Free Classroom Materials - GED - Free Printable

It looks like the worksheet contains two types of problems:

- Problems 1–7: Linear equations to solve for $ x $.
- Problems 8–12: Algebraic expressions (not equations) that need to be simplified.

Let’s go through each problem step by step.

---

Problems 1–7: Solve for $ x $

#### 1. $ 14.13 = x - 4.25 $

Add 4.25 to both sides:
$$
x = 14.13 + 4.25 = 18.38
$$
✔ Answer: $ x = 18.38 $

---

#### 2. $ 14.1 + x = 43.9 $

Subtract 14.1 from both sides:
$$
x = 43.9 - 14.1 = 29.8
$$
✔ Answer: $ x = 29.8 $

---

#### 3. $ 4x - 10 = x - 7 $

Subtract $ x $ from both sides:
$$
3x - 10 = -7
$$

Add 10 to both sides:
$$
3x = 3
$$

Divide by 3:
$$
x = 1
$$
✔ Answer: $ x = 1 $

---

#### 4. $ 2 - x = 5x - 10 $

Move all $ x $-terms to one side and constants to the other.

Add $ x $ to both sides:
$$
2 = 6x - 10
$$

Add 10 to both sides:
$$
12 = 6x
$$

Divide by 6:
$$
x = 2
$$
✔ Answer: $ x = 2 $

---

#### 5. $ 2 + x = -5x - 22 $

Add $ 5x $ to both sides:
$$
2 + 6x = -22
$$

Subtract 2:
$$
6x = -24
$$

Divide by 6:
$$
x = -4
$$
✔ Answer: $ x = -4 $

---

#### 6. $ 2 + 4x = 2x + 20 $

Subtract $ 2x $ from both sides:
$$
2 + 2x = 20
$$

Subtract 2:
$$
2x = 18
$$

Divide by 2:
$$
x = 9
$$
✔ Answer: $ x = 9 $

---

#### 7. $ -1 + 4x = 6x - 7 $

Subtract $ 4x $ from both sides:
$$
-1 = 2x - 7
$$

Add 7:
$$
6 = 2x
$$

Divide by 2:
$$
x = 3
$$
✔ Answer: $ x = 3 $

---

Problems 8–12: Simplify the expressions

These are not equations — they are rational expressions to simplify.

---

#### 8. $ \frac{42x^3}{2x} $

Simplify numerator and denominator:

$$
\frac{42x^3}{2x} = \frac{42}{2} \cdot \frac{x^3}{x} = 21x^{2}
$$
✔ Answer: $ 21x^2 $

---

#### 9. $ \frac{52x^2}{2x} $

$$
\frac{52x^2}{2x} = \frac{52}{2} \cdot \frac{x^2}{x} = 26x
$$
✔ Answer: $ 26x $

---

#### 10. $ \frac{x^2 + 7x + 6}{x + 1} $

Factor the numerator:
$$
x^2 + 7x + 6 = (x + 6)(x + 1)
$$

So:
$$
\frac{(x + 6)(x + 1)}{x + 1} = x + 6 \quad \text{(for } x \ne -1\text{)}
$$
✔ Answer: $ x + 6 $, $ x \ne -1 $

---

#### 11. $ \frac{x + 2}{x^2 + 9x + 18} $

Factor the denominator:
$$
x^2 + 9x + 18 = (x + 3)(x + 6)
$$

So:
$$
\frac{x + 2}{(x + 3)(x + 6)}
$$

No common factors with numerator → cannot simplify further.

✔ Answer: $ \frac{x + 2}{(x + 3)(x + 6)} $

---

#### 12. $ \frac{x^2 + 8x + 15}{x^2 + 6x + 8} $

Factor numerator and denominator:

Numerator:
$$
x^2 + 8x + 15 = (x + 3)(x + 5)
$$

Denominator:
$$
x^2 + 6x + 8 = (x + 2)(x + 4)
$$

So:
$$
\frac{(x + 3)(x + 5)}{(x + 2)(x + 4)}
$$

No common factors → cannot simplify further.

✔ Answer: $ \frac{(x + 3)(x + 5)}{(x + 2)(x + 4)} $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ x = 18.38 $ |
| 2 | $ x = 29.8 $ |
| 3 | $ x = 1 $ |
| 4 | $ x = 2 $ |
| 5 | $ x = -4 $ |
| 6 | $ x = 9 $ |
| 7 | $ x = 3 $ |
| 8 | $ 21x^2 $ |
| 9 | $ 26x $ |
| 10 | $ x + 6 $, $ x \ne -1 $ |
| 11 | $ \frac{x + 2}{(x + 3)(x + 6)} $ |
| 12 | $ \frac{(x + 3)(x + 5)}{(x + 2)(x + 4)} $ |

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