Algebra Practice Questions — SAT — Page 8

A system of linear equations is shown below.

\begin{aligned} 5x - 2y &= 14 \\ 2x + y &= 11 \end{aligned}

If

(x, y)

is the solution to the system of equations, what is the value of

x

?

3

36

4

2

Show answer & explanation

Answer: 4

Answer

4

The correct answer is

4

. Isolating

y

in the second equation gives

y = 11 - 2x

. Substituting this into the first equation yields

5x - 2(11 - 2x) = 14

. Distributing the

-2

results in

5x - 22 + 4x = 14

. Combining like terms simplifies the equation to

9x - 22 = 14

. Adding

22

to both sides gives

9x = 36

, and dividing by

9

yields

x = 4

.

Step-by-Step Solution

1

Isolate

y

in the second equation.

y = 11 - 2x

This allows for substitution into the first equation.

2

Substitute the expression for

y

into the first equation.

5x - 2(11 - 2x) = 14

This creates an equation with only one variable,

x

.

3

Distribute the

-2

through the parentheses and combine like terms.

9x - 22 = 14

Distributing

-2

to

-2x

yields

+4x

, and combining

5x + 4x

gives

9x

.

4

Add

22

to both sides and divide by

9

.

x = 4

Isolating

x

gives the final value of the variable.

Key Concept

Solving systems of linear equations using the substitution method.

Question 142Question →

Which of the following ordered pairs $(x, y)$ is a solution to the system of inequalities below?

\begin{aligned} y &\geq x + 1 \\ y &> -2x + 4 \end{aligned}

(1, 3)

(3, 1)

(0, 2)

(-1, 5)

Show answer & explanation

Answer: $(1, 3)$

Answer

The ordered pair

(1, 3)

The ordered pair

(1, 3)

is the correct answer because substituting

x = 1

and

y = 3

into the system of inequalities results in two true statements:

3 \geq 1 + 1

simplifies to

3 \geq 2

(true), and

3 > -2(1) + 4

simplifies to

3 > 2

(true).

Step-by-Step Solution

1

Substitute the coordinates of the ordered pair

(1, 3)

into the first inequality,

y \geq x + 1

.

3 \geq 1 + 1 \Rightarrow 3 \geq 2

, which is true.

To be a solution to the system, the ordered pair must satisfy the first inequality.

2

Substitute the coordinates of the ordered pair

(1, 3)

into the second inequality,

y > -2x + 4

.

3 > -2(1) + 4 \Rightarrow 3 > 2

, which is true.

The ordered pair must also satisfy the second inequality to be a solution to the system.

3

Verify that the other ordered pairs do not satisfy both inequalities.

None of the other options satisfy both inequalities:

(3, 1)

fails the first inequality,

(0, 2)

fails the second inequality, and

(-1, 5)

fails the second inequality.

Ensuring that there is exactly one correct solution.

Key Concept

A solution to a system of linear inequalities in two variables is an ordered pair

(x, y)

that makes all inequalities in the system true.

Alternative Method

Alternatively, you can graph the two inequalities on a coordinate plane. The region representing the solution to the system is the overlapping shaded area of both inequalities. By plotting each point, you can visually identify which point lies within this double-shaded region.

Estimated Time:1m 0s

Question 143Question →

In the system of equations below,

c

is a constant.

\begin{aligned} 2x + y &= 15 \\ cx - 2y &= 6 \end{aligned}

If the solution

(x, y)

to the system of equations above lies on the line

y = 3x

in the

xy

-plane, what is the value of the constant

c

?

Show answer & explanation

Answer: 8

Answer

The value of the constant

c

is 8.

Since the solution

(x, y)

lies on the line

y = 3x

, we can substitute

y = 3x

into the first equation,

2x + y = 15

, to get

2x + 3x = 15

, which simplifies to

5x = 15

, yielding

x = 3

. Substituting

x = 3

back into

y = 3x

gives

y = 9

. Thus, the intersection point of the system is

(3, 9)

. Substituting

x = 3

and

y = 9

into the second equation,

cx - 2y = 6

, gives

c(3) - 2(9) = 6

, or

3c - 18 = 6

. Adding 18 to both sides gives

3c = 24

, which simplifies to

c = 8

.

Step-by-Step Solution

1

Substitute the equation of the line

y = 3x

into the first equation of the system,

2x + y = 15

.

2x + 3x = 15

, which simplifies to

5x = 15

. Solving for

x

gives

x = 3

.

Since the solution to the system lies on the line

y = 3x

, the coordinates of the solution must satisfy all three equations.

2

Calculate the corresponding

y

-value using the relation

y = 3x

.

y = 3(3) = 9

.

The

y

-coordinate is three times the

x

-coordinate for any point on the line

y = 3x

.

3

Substitute the point

(3, 9)

into the second equation of the system,

cx - 2y = 6

.

c(3) - 2(9) = 6

, which simplifies to

3c - 18 = 6

. Adding 18 to both sides gives

3c = 24

, so

c = 8

.

Since

(3, 9)

is the solution to the system, it must satisfy the second equation as well.

Key Concept

Solving systems of linear equations using substitution and finding parameter values based on geometric constraints.

Question 144Question →

Which of the following inequalities is equivalent to $-3(x - 4) < 15$ ?

x > -1

x < -1

x > -9

x < -9

Show answer & explanation

Answer: $x > -1$

Answer

The inequality

x > -1

The correct answer is obtained by dividing both sides of the inequality

-3(x - 4) < 15

by

-3

, which reverses the inequality symbol to yield

x - 4 > -5

. Adding

4

to both sides isolates the variable, resulting in

x > -1

. Alternatively, distributing the

-3

first yields

-3x + 12 < 15

, which simplifies to

-3x < 3

, and dividing by

-3

with a sign flip yields

x > -1

.

Step-by-Step Solution

1

Divide both sides of the inequality

-3(x - 4) < 15

by

-3

.

x - 4 > -5

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

2

Add

4

to both sides of the inequality to isolate

x

.

x > -1

Adding a constant to both sides preserves the inequality and isolates the variable

x

.

Key Concept

Solving linear inequalities in one variable involving distribution and negative coefficients.

Question 145Question →

A water tank is being filled at a constant rate. The volume of water in the tank, in gallons, is a linear function of the time, in minutes, since the filling process began. The volume of water in the tank was $24$ gallons after $3$ minutes of filling, and it was $40$ gallons after $7$ minutes of filling. What was the initial volume of water, in gallons, in the tank before the filling process began?

Show answer & explanation

Answer: 12

Answer

The initial volume of water in the tank was 12 gallons.

To find the initial volume of water, we must determine the y-intercept of the linear relationship between the volume and time. First, find the rate of change (slope) using the formula m = \frac{V_2 - V_1}{t_2 - t_1}. Substituting the given values gives m = \frac{40 - 24}{7 - 3} = \frac{16}{4} = 4. Using the slope-intercept form V(t) = mt + b, substitute one of the points, such as (3, 24), to solve for b: 24 = 4(3) + b, which simplifies to b = 12. Thus, the initial volume of water is 12 gallons.

Step-by-Step Solution

1

Calculate the slope (rate of change) of the volume with respect to time.

4

The slope is the change in volume divided by the change in time: \frac{40 - 24}{7 - 3} = \frac{16}{4} = 4.

2

Write the linear function V(t) = mt + b using the calculated slope, and substitute one of the given points to solve for b.

b = 12

Using the point (3, 24) and the slope m = 4, we substitute into V(t) = 4t + b to get 24 = 4(3) + b, which simplifies to 24 = 12 + b, meaning b = 12.

3

Identify the initial volume of water in the tank, which is the value of V(t) when t = 0.

12

When t = 0, V(0) = 4(0) + 12 = 12, which represents the initial volume.

Key Concept

Determining the y-intercept (initial value) of a linear function given two points.

Question 146Question →

In the system of equations below,

k

is a positive integer that is a multiple of 9.

\begin{aligned} 3x + 8y &= 213 \\ x - 3y &= k \end{aligned}

If the system has a solution

(x, y)

such that

x

and

y

are both positive integers, what is the value of

k

?

Show answer & explanation

Answer: 54

Answer

54

Substituting

x = 3y + k

into the first equation yields

17y + 3k = 213

. Solving for

k

gives

k = 71 - \frac{17}{3}y

. For

k

to be a positive integer,

y

must be a positive multiple of 3 less than 12.5. Testing the possible values for

y

(3, 6, 9, 12) yields the possible values of

k

as 54, 37, 20, and 3. Since

k

must be a multiple of 9, the correct value is 54.

Step-by-Step Solution

1

Express

x

in terms of

y

and

k

using the second equation.

x = 3y + k

Isolating

x

allows for easy substitution into the first equation to eliminate one of the variables.

2

Substitute the expression for

x

into the first equation.

17y + 3k = 213

Substituting

x = 3y + k

into

3x + 8y = 213

yields

3(3y + k) + 8y = 213

, which simplifies to

9y + 3k + 8y = 213

and then to

17y + 3k = 213

.

3

Solve for

k

in terms of

y

and analyze the divisibility constraint.

k = 71 - \frac{17}{3}y

, where

y

is a multiple of 3

For

k

to be an integer, the fractional term

\frac{17}{3}y

must resolve to an integer, requiring

y

to be divisible by 3.

4

Apply the constraint that

k

must be a positive integer (

k > 0

).

y < 12.5

, meaning

y

can be 3, 6, 9, or 12

Setting

71 - \frac{17}{3}y > 0

yields

17y < 213

, or

y < 12.53

. The positive integer multiples of 3 in this range are 3, 6, 9, and 12.

5

Find the corresponding values of

k

and select the one that is a multiple of 9.

The possible values for

k

are 54, 37, 20, and 3. The only multiple of 9 is 54.

Testing the possible values of

y

gives:

y=3 \rightarrow k=54

,

y=6 \rightarrow k=37

,

y=9 \rightarrow k=20

, and

y=12 \rightarrow k=3

. Among these, 54 is the only multiple of 9.

Key Concept

Systems of Linear Equations with Integer Constraints

Estimated Time:3m 0s

Question 147Question →

A local logistics company charges a flat account setup fee of $5$ dollars plus $4$ dollars per pound for the first $6$ pounds of a package shipment. For any weight beyond $6$ pounds, the company charges $3$ dollars per pound. If the total cost to ship a package is $50$ dollars, what is the total weight of the package, in pounds?

7

9

13

1

Show answer & explanation

Answer: 13

Answer

The correct answer is 13, which represents the total weight of the package in pounds.

To find the total weight, we first determine the cost of the first

6

pounds, which is the

5

dollars flat fee plus

4

dollars per pound, totaling

29

dollars. Subtracting this baseline cost from the total cost of

50

dollars leaves

21

dollars for the weight exceeding

6

pounds. Since the rate for the excess weight is

3

dollars per pound, the excess weight is

21

divided by

3

, which is

7

pounds. Adding the initial

6

pounds gives a total weight of

13

pounds.

Step-by-Step Solution

1

Calculate the total cost of shipping the first 6 pounds of the package.

The cost for the first

6

pounds, including the flat setup fee, is

5 + 6 \times 4 = 29

dollars.

This establishes the baseline cost before the rate of

3

dollars per pound is applied to any weight exceeding

6

pounds.

2

Set up a linear equation representing the total shipping cost of 50 dollars for a package of weight w, where w is greater than 6.

The equation is

29 + 3(w - 6) = 50

.

The total cost is the sum of the cost for the first

6

pounds (

29

dollars) and the cost of the excess weight (

w - 6

) at the rate of

3

dollars per pound.

3

Solve the linear equation for w by isolating the variable.

3(w - 6) = 21 \implies w - 6 = 7 \implies w = 13

.

Subtracting

29

from both sides and then dividing by

3

isolates the term representing the excess weight, and adding

6

yields the total weight.

Key Concept

Linear Equations in One Variable

Question 148Question →

In the $xy$ -plane, the graph of the linear equation $ax + by = c$ , where $a$ , $b$ , and $c$ are non-zero constants, has a positive slope and a negative $y$ -intercept. If the line passes through the point $(b, a)$ , which of the following inequalities must be true?

ab > 0

ac > 0

bc > 0

ac < 0

Show answer & explanation

Answer: $ac > 0$

Answer

The inequality

ac > 0

must be true.

The correct inequality is

ac > 0

. Rearranging

ax + by = c

into slope-intercept form gives

y = -\frac{a}{b}x + \frac{c}{b}

. A positive slope means

-\frac{a}{b} > 0

, so

\frac{a}{b} < 0

, indicating that

a

and

b

have opposite signs (

ab < 0

). A negative

y

-intercept means

\frac{c}{b} < 0

, indicating that

b

and

c

have opposite signs (

bc < 0

). Substituting the point

(b, a)

into the line's equation gives

2ab = c

. Since

ab < 0

, it follows that

c < 0

. Using the relation

bc < 0

with

c < 0

, we find

b > 0

. Using

ab < 0

with

b > 0

, we find

a

is negative. Since both

a

and

c

are negative, their product

ac

must be positive.

Step-by-Step Solution

1

Rearrange the standard form equation

ax + by = c

into slope-intercept form.

y = -\frac{a}{b}x + \frac{c}{b}

To identify the expressions for the slope and the

y

-intercept in terms of the constants

a

,

b

, and

c

.

2

Determine the sign relationships between the coefficients based on the given signs of the slope and

y

-intercept.

Since the slope is positive,

-\frac{a}{b} > 0 \implies ab < 0

. Since the

y

-intercept is negative,

\frac{c}{b} < 0 \implies bc < 0

.

This establishes that

a

and

b

have opposite signs, and

b

and

c

have opposite signs.

3

Substitute the coordinates of the point

(b, a)

into the original equation

ax + by = c

.

a(b) + b(a) = c \implies 2ab = c

Since the point lies on the line, substituting its coordinates must satisfy the equation.

4

Analyze the sign of

c

using the relation

c = 2ab

.

Since

ab < 0

, it follows that

2ab < 0

, which means

c < 0

.

To determine whether the constant

c

is positive or negative.

5

Determine the individual signs of

b

and

a

.

Since

bc < 0

and

c < 0

,

b

must be positive (

b > 0

). Since

ab < 0

and

b > 0

,

a

must be negative (

a < 0

).

To find the sign of each coefficient individually.

6

Determine the sign of the product

ac

.

Since both

a < 0

and

c < 0

, their product

ac

must be positive (

ac > 0

).

To evaluate which inequality must be true.

Key Concept

Analyzing slope, intercepts, and points on a linear equation in two variables using sign analysis of its constants.

Alternative Method

Instead of solving for the individual signs of all variables, we can choose test values that satisfy the conditions. For example, let

a = -1

and

b = 2

, which gives a positive slope of

-\frac{a}{b} = \frac{1}{2}

. Since the line passes through

(b, a) = (2, -1)

, we can find

c

:

ax + by = c \implies (-1)(2) + (2)(-1) = -4

, so

c = -4

. The

y

-intercept is

\frac{c}{b} = \frac{-4}{2} = -2

, which is negative, satisfying all conditions. Evaluating the product of

a

and

c

gives

(-1)(-4) = 4 > 0

, confirming that the product of these constants must be positive.

Estimated Time:3m 0s

Question 149Question →

In the $xy$ -plane, a line with a positive slope $m$ passes through the point $(4, -3)$ and intersects the $x$ -axis at $(p, 0)$ and the $y$ -axis at $(0, q)$ , where $p$ and $q$ are non-zero constants. If $p + q = 5$ , what is the value of $m$ ?

Show answer & explanation

Answer: 0.5

Answer

The value of

m

is

0.5

(or

1/2

).

To find the slope

m

, write the equation of the line in point-slope form:

y - (-3) = m(x - 4)

, which simplifies to

y = mx - 4m - 3

. The

y

-intercept is found by setting

x = 0

, giving

q = -4m - 3

. The

x

-intercept is found by setting

y = 0

, giving

p = 4 + \frac{3}{m}

. Substituting these expressions into

p + q = 5

gives

4 + \frac{3}{m} - 4m - 3 = 5

, which simplifies to

-4m - 4 + \frac{3}{m} = 0

. Multiplying this equation by

-m

yields the quadratic equation

4m^2 + 4m - 3 = 0

. Factoring the quadratic gives

(2m - 1)(2m + 3) = 0

. Since the line is defined to have a positive slope, the value of

m

must be positive, which is

0.5

(or

1/2

).

Step-by-Step Solution

1

Write the general equation of a line passing through

(4, -3)

with slope

m

.

y = mx - 4m - 3

Using the point-slope form

y - y_1 = m(x - x_1)

with

(x_1, y_1) = (4, -3)

establishes the relationship between

x

,

y

, and

m

.

2

Determine the value of

q

by finding the

y

-intercept of the line.

q = -4m - 3

The

y

-intercept

(0, q)

occurs when

x = 0

.

3

Determine the value of

p

by finding the

x

-intercept of the line.

p = 4 + \frac{3}{m}

The

x

-intercept

(p, 0)

occurs when

y = 0

.

4

Substitute the expressions for

p

and

q

into the given equation

p + q = 5

.

4 + \frac{3}{m} - 4m - 3 = 5

This sets up a single equation with the variable

m

using the given relationship.

5

Simplify the equation and rewrite it in standard quadratic form.

4m^2 + 4m - 3 = 0

Multiplying the simplified expression

-4m - 4 + \frac{3}{m} = 0

by

-m

eliminates the fraction and aligns it to standard quadratic form.

6

Factor the quadratic equation to find the possible values of

m

.

m = 0.5

or

m = -1.5

Factoring

(2m - 1)(2m + 3) = 0

gives the roots of the equation.

7

Filter the results based on the positive slope constraint.

m = 0.5

The problem states that the line has a positive slope, so we must reject the negative root.

Key Concept

Using linear intercepts to solve system constraints

Alternative Method

Instead of using point-slope form, use the intercept form of a linear equation:

\frac{x}{p} + \frac{y}{q} = 1

. Since the line passes through the point

(4, -3)

, substitute

x = 4

and

y = -3

to get

\frac{4}{p} - \frac{3}{q} = 1

. Given

p + q = 5

, substitute

p = 5 - q

into the equation to get

\frac{4}{5 - q} - \frac{3}{q} = 1

. Multiply both sides by the common denominator

q(5 - q)

to obtain

4q - 3(5 - q) = q(5 - q) \implies 7q - 15 = 5q - q^2 \implies q^2 + 2q - 15 = 0

. Factoring gives

(q + 5)(q - 3) = 0

. This yields

q = -5

or

q = 3

. If

q = 3

, then

p = 2

, and the slope is

m = -\frac{q}{p} = -\frac{3}{2}

(negative). If

q = -5

, then

p = 10

, and the slope is

m = -\frac{q}{p} = -\frac{-5}{10} = 0.5

(positive). Since the slope must be positive, the slope of the line is

0.5

.

Estimated Time:2m 30s

Question 150Question →

In a certain video game, a player earns 15 points for completing a level and loses 3 points for each hint they use. If a player wants to score at least 6 points on a level, what is the maximum number of hints they can use?

Show answer & explanation

Answer: 3

Answer

The maximum number of hints the player can use is 3.

To find the maximum number of hints, we construct the inequality representing the player's score:

15 - 3h \geq 6

, where

h

is the number of hints used. Subtracting 15 from both sides gives

-3h \geq -9

. Dividing both sides by

-3

and reversing the inequality sign yields

h \leq 3

. This means the player can use at most 3 hints to achieve a score of at least 6 points. Thus, the maximum number of hints is 3.

Step-by-Step Solution

1

Set up the inequality representing the score requirement.

15 - 3h \geq 6

The starting score is 15, and 3 points are lost for each hint

h

. The final score must be at least (greater than or equal to) 6.

2

Subtract 15 from both sides of the inequality.

-3h \geq -9

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

3

Divide both sides by -3 and flip the inequality sign.

h \leq 3

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

Key Concept

Solving multi-step linear inequalities with negative coefficients

Estimated Time:45s

Question 151Question →

A local store sells two types of coffee beans: Arabica and Robusta. On Monday, the store sold $8$ pounds of Arabica coffee and $5$ pounds of Robusta coffee for a total of $\$62.00$ . On Tuesday, the store sold $4$ pounds of Arabica coffee and $7$ pounds of Robusta coffee for a total of $\$58.00$ . What is the cost, in dollars, of one pound of Robusta coffee?

Show answer & explanation

Answer: 6

Answer

The cost of one pound of Robusta coffee is 6 dollars.

The correct answer is 6. By setting up the system of equations

8a + 5r = 62

and

4a + 7r = 58

, we can multiply the second equation by 2 to get

8a + 14r = 116

. Subtracting the first equation from this yields

9r = 54

, which simplifies to

r = 6

.

Step-by-Step Solution

1

Define variables for the price per pound of Arabica coffee (

a

) and Robusta coffee (

r

), then set up a system of linear equations based on the given store sales information.

Equation 1:

8a + 5r = 62

Equation 2:

4a + 7r = 58

This translates the word problem context into a solvable system of mathematical equations.

2

Multiply the second equation by

2

to make the coefficients of

a

equal in both equations.

8a + 14r = 116

This prepares the system for solving by the elimination method.

3

Subtract the first equation (

8a + 5r = 62

) from the new equation (

8a + 14r = 116

) to eliminate

a

and solve for

r

.

9r = 54

, which simplifies to

r = 6

Subtracting the equations eliminates one variable, leaving a single-variable equation that is easily solved.

Key Concept

Solving systems of two linear equations in two variables using the elimination method.

Estimated Time:1m 30s

Question 152Question →

In the system of equations below, $c$ is a constant.

\begin{aligned} 3x - 2y &= 3c - 1 \\ x + 3y &= 2c - 8 \end{aligned}

If the solution $(x, y)$ to the system satisfies $x + y = 2$ , what is the value of $c$ ?

4

16

3

-1

Show answer & explanation

Answer: 4

Answer

The correct value of the constant

c

is 4.

To find the value of

c

, the linear constraint

x + y = 2

is rearranged to

x = 2 - y

. Substituting this expression into the first equation yields

3(2 - y) - 2y = 3c - 1

, which simplifies to

6 - 5y = 3c - 1

, or

y = \frac{7 - 3c}{5}

. Substituting it into the second equation yields

(2 - y) + 3y = 2c - 8

, which simplifies to

2 + 2y = 2c - 8

, or

y = c - 5

. Equating these two expressions gives

c - 5 = \frac{7 - 3c}{5}

. Multiplying both sides by

5

results in

5c - 25 = 7 - 3c

. Adding

3c

and

25

to both sides yields

8c = 32

, which gives

c = 4

.

Step-by-Step Solution

1

Express

x

in terms of

y

using the given constraint

x + y = 2

.

x = 2 - y

This allows us to substitute the relation into the system of equations and reduce the variables from two to one.

2

Substitute

x = 2 - y

into the first equation,

3x - 2y = 3c - 1

, and isolate

y

.

3(2 - y) - 2y = 3c - 1 \Rightarrow 6 - 5y = 3c - 1 \Rightarrow 5y = 7 - 3c \Rightarrow y = \frac{7 - 3c}{5}

To represent

y

as a function of the parameter

c

.

3

Substitute

x = 2 - y

into the second equation,

x + 3y = 2c - 8

, and isolate

y

.

(2 - y) + 3y = 2c - 8 \Rightarrow 2 + 2y = 2c - 8 \Rightarrow 2y = 2c - 10 \Rightarrow y = c - 5

To obtain another independent expression for

y

in terms of

c

.

4

Equate the two expressions for

y

and solve for

c

.

c - 5 = \frac{7 - 3c}{5} \Rightarrow 5(c - 5) = 7 - 3c \Rightarrow 5c - 25 = 7 - 3c \Rightarrow 8c = 32 \Rightarrow c = 4

Since both expressions represent the same value

y

, they must be equal, allowing us to determine the constant

c

.

Key Concept

Solving a system of linear equations containing unknown parameters under given linear constraints.

Alternative Method

Alternatively, substitute the relationship

x = 2 - y

into both equations to write them as a system in terms of

y

and

c

:

3(2 - y) - 2y = 3c - 1 \Rightarrow 5y + 3c = 7

and

(2 - y) + 3y = 2c - 8 \Rightarrow 2y - 2c = -10

. This simplified system can be solved for

c

by multiplying the first equation by 2 and the second equation by 5 to eliminate

y

:

2(5y + 3c) - 5(2y - 2c) = 2(7) - 5(-10) \Rightarrow 6c + 10c = 14 + 50 \Rightarrow 16c = 64 \Rightarrow c = 4

.

Estimated Time:2m 0s

Question 153Question →

For which values of $x$ is the inequality $2 - 3(x + 4) \geq 11$ true?

x \leq -7

x \geq -7

x \leq 1

x \leq -1

Show answer & explanation

Answer: $x \leq -7$

Answer

The correct inequality is

x \leq -7

.

Subtracting 2 from both sides of

2 - 3(x + 4) \geq 11

gives

-3(x + 4) \geq 9

. Dividing both sides by the negative number

-3

requires reversing the inequality sign, yielding

x + 4 \leq -3

. Subtracting 4 from both sides completes the isolation of

x

, yielding the correct solution

x \leq -7

.

Step-by-Step Solution

1

Subtract 2 from both sides of the inequality.

-3(x + 4) \geq 9

This isolates the term containing the parentheses on one side of the inequality.

2

Divide both sides of the inequality by

-3

and reverse the direction of the inequality sign.

x + 4 \leq -3

Dividing an inequality by a negative number requires reversing the direction of the inequality symbol to maintain equivalence.

3

Subtract 4 from both sides of the inequality.

x \leq -7

This isolates the variable

x

on one side of the inequality, providing the final solution set.

Key Concept

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

Question 154Question →

In the $xy$ -plane, the graph of a linear function $f$ has a $y$ -intercept of $(0, r)$ and an $x$ -intercept of $(s, 0)$ , where $r$ and $s$ are positive constants. The line $y = -2x$ is perpendicular to the line that passes through the origin $(0,0)$ and the midpoint of the segment connecting the two intercepts of $f$ . If $r = 12$ , what is the value of $s$ ?

Show answer & explanation

Answer: 24

Answer

24

The midpoint of the segment connecting

(0, 12)

and

(s, 0)

is

(\frac{s}{2}, 6)

. The line passing through the origin

(0, 0)

and this midpoint has a slope of

\frac{6}{s/2} = \frac{12}{s}

. Since this line is perpendicular to the line

y = -2x

, which has a slope of

-2

, its slope must be the negative reciprocal of

-2

, which is

\frac{1}{2}

. Equating these two slopes, we get

\frac{12}{s} = \frac{1}{2}

, which simplifies to

s = 24

.

Step-by-Step Solution

1

Find the coordinates of the intercepts and their midpoint.

The intercepts are

(0, 12)

and

(s, 0)

, and their midpoint is

(\frac{s}{2}, 6)

.

The intercepts of the function

f

form a line segment whose midpoint must be calculated.

2

Calculate the slope of the line passing through the origin and the midpoint.

The slope is

\frac{12}{s}

.

A line passing through the origin

(0,0)

and a point

(x_1, y_1)

has a slope of

\frac{y_1}{x_1}

.

3

Relate the slope of the line to the perpendicular line

y = -2x

.

The slope of the line must be

\frac{1}{2}

.

Perpendicular lines have slopes that are negative reciprocals of each other, and the negative reciprocal of

-2

is

\frac{1}{2}

.

4

Solve for the value of

s

.

s = 24

.

Setting the two expressions for the slope equal,

\frac{12}{s} = \frac{1}{2}

, yields

s = 24

.

Key Concept

Linear Functions and Graphs

Estimated Time:2m 30s

Question 155Question →

A student wants to buy $x$ notebooks and $y$ pens. The student must buy at least $3$ notebooks. Each notebook costs $3$ dollars and each pen costs $2$ dollars. If the student can spend a maximum of $15$ dollars, what is the maximum number of pens the student can buy?

Show answer & explanation

Answer: 3

Answer

3

The correct answer is 3. The constraints are represented by the system of inequalities

x \geq 3

and

3x + 2y \leq 15

. To find the maximum number of pens

y

, we minimize the number of notebooks

x

by setting

x = 3

. Substituting

x = 3

into the budget inequality gives

3(3) + 2y \leq 15

, which simplifies to

9 + 2y \leq 15

. Subtracting

9

from both sides results in

2y \leq 6

. Dividing by

2

yields

y \leq 3

. Therefore, the maximum number of pens the student can buy is 3.

Step-by-Step Solution

1

Set up the system of inequalities from the given constraints.

The system of inequalities is

x \geq 3

and

3x + 2y \leq 15

, where

x

represents the number of notebooks and

y

represents the number of pens.

To represent the minimum number of notebooks required and the total budget limit.

2

Substitute the minimum possible value of

x

into the budget inequality to maximize

y

.

Setting

x = 3

gives

3(3) + 2y \leq 15

, which simplifies to

9 + 2y \leq 15

.

Since the cost per notebook is positive, minimizing the number of notebooks leaves the maximum amount of budget for purchasing pens.

3

Solve the inequality for

y

.

2y \leq 6 \implies y \leq 3

.

To find the upper bound for the number of pens the student can buy.

Key Concept

Solving a system of linear inequalities in a real-world context to find a maximum value.

Question 156Question →

In the $xy$ -plane, a point $(x, y)$ is a solution to the system of inequalities below.

y \leq -3x + 14

y \leq 4x - 7

What is the maximum possible value of $y$ ?

Show answer & explanation

Answer: 5

Answer

The maximum possible value of

y

is 5.

The maximum value of

y

is found at the intersection of the two boundary lines,

y = -3x + 14

and

y = 4x - 7

. Solving the system of equations gives

x = 3

and

y = 5

. Because the inequalities are both of the 'less than or equal to' type, any point in the solution set must lie on or below both lines. For

x < 3

, the value of

y

is limited by

y \leq 4x - 7

, which is strictly less than 5. For

x > 3

, the value of

y

is limited by

y \leq -3x + 14

, which is also strictly less than 5. Thus, the maximum value of

y

is 5.

Step-by-Step Solution

1

Set the two boundary equations equal to find their intersection point.

-3x + 14 = 4x - 7

The maximum value of

y

under these constraints occurs at the intersection of the two boundary lines.

2

Solve the equation for

x

.

x = 3

Isolating

x

determines the x-coordinate of the intersection point.

3

Substitute

x = 3

back into one of the boundary equations to solve for

y

.

y = 5

This yields the y-coordinate of the intersection point, which is the upper bound for

y

in the solution set.

Key Concept

Determining the maximum value of a variable in the solution set of a system of linear inequalities by finding the intersection of their boundary lines.

Estimated Time:1m 30s

Question 157Question →

In the $xy$ -plane, a line has an $x$ -intercept of $(k + 2, 0)$ and a $y$ -intercept of $(0, 2k + 2)$ , where $k$ is a positive constant. If the line passes through the point $(3, 5)$ , what is the value of $k$ ?

1.5

2.5

4

6.5

Show answer & explanation

Answer: 4

Answer

4

The correct answer is the value 4. By substituting the point

(3, 5)

into the intercept form of the line equation

\frac{x}{k+2} + \frac{y}{2k+2} = 1

, we obtain

\frac{3}{k+2} + \frac{5}{2k+2} = 1

. Multiplying by the common denominator

(k+2)(2k+2)

and simplifying results in the quadratic equation

2k^2 - 5k - 12 = 0

. Factoring this equation yields

(2k+3)(k-4) = 0

. Since

k

must be a positive constant, we select

k = 4

.

Step-by-Step Solution

1

Write the equation of the line using intercept form.

The equation of a line with

x

-intercept

(k+2, 0)

and

y

-intercept

(0, 2k+2)

is

\frac{x}{k+2} + \frac{y}{2k+2} = 1

.

Intercept form

\frac{x}{a} + \frac{y}{b} = 1

is the most direct way to represent a line when both intercepts are given.

2

Substitute the given point

(3, 5)

into the line's equation.

Substituting

x = 3

and

y = 5

yields the equation

\frac{3}{k+2} + \frac{5}{2k+2} = 1

.

Since the line passes through

(3, 5)

, these coordinates must satisfy the equation of the line.

3

Eliminate the denominators by multiplying both sides of the equation by the least common denominator.

Multiplying by

(k+2)(2k+2)

gives

3(2k+2) + 5(k+2) = (k+2)(2k+2)

.

This clears the fractions and allows us to rewrite the equation in polynomial form.

4

Expand both sides of the equation and combine like terms.

Expanding both sides gives

6k + 6 + 5k + 10 = 2k^2 + 2k + 4k + 4 \implies 11k + 16 = 2k^2 + 6k + 4

.

To solve a polynomial equation, we need to simplify the algebraic expressions on both sides.

5

Rearrange the equation into standard quadratic form.

Moving all terms to one side gives

2k^2 - 5k - 12 = 0

.

Standard quadratic form

Ax^2 + Bx + C = 0

is required to apply factoring or the quadratic formula.

6

Factor and solve the quadratic equation.

(2k+3)(k-4) = 0

, which gives

k = -1.5

or

k = 4

.

Factoring determines the values of

k

that satisfy the quadratic relation.

7

Apply the constraint that

k

is a positive constant.

Since

k > 0

, we discard

k = -1.5

, leaving

k = 4

.

The problem specifies that

k

must be a positive constant.

Key Concept

Using the intercepts of a line to formulate its equation and solving the resulting rational/quadratic equation given a point on the line.

Alternative Method

Instead of using the intercept form of the line, you can equate the slope calculated between the y-intercept

(0, 2k+2)

and the point

(3,5)

to the slope calculated between the point

(3,5)

and the x-intercept

(k+2,0)

. This yields the equation

\frac{5 - (2k+2)}{3 - 0} = \frac{0 - 5}{(k+2) - 3}

, which simplifies to

\frac{3 - 2k}{3} = \frac{-5}{k - 1}

. Cross-multiplying and simplifying leads to the same quadratic equation,

2k^2 - 5k - 12 = 0

.

Estimated Time:2m 30s

Question 158Question →

In the $xy$ -plane, the graph of the linear function $f$ passes through the point $(2, 5)$ and has a $y$ -intercept of $(0, b)$ , where $b$ is a constant. The function $g$ is defined by $g(x) = f(x) - 4$ . If the $x$ -intercept of the graph of $g$ is $(6, 0)$ , what is the value of $b$ ?

10

\frac{19}{2}

\frac{11}{2}

\frac{11}{4}

Show answer & explanation

Answer: $\frac{11}{2}$

Answer

The correct value of

b

is

\frac{11}{2}

.

The correct answer is the value

\frac{11}{2}

. By writing

f(x) = mx + b

and using the point

(2, 5)

, we get

m = \frac{5-b}{2}

. Applying the definition of

g(x) = f(x) - 4

gives

g(x) = mx + b - 4

. Since the graph of

g

has an

x

-intercept of

(6, 0)

, substituting

x = 6

and

g(6) = 0

gives

6m + b - 4 = 0

. Substituting the expression for

m

yields

3(5-b) + b - 4 = 0

, which simplifies to

11 - 2b = 0

, and thus

b = \frac{11}{2}

.

Step-by-Step Solution

1

Write the linear function

f(x)

in slope-intercept form using the given

y

-intercept

(0, b)

.

f(x) = mx + b

, where

m

is the slope of the line.

This sets up the general equation of the line with unknown parameters

m

and

b

.

2

Substitute the coordinates of the point

(2, 5)

into the equation for

f(x)

to express

m

in terms of

b

.

5 = 2m + b

, which simplifies to

m = \frac{5 - b}{2}

.

Since the point lies on the graph of

f

, its coordinates must satisfy the equation.

3

Define

g(x)

using the relationship

g(x) = f(x) - 4

and apply the

x

-intercept

(6, 0)

.

g(x) = mx + b - 4

. Since the

x

-intercept is

(6, 0)

, we have

g(6) = 0

, which gives

6m + b - 4 = 0

.

The

x

-intercept is the point where the output of the function is zero.

4

Substitute the expression for

m

from Step 2 into the equation from Step 3 and solve for

b

.

6\left(\frac{5-b}{2}\right) + b - 4 = 0 \Rightarrow 3(5-b) + b - 4 = 0 \Rightarrow 15 - 3b + b - 4 = 0 \Rightarrow 11 - 2b = 0 \Rightarrow b = \frac{11}{2}

.

This solves the system of equations to determine the value of the constant

b

.

Key Concept

Linear Functions and Graphs

Question 159Question →

The table below shows some values of the linear function $h$ .

$x$	$h(x)$
$2$	$k - 4$
$5$	$k + 8$
$8$	$2k + 2$

If $k$ is a constant, what is the value of $k$ ?

Show answer & explanation

Answer: 18

Answer

The value of the constant

k

is

18

.

Since the function

h

is linear, its rate of change (slope) is constant. Calculating the slope using the first two coordinate pairs

(2, k - 4)

and

(5, k + 8)

gives

\frac{(k + 8) - (k - 4)}{5 - 2} = \frac{12}{3} = 4

. Using the next two coordinate pairs

(5, k + 8)

and

(8, 2k + 2)

gives the slope as

\frac{(2k + 2) - (k + 8)}{8 - 5} = \frac{k - 6}{3}

. Setting these two slope values equal to each other gives the equation

\frac{k - 6}{3} = 4

. Multiplying both sides by

3

results in

k - 6 = 12

, and adding

6

to both sides yields

k = 18

.

Step-by-Step Solution

1

Identify that the rate of change (slope) of a linear function is constant between any two points.

The slope calculated from the first two points must equal the slope calculated from the second and third points.

This relationship allows us to set up an algebraic equation to solve for the unknown constant

k

.

2

Calculate the slope using the first two points:

(2, k - 4)

and

(5, k + 8)

.

Slope =

\frac{(k + 8) - (k - 4)}{5 - 2} = \frac{k + 8 - k + 4}{3} = \frac{12}{3} = 4

.

This simplifies to a constant numerical value of

4

for the slope of the function.

3

Calculate the slope using the second and third points:

(5, k + 8)

and

(8, 2k + 2)

.

Slope =

\frac{(2k + 2) - (k + 8)}{8 - 5} = \frac{2k + 2 - k - 8}{3} = \frac{k - 6}{3}

.

This provides a second expression for the slope in terms of the variable

k

.

4

Equate the two slope expressions and solve for

k

.

\frac{k - 6}{3} = 4 \implies k - 6 = 12 \implies k = 18

.

Setting the two expressions equal and solving isolating

k

gives the correct value of

18

.

Key Concept

A linear function has a constant rate of change (slope) between any two points on its graph.

Question 160Question →

A local delivery service charges a flat fee of $12$ dollars plus $1.50$ dollars per mile to deliver a package. If a customer wants to spend no more than $30$ dollars for a package delivery, what is the maximum number of miles the delivery service can travel?

Show answer & explanation

Answer: 12

Answer

The maximum number of miles the delivery service can travel is 12.

The total delivery cost is the sum of the flat fee (

12

dollars) and the rate per mile (

1.50

dollars multiplied by

m

miles), which is represented by

1.50m + 12

. Since the total cost cannot exceed

30

dollars, the inequality is

1.50m + 12 \leq 30

. Subtracting

12

from both sides gives

1.50m \leq 18

. Dividing both sides by

1.50

yields

m \leq 12

. Thus, the maximum distance the delivery service can travel is

12

miles.

Step-by-Step Solution

1

Set up the inequality representing the delivery cost constraint.

1.50m + 12 \leq 30

The total cost of the delivery is the flat fee of

12

dollars plus

1.50

dollars per mile,

m

, which must be less than or equal to the budget of

30

dollars.

2

Subtract 12 from both sides of the inequality.

1.50m \leq 18

To isolate the variable term, subtract the constant flat fee from both sides of the inequality.

3

Divide both sides of the inequality by 1.50.

m \leq 12

Dividing by the per-mile rate calculates the maximum distance constraint on the variable

m

.

Key Concept

Setting up and solving a one-variable linear inequality to determine a maximum boundary value in context.

Algebra

Answer

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Alternative Method

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