer & explanation

Answer: 35

Answer

The maximum weight of the contents that can be placed inside the box is 35 ounces.

To find the maximum weight of the contents, let

w

represent the weight of the contents in ounces. The total weight of the package is the sum of the content weight and the empty box weight, or

w + 14

ounces. The total shipping cost is the flat rate of

15.50 plus

0.50 times the total weight, which is represented by the expression

15.50 + 0.50(w + 14)

. Since the customer wants the cost to be at most

40.00, we write the inequality

15.50 + 0.50(w + 14) \leq 40.00

. Distributing

0.50

yields

15.50 + 0.50w + 7 \leq 40.00

. Combining the constant terms gives

22.50 + 0.50w \leq 40.00

. Subtracting

22.50

from both sides results in

0.50w \leq 17.50

. Finally, dividing both sides by

0.50

gives

w \leq 35$. Thus, the maximum weight of the contents is 35 ounces.

Step-by-Step Solution

1

Define the variable and write the inequality representing the total cost constraints.

15.50 + 0.50(w + 14) \leq 40.00

The total weight is the sum of the contents (

w

) and the empty box (14 ounces), and the cost is

15.50 plus

0.50 per ounce, which cannot exceed $40.00.

2

Distribute the rate of $0.50 per ounce across the weight terms and simplify.

22.50 + 0.50w \leq 40.00

Applying the distributive property gives

0.50w + 7.00

, and adding the flat rate

15.50

simplifies the left side.

3

Isolate the variable term by subtracting

22.50

from both sides.

0.50w \leq 17.50

This isolates the variable term

0.50w

on one side of the inequality.

4

Solve for

w

by dividing both sides by

0.50

.

w \leq 35

Dividing by

0.50

(or multiplying by 2) yields the maximum weight limit for the contents.

Key Concept

Solving linear inequalities in one variable using distributive properties and combining like terms.

Question 431Question →

An agricultural irrigation system draws water from a storage tank at a constant rate. The volume of water, $V$ , in gallons, remaining in the tank $m$ minutes after the irrigation begins is given by the equation $V = 12,000 - 45m$ . According to the model, after how many minutes of irrigation will there be exactly $8,400$ gallons of water remaining in the tank?

Show answer & explanation

Answer: 80

Answer

The irrigation system will have exactly

8,400

gallons of water remaining after

80

minutes.

To find the number of minutes,

m

, when the volume of water remaining is exactly

8,400

gallons, substitute

8,400

for

V

in the given linear model:

8,400 = 12,000 - 45m

. Subtracting

12,000

from both sides of the equation yields

-3,600 = -45m

. Dividing both sides of the equation by

-45

yields

m = 80

. Therefore, after

80

minutes, there will be exactly

8,400

gallons of water remaining in the tank.

Step-by-Step Solution

1

Substitute the target volume of

8,400

gallons for

V

in the linear model equation.

8,400 = 12,000 - 45m

We want to find the number of minutes,

m

, when the remaining volume,

V

, is exactly

8,400

gallons.

2

Subtract

12,000

from both sides of the equation.

-3,600 = -45m

This isolates the variable term

-45m

on one side of the equation.

3

Divide both sides by

-45

to solve for

m

.

m = 80

This isolates

m

to find the number of minutes elapsed.

Key Concept

Solving linear equations in context

Question 432Question →

A commercial drone descends from a mapping mission at a constant rate. The altitude of the drone, $A$ , in meters, $t$ seconds after it begins its descent is modeled by the equation $A = -4.5t + 360$ . The graph of this equation in the $tA$ -plane is shown below.

Which of the following is the best interpretation of the $t$ -intercept of the graph in this context?

The number of seconds it takes for the drone to reach sea level.

The altitude, in meters, of the drone when it begins its descent.

The rate, in meters per second, at which the drone descends.

The number of seconds it takes for the drone to descend 1 meter.

Show answer & explanation

Answer: The number of seconds it takes for the drone to reach sea level.

Answer

The

t

-intercept represents the number of seconds it takes for the drone to reach sea level.

The correct answer is the option stating that the intercept represents the number of seconds it takes for the drone to reach sea level. The

t

-intercept is the point on the graph where

A = 0

. In this context,

A = 0

means the drone's altitude is

0

meters (sea level), and

t

represents the time elapsed in seconds. Setting

A = 0

in the equation

A = -4.5t + 360

gives

t = 80

, meaning it takes

80

seconds for the drone to reach sea level.

Step-by-Step Solution

1

Identify the meaning of the

t

-intercept on the graph.

The

t

-intercept is the point on the graph where the vertical coordinate,

A

, is equal to

0

.

By definition, the horizontal intercept of a function occurs when the dependent variable (altitude,

A

) is zero.

2

Relate the condition

A = 0

to the context.

An altitude of

A = 0

meters corresponds to sea level.

The variable

A

represents altitude in meters, so

A = 0

represents the level where altitude is zero, which is sea level.

3

Interpret the corresponding value of

t

.

The

t

-coordinate represents the time, in seconds, when the altitude is 0.

Since

t

represents the number of seconds after the descent began, the

t

-intercept

(80, 0)

indicates that it takes

80

seconds for the drone to reach sea level.

Key Concept

Interpreting the intercepts of a linear relationship in context.

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

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