Linear Inequalities in One Variable Practice Questions — SAT — Page 2

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 22Question →

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 23Question →

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 24Question →

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 25Question →

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 26Question →

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 27Question →

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

x \le 100

x \ge 150

x \ge 10

x \ge 100

Show answer & explanation

Answer: $x \ge 100$

Answer

The correct answer is

x \ge 100

.

The correct answer is the inequality stating that

x

is greater than or equal to

100

. Distributing

-3

to both terms inside the parentheses gives

1,200 - 6x - 150 \le 450

. Combining the constant terms on the left side results in

1,050 - 6x \le 450

. Subtracting

1,050

from both sides gives

-6x \le -600

. Dividing both sides by

-6

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

x \ge 100

.

Step-by-Step Solution

1

Substitute the formula for the remaining budget

B

into the inequality

B \le 450

.

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

To find the possible values of

x

when the budget is at most

450

.

2

Distribute

-3

to both terms inside the parentheses.

1,200 - 6x - 150 \le 450

To simplify the expression by removing the parentheses.

3

Combine the constant terms

1,200

and

-150

on the left side.

1,050 - 6x \le 450

To simplify the inequality further before isolating the variable.

4

Subtract

1,050

from both sides of the inequality.

-6x \le -600

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

5

Divide both sides of the inequality by

-6

and reverse the inequality sign.

x \ge 100

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

Key Concept

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

Question 28Question →

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

Show answer & explanation

Answer: 11

Answer

The maximum possible value of

x

is

11

.

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

-2

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

x \leq 11

. Thus, the maximum possible value of

x

is

11

.

Step-by-Step Solution

1

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

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

Multiplying

-3

by each term inside the parentheses

(2x - 5)

yields

-6x

and

+15

.

2

Combine the variable terms on the left side.

-2x + 15 \geq -7

Combining

-6x

and

4x

gives

-2x

.

3

Subtract

15

from both sides of the inequality.

-2x \geq -22

To isolate the variable term

-2x

on the left side.

4

Divide both sides by

-2

and flip the inequality sign.

x \leq 11

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

5

Determine the maximum value from the solution set.

11

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

11

, so the greatest value is

11

.

Key Concept

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

Question 29Question →

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

Show answer & explanation

Answer: 25

Answer

The minimum possible value of

x

is 25.

To find the minimum possible value of

x

, solve the inequality

450 - 8(15 + x) \leq 130

. First, distribute the

-8

to obtain

450 - 120 - 8x \leq 130

. Simplify the constant terms on the left side to get

330 - 8x \leq 130

. Subtract 330 from both sides, yielding

-8x \leq -200

. Finally, divide both sides by

-8

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

x \geq 25

. The minimum possible value of

x

is 25.

Step-by-Step Solution

1

Distribute the factor of

-8

to both terms inside the parentheses.

450 - 120 - 8x \leq 130

To remove the parentheses and prepare to combine like terms.

2

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

330 - 8x \leq 130

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

3

Subtract 330 from both sides of the inequality.

-8x \leq -200

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

4

Divide both sides by

-8

and reverse the direction of the inequality symbol.

x \geq 25

Dividing by a negative value reverses the inequality sign. Since

x

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

Key Concept

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

Question 30Question →

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

x > \frac{25}{3}

x < \frac{25}{9}

x < 3

x < \frac{25}{3}

Show answer & explanation

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

Answer

x < \frac{25}{3}

The correct inequality is obtained by first distributing

-2

to the terms inside the parentheses to get

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

. Combining the constants on the left yields

-6x + 13 > -3x - 12

. Adding

3x

to both sides results in

-3x + 13 > -12

, and subtracting 13 from both sides gives

-3x > -25

. Finally, dividing by

-3

and flipping the inequality sign results in the correct solution.

Step-by-Step Solution

1

Distribute

-2

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

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

Applying the distributive property simplifies the expression.

2

Combine the constant terms on the left side.

-6x + 13 > -3x - 12

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

3

Add

3x

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

-3x + 13 > -12

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

4

Subtract 13 from both sides to isolate the variable term.

-3x > -25

This isolates the term containing

x

.

5

Divide both sides by

-3

and reverse the direction of the inequality sign.

x < \frac{25}{3}

Dividing by a negative number requires reversing the inequality sign.

Key Concept

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

Question 31Question →

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

Show answer & explanation

Answer: 26

Answer

The minimum number of people in the group is 26.

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

125 + 9.5p < 50 + 12.5p

. Subtracting

9.5p

from both sides gives

125 < 50 + 3p

. Subtracting

50

from both sides yields

75 < 3p

. Dividing by

3

gives

p > 25

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

p

that is strictly greater than

25

is

26

.

Step-by-Step Solution

1

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

125 + 9.5p < 50 + 12.5p

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

2

Isolate the variable

p

by subtracting

9.5p

and

50

from both sides of the inequality.

p > 25

Subtracting

9.5p

yields

125 < 50 + 3p

. Subtracting

50

yields

75 < 3p

. Dividing by

3

yields

p > 25

.

3

Identify the minimum integer value of

p

that satisfies the inequality.

26

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

25

is

26

.

Key Concept

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

Estimated Time:1m 30s

Question 32Question →

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

x \leq -2

x \geq -2

x \leq -4

x \geq -4

Show answer & explanation

Answer: $x \leq -2$

Answer

The correct solution is

x

is less than or equal to

-2

.

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

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

. Combining the constants on the left side yields

7 - 6x \geq -4x + 11

. Adding four

x

to both sides results in

7 - 2x \geq 11

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

-2x \geq 4

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

x

is less than or equal to negative two.

Step-by-Step Solution

1

Distribute the

-2

to the terms inside the parentheses.

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

To simplify the expression by removing the parentheses.

2

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

7 - 6x \geq -4x + 11

To group like terms together before isolating the variable.

3

Add

4x

to both sides of the inequality.

7 - 2x \geq 11

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

4

Subtract

7

from both sides of the inequality.

-2x \geq 4

To isolate the variable term.

5

Divide both sides by

-2

and reverse the inequality sign.

x \leq -2

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

Key Concept

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

Estimated Time:1m 30s

Question 33Question →

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

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

Distributing the

-5

gives

-10x + 45 + 3 \geq 18

. Combining constants yields

-10x + 48 \geq 18

. Subtracting 48 from both sides gives

-10x \geq -30

. Finally, dividing both sides by

-10

and reversing the inequality sign gives

x \leq 3

. The maximum possible value is therefore 3.

Step-by-Step Solution

1

Distribute

-5

to the terms inside the parentheses.

-10x + 45 + 3 \geq 18

Simplify the expression by expanding the parentheses.

2

Combine the constant terms

45

and

3

on the left side.

-10x + 48 \geq 18

Group like terms together.

3

Subtract

48

from both sides of the inequality.

-10x \geq -30

Isolate the variable term.

4

Divide both sides by

-10

and reverse the inequality symbol.

x \leq 3

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

Key Concept

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

Question 34Question →

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

x > -\frac{1}{2}

x < -\frac{1}{2}

x < -\frac{7}{2}

x < -2

Show answer & explanation

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

Answer

The inequality is satisfied for all values of

x

such that

x < -\frac{1}{2}

.

Distributing

-\frac{2}{3}

across

(6x - 9)

yields

-4x + 6

. Adding 4 gives

-4x + 10 > 12

. Subtracting 10 from both sides results in

-4x > 2

. Dividing by

-4

and reversing the inequality sign yields

x < -\frac{1}{2}

.

Step-by-Step Solution

1

Distribute

-\frac{2}{3}

to both terms inside the parentheses:

(6x - 9)

.

-4x + 6 + 4 > 12

, which simplifies to

-4x + 10 > 12

.

Applying the distributive property removes the parentheses.

2

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

x

.

-4x > 2

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

3

Divide both sides by

-4

and reverse the inequality sign.

x < -\frac{1}{2}

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

Key Concept

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

Estimated Time:1m 15s

Question 35Question →

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 36Question →

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 37Question →

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 38Question →

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 39Question →

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 40Question →

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.

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