Inequalities Word Problems worksheet with four math questions related to real-world scenarios.

Explanation:
Let’s solve each problem one by one, step by step.

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Problem 1:
*The sum of twice a number and 5 is at most 15. What are the possible values for the number?*

Let the number be $ x $.
“Twice a number” = $ 2x $
“Sum of twice a number and 5” = $ 2x + 5 $
“At most 15” means ≤ 15.

So the inequality is:
$$
2x + 5 \leq 15
$$

Subtract 5 from both sides:
$$
2x \leq 10
$$

Divide both sides by 2:
$$
x \leq 5
$$

✔ So all numbers less than or equal to 5 work.
Possible values: any real number ≤ 5 (e.g., 5, 4, 0, −10, etc.). If only integers are expected, then integers ≤ 5.

But since the question says “possible values”, and doesn’t restrict to integers, the answer is:
All real numbers $ x $ such that $ x \leq 5 $.

We’ll write this in interval notation or inequality form as needed later.

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Problem 2:
*Two-thirds of a number plus 5 is greater than 12. Find the number.*

Let the number be $ x $.
“Two-thirds of a number” = $ \frac{2}{3}x $
“Plus 5” = $ \frac{2}{3}x + 5 $
“Greater than 12” → > 12

Inequality:
$$
\frac{2}{3}x + 5 > 12
$$

Subtract 5:
$$
\frac{2}{3}x > 7
$$

Multiply both sides by $ \frac{3}{2} $ (the reciprocal of $ \frac{2}{3} $) to isolate $ x $:
$$
x > 7 \cdot \frac{3}{2} = \frac{21}{2} = 10.5
$$

✔ So the number must be greater than 10.5.

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Problem 3:
*The quotient of a number and −15 is no greater than 450. What are the possible values for the number?*

Let the number be $ x $.
“Quotient of a number and −15” = $ \frac{x}{-15} $
“No greater than 450” means ≤ 450.

So:
$$
\frac{x}{-15} \leq 450
$$

Now, multiply both sides by −15.
⚠️ Important: When multiplying or dividing an inequality by a negative number, the inequality sign flips.

So:
$$
x \geq 450 \cdot (-15)
$$

Calculate:
$$
450 \cdot 15 = (400 + 50) \cdot 15 = 6000 + 750 = 6750
\Rightarrow 450 \cdot (-15) = -6750
$$

So:
$$
x \geq -6750
$$

✔ Possible values: all real numbers ≥ −6750.

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Problem 4:
*A store is offering a rebate of $30 on all color printers. David is looking for different color printers that range in price from $175 to $260. How much does David need to spend after the rebate?*

Rebate = $30 off.
So after rebate, price = original price − 30.

Minimum original price = $175 → after rebate: $175 − 30 = $145
Maximum original price = $260 → after rebate: $260 − 30 = $230

So David will spend between $145 and $230, inclusive.

The question asks: *How much does David need to spend after the rebate?*
Since printers range from $175 to $260, after rebate the range is $145 to $230.

So answer: from $145 to $230, or in inequality: $145 \leq \text{cost} \leq 230$.

But if they want a single number — no, it's a range. So we report the range.

Now let’s double-check all calculations:

1. $2x + 5 \leq 15$ → $2x \leq 10$ → $x \leq 5$ ✔
2. $\frac{2}{3}x + 5 > 12$ → $\frac{2}{3}x > 7$ → $x > 10.5$ ✔
3. $\frac{x}{-15} \leq 450$ → multiply by −15 (flip): $x \geq -6750$ ✔
Check: try $x = -6750$: $-6750 / -15 = 450$ → equals 450, allowed (≤ 450). Try $x = -6751$: $-6751 / -15 ≈ 450.066... > 450$ → not allowed. So yes, $x \geq -6750$ is correct.
4. $175−30=145$, $260−30=230$ ✔

All correct.

Final Answer:
1) $ x \leq 5 $
2) $ x > 10.5 $
3) $ x \geq -6750 $
4) Between \$145 and \$230, inclusive