Math worksheet with algebraic equations and a riddle theme to engage students in solving inequalities.

It looks like you've uploaded a worksheet titled "HW 25: Solving Inequalities", and it contains various inequality problems to solve and graph. However, since I can't view images directly, I'll help you solve the problems based on what you've described or transcribed.

Let me go through each problem step by step and explain how to solve them. If you have specific questions about any of them, let me know!

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Problem 1:

Solve and Graph:
$ 3x + 2 > 4x + 10 $

Step-by-step Solution:

1. Subtract $3x$ from both sides:
$$
2 > x + 10
$$

2. Subtract 10 from both sides:
$$
-8 > x \quad \text{or} \quad x < -8
$$

Graph: Draw an open circle at $-8$ and shade to the left.

✔ Answer: $ x < -8 $

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Problem 2:

$ 6(x + 3) < 4(x - 7) + 3 $

Step-by-step:

1. Expand both sides:
$$
6x + 18 < 4x - 28 + 3
$$
$$
6x + 18 < 4x - 25
$$

2. Subtract $4x$:
$$
2x + 18 < -25
$$

3. Subtract 18:
$$
2x < -43
$$

4. Divide by 2:
$$
x < -\frac{43}{2} = -21.5
$$

✔ Answer: $ x < -21.5 $

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Problem 3:

$ 2(3x - 4) > 9(x + 1) $

1. Expand:
$$
6x - 8 > 9x + 9
$$

2. Subtract $6x$:
$$
-8 > 3x + 9
$$

3. Subtract 9:
$$
-17 > 3x
$$

4. Divide by 3:
$$
x < -\frac{17}{3} \approx -5.67
$$

✔ Answer: $ x < -\frac{17}{3} $

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Problem 4:

$ 3x + 40 ≥ 35 - x $

1. Add $x$ to both sides:
$$
4x + 40 ≥ 35
$$

2. Subtract 40:
$$
4x ≥ -5
$$

3. Divide by 4:
$$
x ≥ -\frac{5}{4} = -1.25
$$

✔ Answer: $ x ≥ -1.25 $

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Problem 5:

$ 2x + 14 - 2(x - 5) < -40 $

1. Expand:
$$
2x + 14 - 2x + 10 < -40
$$

2. Simplify:
$$
24 < -40 \quad \text{✘ FALSE}
$$

This is a contradiction — no solution.

✔ Answer: No solution (empty set)

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Problem 6:

$ 3(x - 7) > 18 $

1. Distribute:
$$
3x - 21 > 18
$$

2. Add 21:
$$
3x > 39
$$

3. Divide:
$$
x > 13
$$

✔ Answer: $ x > 13 $

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Problem 7:

$ 35 + 2(3x + 3) > 8(1 - x) $

1. Expand:
$$
35 + 6x + 6 > 8 - 8x
$$
$$
41 + 6x > 8 - 8x
$$

2. Add $8x$:
$$
41 + 14x > 8
$$

3. Subtract 41:
$$
14x > -33
$$

4. Divide:
$$
x > -\frac{33}{14} \approx -2.36
$$

✔ Answer: $ x > -\frac{33}{14} $

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Problem 8:

$ 35 + 70(4x + 1) $

Wait — this is not an inequality. It's an expression.

But if it’s meant to be part of an inequality (e.g., $ 35 + 70(4x + 1) > 0 $), please clarify.

Assuming it's just simplifying:

$$
35 + 70(4x + 1) = 35 + 280x + 70 = 280x + 105
$$

But without an inequality sign, we can’t solve.

⚠️ Clarification needed — is there a missing inequality?

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Problem 9:

$ 58 - 3(5) ≥ 3 + 14(4x + 1) $

Wait — this has no variable in the first term?

Let’s simplify:

Left side:
$$
58 - 15 = 43
$$

Right side:
$$
3 + 14(4x + 1) = 3 + 56x + 14 = 56x + 17
$$

So:
$$
43 ≥ 56x + 17
$$

Subtract 17:
$$
26 ≥ 56x
$$

Divide:
$$
x ≤ \frac{26}{56} = \frac{13}{28} \approx 0.464
$$

✔ Answer: $ x ≤ \frac{13}{28} $

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Problem 10:

$ 4(2x - 9) > 4 + 11x $

1. Expand:
$$
8x - 36 > 4 + 11x
$$

2. Subtract $8x$:
$$
-36 > 4 + 3x
$$

3. Subtract 4:
$$
-40 > 3x
$$

4. Divide:
$$
x < -\frac{40}{3} \approx -13.33
$$

✔ Answer: $ x < -\frac{40}{3} $

---

Problem 11:

$ 7(3x - 4) - (x + 15) < 60 $

1. Expand:
$$
21x - 28 - x - 15 < 60
$$
$$
20x - 43 < 60
$$

2. Add 43:
$$
20x < 103
$$

3. Divide:
$$
x < \frac{103}{20} = 5.15
$$

✔ Answer: $ x < 5.15 $

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Problem 12:

$ 8(4x - 4) + 12(2x - 5) $

Again — this is an expression, not an inequality.

If it's meant to be $ \geq 0 $, for example:

$$
8(4x - 4) + 12(2x - 5) = 32x - 32 + 24x - 60 = 56x - 92
$$

Then:
$$
56x - 92 \geq 0 \Rightarrow x \geq \frac{92}{56} = \frac{23}{14} \approx 1.64
$$

But again — need clarification.

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Problem 13 (Word Problem):

> Suppose you write a book. The printer charges $8 per book to print it, and you spend $4000 on advertising. You sell the book for $12 a copy. How many copies must you sell so that your income from sales is greater than your total cost?

Let $ x $ = number of books sold

- Income: $ 12x $
- Cost: $ 8x + 4000 $ (printing + ad)

We want:
$$
12x > 8x + 4000
$$

Subtract $8x$:
$$
4x > 4000
$$

Divide:
$$
x > 1000
$$

✔ Answer: You must sell more than 1000 copies.

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Summary of Answers:

| Problem | Answer |
|--------|--------|
| 1 | $ x < -8 $ |
| 2 | $ x < -21.5 $ |
| 3 | $ x < -\frac{17}{3} $ |
| 4 | $ x \geq -1.25 $ |
| 5 | No solution |
| 6 | $ x > 13 $ |
| 7 | $ x > -\frac{33}{14} $ |
| 8 | Expression: $ 280x + 105 $ → needs inequality |
| 9 | $ x \leq \frac{13}{28} $ |
| 10 | $ x < -\frac{40}{3} $ |
| 11 | $ x < 5.15 $ |
| 12 | Expression: $ 56x - 92 $ → needs inequality |
| 13 | $ x > 1000 $ |

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Let me know if you'd like me to graph any of these inequalities or help with formatting for your homework!