Algebraic fraction operations: addition and subtraction of expressions with variables.

You’ve uploaded an image that shows 10 algebraic expressions involving addition and subtraction of fractions with variables, along with their solutions written in red. The task appears to be to verify whether the given solutions are correct, and if not, explain the correct solution.

Let’s go through each one step-by-step.

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## 🔢 Left Column: Expressions with `x` in numerator

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1. `3x/2 + 4x/5`

✔ Given answer: `23x/10`

Check:
LCM of 2 and 5 is 10.
- `3x/2 = (3x * 5)/10 = 15x/10`
- `4x/5 = (4x * 2)/10 = 8x/10`
- Sum: `15x/10 + 8x/10 = 23x/10`

✔️ Correct

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2. `x/2 + 4x/5`

✔ Given answer: `13x/10`

Check:
LCM = 10
- `x/2 = 5x/10`
- `4x/5 = 8x/10`
- Sum: `5x/10 + 8x/10 = 13x/10`

✔️ Correct

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3. `x/2 + x/5`

✔ Given answer: `7x/10`

Check:
LCM = 10
- `x/2 = 5x/10`
- `x/5 = 2x/10`
- Sum: `5x/10 + 2x/10 = 7x/10`

✔️ Correct

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4. `x/2 - x/5`

✔ Given answer: `3x/10`

Check:
LCM = 10
- `x/2 = 5x/10`
- `x/5 = 2x/10`
- Difference: `5x/10 - 2x/10 = 3x/10`

✔️ Correct

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5. `x/5 - x/2`

✔ Given answer: `-3x/10`

Check:
LCM = 10
- `x/5 = 2x/10`
- `x/2 = 5x/10`
- Difference: `2x/10 - 5x/10 = -3x/10`

✔️ Correct

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## 🔢 Right Column: Expressions with `x` in denominator

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6. `5x/5 - x/2`

✔ Given answer: `x/2`

Check:
Simplify `5x/5 = x`
So: `x - x/2 = (2x/2 - x/2) = x/2`

✔️ Correct

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7. `5/x + 2/x`

✔ Given answer: `7/x`

Check:
Same denominator → `(5 + 2)/x = 7/x`

✔️ Correct

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8. `5/x² + 2/x`

✔ Given answer: `(2x + 5)/x²`

Check:
Common denominator = `x²`
- `5/x²` stays as is
- `2/x = 2x/x²`
- Sum: `(5 + 2x)/x² = (2x + 5)/x²`

✔️ Correct

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9. `5/(2x²) - 2/x`

✔ Given answer: `(5 - 4x)/(2x²)`

Check:
Common denominator = `2x²`
- `5/(2x²)` stays
- `2/x = (2 * 2x)/(2x²) = 4x/(2x²)`
- So: `5/(2x²) - 4x/(2x²) = (5 - 4x)/(2x²)`

✔️ Correct

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10. `6/(2x²) - 2/x`

✔ Given answer: `(3 - 2x)/x²`

Check:
First, simplify `6/(2x²) = 3/x²`
Now: `3/x² - 2/x`

Common denominator = `x²`
- `3/x²` stays
- `2/x = 2x/x²`
- So: `3/x² - 2x/x² = (3 - 2x)/x²`

✔️ Correct

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## ✔ Final Verdict:

All 10 solutions provided in red are CORRECT.

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## 🧠 Explanation Summary:

To add or subtract algebraic fractions:

1. Find the Least Common Denominator (LCD) — usually the LCM of denominators.
2. Rewrite each fraction with the LCD by multiplying numerator and denominator appropriately.
3. Combine numerators — keep denominator the same.
4. Simplify if possible (like reducing coefficients or factoring).

In cases where denominators have variables (like `x`, `x²`), treat them like numbers — find common denominator by taking the highest power of each factor.

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## 💡 Pro Tip:

Always check your work by plugging in a simple value for `x` (e.g., `x = 10`) into both the original expression and your simplified answer. If they match, you’re likely correct!

Example for #1:
Original: `3(10)/2 + 4(10)/5 = 15 + 8 = 23`
Answer: `23(10)/10 = 23` → ✔️ matches.

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✔ All answers are verified and correct. No errors found.

@fortyninecubed did a great job! 🎉