One Variable Inequalities Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve each of these one-variable inequality word problems step by step, write the inequality, solve it, and describe how to graph the solution.

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Problem 1:

> A boat can hold at most 1,000 pounds. If there is 300 pounds of equipment plus 25-pound boxes, what is the maximum number of boxes the ship can carry?

#### Step 1: Define the variable
Let $ x $ = number of 25-pound boxes.

#### Step 2: Write the inequality
Total weight = equipment + boxes
$ 300 + 25x \leq 1000 $

#### Step 3: Solve
$$
25x \leq 700 \\
x \leq 28
$$

#### ✔ Answer:
The boat can carry at most 28 boxes.

#### Graph:
Draw a number line. Place a closed circle at 28 (since it’s "at most"), and shade everything to the left (all values less than or equal to 28).

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Problem 2:

> A hotel costs $100 per night plus a $35 hotel fee. If George has $500, what is the maximum number of nights he could stay?

#### Step 1: Define the variable
Let $ n $ = number of nights.

#### Step 2: Write the inequality
Cost = $100n + 35 \leq 500$

#### Step 3: Solve
$$
100n \leq 465 \\
n \leq 4.65
$$

Since $ n $ must be a whole number (can't stay a fraction of a night), the maximum number of full nights is 4.

✔ Answer: George can stay at most 4 nights.

#### Graph:
Closed circle at 4, shade left (all values ≤ 4). But since we're dealing with whole nights, only integers from 0 to 4 are valid.

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Problem 3:

> The school paper says that no less than a third of the juniors went to prom. If 37 juniors went, what’s the minimum amount of juniors at the school?

#### Step 1: Define the variable
Let $ j $ = total number of juniors.

#### Step 2: Write the inequality
"no less than a third" means $ \frac{1}{3}j \leq 37 $? Wait — actually, "no less than" means at least, so:

$$
\frac{1}{3}j \geq 37
$$

#### Step 3: Solve
$$
j \geq 37 \times 3 = 111
$$

✔ Answer: There are at least 111 juniors in the school.

#### Graph:
Closed circle at 111, shade to the right (all values ≥ 111).

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Problem 4:

> Kyle’s doctor told him that in order to lose weight, he needs to eat less than 2,000 calories a day. He eats 875 calories for breakfast and then 3 protein smoothies. How many calories can the protein smoothies have to reach his goal?

#### Step 1: Define the variable
Let $ s $ = calories in each smoothie.

So 3 smoothies = $ 3s $

#### Step 2: Write the inequality
Total calories < 2000
$$
875 + 3s < 2000
$$

#### Step 3: Solve
$$
3s < 1125 \\
s < 375
$$

✔ Answer: Each smoothie must have less than 375 calories.

#### Graph:
Open circle at 375 (since it’s “less than”, not “equal”), shade to the left.

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Problem 5:

> 2 less than 3 times a number is greater than 10.

#### Step 1: Define the variable
Let $ x $ = the number.

#### Step 2: Translate
"3 times a number" = $ 3x $
"2 less than" = $ 3x - 2 $
"is greater than 10" → $ 3x - 2 > 10 $

#### Step 3: Solve
$$
3x > 12 \\
x > 4
$$

✔ Answer: The number must be greater than 4.

#### Graph:
Open circle at 4, shade to the right.

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Problem 6:

> Ben is twice his brother’s age. If the sum of their ages is below 81, how old could Ben’s brother be?

#### Step 1: Define the variable
Let $ b $ = brother’s age.
Then Ben’s age = $ 2b $

#### Step 2: Write the inequality
Sum of ages < 81
$$
b + 2b < 81 \\
3b < 81
$$

#### Step 3: Solve
$$
b < 27
$$

✔ Answer: Ben’s brother must be less than 27 years old.

#### Graph:
Open circle at 27, shade to the left.

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✔ Final Answers Summary:

| Problem | Inequality | Solution | Graph Description |
|--------|------------|---------|-------------------|
| 1 | $ 300 + 25x \leq 1000 $ | $ x \leq 28 $ | Closed circle at 28, shade left |
| 2 | $ 100n + 35 \leq 500 $ | $ n \leq 4.65 $ → $ n \leq 4 $ | Closed circle at 4, shade left |
| 3 | $ \frac{1}{3}j \geq 37 $ | $ j \geq 111 $ | Closed circle at 111, shade right |
| 4 | $ 875 + 3s < 2000 $ | $ s < 375 $ | Open circle at 375, shade left |
| 5 | $ 3x - 2 > 10 $ | $ x > 4 $ | Open circle at 4, shade right |
| 6 | $ b + 2b < 81 $ | $ b < 27 $ | Open circle at 27, shade left |

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