Linear Inequalities Worksheet with Answers

A theater is selling adult tickets and child tickets for a weekend show. Each adult ticket costs $18, and each child ticket costs $11. Let $a$ represent the number of adult tickets sold and $c$ represent the number of child tickets sold. If the total amount collected from ticket sales must be at least $540, which inequality represents this situation?

$18a+11c<540$
$11a+18c\ge540$
$18a+11c\ge540$
$11a+18c<540$

A community garden can grow no more than 18 tomato plants and pepper plants combined. Let $t$ represent the number of tomato plants and let $p$ represent the number of pepper plants. Which inequality models this situation?

$t+p\ge18$
$t-p\le18$
$t+p\le18$
$t-p\ge18$

A community garden can grow no fewer than 18 tomato plants and pepper plants combined. Let $t$ represent the number of tomato plants and let $p$ represent the number of pepper plants. Which inequality represents this situation?

$t+p\le 18$
$t-p\ge 18$
$t+p\ge 18$
$t-p\le 18$

A community center requires at least $75$ volunteer hours each month to qualify for a city grant. This month, the center has recorded $63$ volunteer hours so far. If $x$ represents the additional volunteer hours needed, which value of $x$ is the least that will allow the center to qualify for the grant?

A factory machine fills between 18 and 24 boxes each hour, inclusive. The machine runs for 7 hours, and $b$ represents the total number of boxes filled during that time. Which inequality represents the possible values of $b$?

$18+7 \leq b \leq 24+7$
$18 \leq 7b \leq 24$
$(18)(7) \leq b \leq (24)(7)$
$18 \leq 7+b \leq 24$

A delivery truck transports between 18 and 24 packages each hour, inclusive. After 7 hours, the total number of packages transported is represented by $p$. Which inequality represents the possible values of $p$?

$18+7 \leq p \leq 24+7$
$18 \leq 7p \leq 24$
$(18)(7) \leq p \leq (24)(7)$
$18 \leq p+7 \leq 24$

A community art center charges a one-time registration fee of $18.50 for a workshop series. In addition, each workshop session costs $7.25. Maya has at most $85 to spend on the series.

If $w$ represents the number of workshop sessions Maya can attend, what is the greatest whole-number value of $w$ that satisfies the budget constraint?

A community art studio charges a one-time registration fee of $\$18.50$ plus $\$7.25$ for each pottery class a student attends. A student has no more than $\$95$ to spend on the studio.

What is the greatest whole number of pottery classes the student can attend without exceeding the budget?

Which table contains only ordered pairs $(x,y)$ that satisfy the inequality $y<2x-3$?

x y
-1 -5
3 2
5 6
x y
2 0
4 3
6 10
x y
2 -1
4 2
6 8
x y
2 4
4 7
6 12

x	y
-1	-5
3	2
5	6

x	y
2	0
4	3
6	10

x	y
2	4
4	7
6	12

A theater manager is ordering snacks for an event. Bags of popcorn cost 4 dollars each, and bottles of juice cost 3 dollars each. The manager wants to spend at most 90 dollars, so the inequality $4p+3j\leq 90$ is used, where $p$ is the number of bags of popcorn and $j$ is the number of bottles of juice.

Which choice best describes the meaning of the expression $3j$ in this situation?

The amount spent on one bottle of juice
The total amount spent on bags of popcorn
The total amount spent on bottles of juice
The total number of bottles of juice and bags of popcorn combined

A community theater is selling adult tickets and student tickets for a show. Let $a$ represent the number of adult tickets sold and $s$ represent the number of student tickets sold. The theater wants the total ticket revenue to be at least $900, so the inequality $18a+12s\geq 900$ is used.

Which choice best describes the meaning of the term $12s$ in this context?

The amount charged for one student ticket
The total amount of money earned from selling adult tickets
The total amount of money earned from selling student tickets
The total number of student tickets and adult tickets sold

A community theater sold 145 standard tickets and 210 balcony tickets for an upcoming performance. Let $p$ represent the price, in dollars, of one balcony ticket. The price of a standard ticket was $6 more than the price of a balcony ticket. The theater collected at least $5,910 from these ticket sales.

Which inequality represents this situation?

$145(p-6)+210p\le5910$
$145(p+6)+210p\le5910$
$145(p+6)+210p\ge5910$
$145(p-6)+210p\ge5910$

Answer key and explanations

1. Answer

$18a+11c\ge540$

Each adult ticket contributes 18 dollars, so the amount from adult tickets is $18a$. Each child ticket contributes 11 dollars, so the amount from child tickets is $11c$. Adding these gives the total revenue: $18a+11c$. The phrase “at least $540” means the total is greater than or equal to 540, so the correct inequality is $18a+11c\ge540$.

2. Answer

$t+p\le18$

The phrase “no more than 18” means the total cannot exceed 18, so the inequality should use $\le$. Because the total number of plants is the sum of tomato plants and pepper plants, the correct expression is $t+p\le18$.

3. Answer

$t+p\ge 18$

The phrase “no fewer than 18” means the total number must be at least 18. Since the total combines tomato plants and pepper plants, the variables should be added: $t+p$. Therefore, the correct inequality is $t+p\ge 18$.

4. Answer

The situation can be modeled with the inequality $63 + x \ge 75$ because the total volunteer hours must be at least 75.

Subtract 63 from both sides:

$x \ge 12$

The least additional number of volunteer hours needed is $12$.

5. Answer

$(18)(7) \leq b \leq (24)(7)$

The machine fills between 18 and 24 boxes per hour. Since it runs for 7 hours, both ends of the interval must be multiplied by 7. This gives $(18)(7) \leq b \leq (24)(7)$. Because 7 is positive, the inequality directions stay the same.

6. Answer

$(18)(7) \leq p \leq (24)(7)$

The truck transports between 18 and 24 packages each hour. Since the truck works for 7 hours, multiply both ends of the interval by 7.

$(18)(7) \leq p \leq (24)(7)$

Because 7 is positive, the inequality directions stay the same. Therefore, the correct choice is C.

7. Answer

The total cost is the one-time registration fee plus the cost per workshop session, so the inequality is $18.50+7.25w\leq 85$.

Subtract 18.50 from both sides:

$7.25w\leq 66.5$

Divide both sides by 7.25:

$w\leq \frac{66.5}{7.25}\approx 9.17$

Because Maya can only attend a whole number of workshop sessions and cannot exceed the budget, the greatest possible whole-number value is $9$.

8. Answer

Let $c$ represent the number of pottery classes. The total cost is the registration fee plus the cost per class, so the inequality is:

$18.50+7.25c\leq95$

Subtract $18.50$ from both sides:

$7.25c\leq76.50$

Divide both sides by $7.25$:

$c\leq10.55...$

Since the number of classes must be a whole number and the budget cannot be exceeded, the greatest possible number of classes is $10$.

9. Answer

x	y
2	-1
4	2
6	8

Substitute each ordered pair into the inequality $y<2x-3$.

Choice A: For $(-1,-5)$, $-5<2(-1)-3=-5$ is false because the inequality is strict and $-5$ is equal to $-5$, not less than it. So A is not correct.

Choice B: For $(2,0)$, $0<2(2)-3=1$ is true. For $(4,3)$, $3<5$ is true. For $(6,10)$, $10<9$ is false. So B is not correct.

Choice C: For $(2,-1)$, $-1<1$ is true. For $(4,2)$, $2<5$ is true. For $(6,8)$, $8<9$ is true. All rows satisfy the inequality.

Choice D: For $(2,4)$, $4<1$ is false, so D is not correct.

Therefore, the correct answer is Choice C.

10. Answer

The total amount spent on bottles of juice

In the inequality $4p+3j\leq 90$, the coefficient 3 represents the cost of one bottle of juice, and $j$ represents the number of bottles of juice. Therefore, $3j$ represents the total amount of money spent on juice bottles.

11. Answer

The total amount of money earned from selling student tickets

In the inequality $18a+12s\geq 900$, the coefficient 12 represents the price of one student ticket in dollars, and $s$ represents the number of student tickets sold. Therefore, the product $12s$ represents the total amount of money earned from selling student tickets.

12. Answer

$145(p+6)+210p\ge5910$

Since $p$ represents the price of one balcony ticket, the price of one standard ticket is $p+6$. Revenue from the 145 standard tickets is $145(p+6)$, and revenue from the 210 balcony tickets is $210p$. Because the theater collected at least $5,910, the total revenue must be greater than or equal to 5,910. Therefore, the correct inequality is $145(p+6)+210p\ge5910$.