Algebraic expressions and their simplified solutions, showing addition and subtraction of fractions with variables.

Let’s go through each problem one by one. We’ll simplify the expressions step by step, just like you would on homework.

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Problem 1:
\[
\frac{3x}{2} + \frac{4x}{5}
\]

To add fractions, we need a common denominator. The least common denominator of 2 and 5 is 10.

Convert each fraction:

- \(\frac{3x}{2} = \frac{3x \cdot 5}{2 \cdot 5} = \frac{15x}{10}\)
- \(\frac{4x}{5} = \frac{4x \cdot 2}{5 \cdot 2} = \frac{8x}{10}\)

Now add:

\[
\frac{15x}{10} + \frac{8x}{10} = \frac{23x}{10}
\]

✔ Matches the red answer.

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Problem 2:
\[
\frac{x}{2} + \frac{4x}{5}
\]

Common denominator = 10

- \(\frac{x}{2} = \frac{5x}{10}\)
- \(\frac{4x}{5} = \frac{8x}{10}\)

Add:

\[
\frac{5x}{10} + \frac{8x}{10} = \frac{13x}{10}
\]

✔ Correct.

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Problem 3:
\[
\frac{x}{2} + \frac{x}{5}
\]

Common denominator = 10

- \(\frac{x}{2} = \frac{5x}{10}\)
- \(\frac{x}{5} = \frac{2x}{10}\)

Add:

\[
\frac{5x}{10} + \frac{2x}{10} = \frac{7x}{10}
\]

✔ Correct.

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Problem 4:
\[
\frac{x}{2} - \frac{x}{5}
\]

Common denominator = 10

- \(\frac{x}{2} = \frac{5x}{10}\)
- \(\frac{x}{5} = \frac{2x}{10}\)

Subtract:

\[
\frac{5x}{10} - \frac{2x}{10} = \frac{3x}{10}
\]

✔ Correct.

---

Problem 5:
\[
\frac{x}{5} - \frac{x}{2}
\]

Common denominator = 10

- \(\frac{x}{5} = \frac{2x}{10}\)
- \(\frac{x}{2} = \frac{5x}{10}\)

Subtract:

\[
\frac{2x}{10} - \frac{5x}{10} = -\frac{3x}{10}
\]

✔ Correct.

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Problem 6:
\[
\frac{5x}{5} - \frac{x}{2}
\]

First, simplify \(\frac{5x}{5} = x\)

So now: \(x - \frac{x}{2}\)

Write \(x\) as \(\frac{2x}{2}\):

\[
\frac{2x}{2} - \frac{x}{2} = \frac{x}{2}
\]

✔ Correct.

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Problem 7:
\[
\frac{5}{x} + \frac{2}{x}
\]

Same denominator → just add numerators:

\[
\frac{5 + 2}{x} = \frac{7}{x}
\]

✔ Correct.

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Problem 8:
\[
\frac{5}{x^2} + \frac{2}{x}
\]

Need common denominator. Since \(x^2\) is bigger, use that.

Convert \(\frac{2}{x}\) to have denominator \(x^2\):

\[
\frac{2}{x} = \frac{2 \cdot x}{x \cdot x} = \frac{2x}{x^2}
\]

Now add:

\[
\frac{5}{x^2} + \frac{2x}{x^2} = \frac{2x + 5}{x^2}
\]

✔ Correct.

---

Problem 9:
\[
\frac{5}{2x^2} - \frac{2}{x}
\]

Common denominator = \(2x^2\)

Convert \(\frac{2}{x}\) to denominator \(2x^2\):

Multiply numerator and denominator by \(2x\):

\[
\frac{2}{x} = \frac{2 \cdot 2x}{x \cdot 2x} = \frac{4x}{2x^2}
\]

Now subtract:

\[
\frac{5}{2x^2} - \frac{4x}{2x^2} = \frac{5 - 4x}{2x^2}
\]

✔ Correct.

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Problem 10:
\[
\frac{6}{2x^2} - \frac{2}{x}
\]

First, simplify \(\frac{6}{2x^2} = \frac{3}{x^2}\)

Now: \(\frac{3}{x^2} - \frac{2}{x}\)

Common denominator = \(x^2\)

Convert \(\frac{2}{x} = \frac{2x}{x^2}\)

Subtract:

\[
\frac{3}{x^2} - \frac{2x}{x^2} = \frac{3 - 2x}{x^2}
\]

✔ Correct.

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All answers match the red ones shown. So if this was your homework, you got them all right!

Final Answer:
All given solutions are correct.