Set of five algebraic equations requiring solving for the variable x.

Let’s solve each equation step by step.

---

1.
\[
\frac{x}{3} + \frac{x}{4} = 14
\]

✔ Step 1: Find the LCD (Least Common Denominator) of 3 and 4 → 12

✔ Step 2: Multiply every term by 12 to eliminate denominators:

\[
12 \cdot \left( \frac{x}{3} \right) + 12 \cdot \left( \frac{x}{4} \right) = 12 \cdot 14
\]

\[
4x + 3x = 168
\]

\[
7x = 168
\]

✔ Step 3: Solve for x

\[
x = \frac{168}{7} = 24
\]

✔ Answer: \( \boxed{24} \)

---

2.
\[
\frac{8 - 3x}{5x + 31} = \frac{2}{3}
\]

✔ Step 1: Cross-multiply

\[
3(8 - 3x) = 2(5x + 31)
\]

✔ Step 2: Expand both sides

Left: \( 24 - 9x \)
Right: \( 10x + 62 \)

So:

\[
24 - 9x = 10x + 62
\]

✔ Step 3: Move all x terms to one side, constants to the other

Subtract 24 from both sides:

\[
-9x = 10x + 38
\]

Subtract 10x from both sides:

\[
-19x = 38
\]

✔ Step 4: Solve for x

\[
x = \frac{38}{-19} = -2
\]

✔ Answer: \( \boxed{-2} \)

---

3.
\[
\frac{x}{5} + \frac{x}{4} = 18
\]

✔ Step 1: LCD of 5 and 4 is 20

Multiply every term by 20:

\[
20 \cdot \frac{x}{5} + 20 \cdot \frac{x}{4} = 20 \cdot 18
\]

\[
4x + 5x = 360
\]

\[
9x = 360
\]

✔ Step 2: Solve for x

\[
x = \frac{360}{9} = 40
\]

✔ Answer: \( \boxed{40} \)

---

4.
\[
\frac{2x + 3}{3 + x} = \frac{3}{2}
\]

✔ Step 1: Cross-multiply

\[
2(2x + 3) = 3(3 + x)
\]

✔ Step 2: Expand both sides

Left: \( 4x + 6 \)
Right: \( 9 + 3x \)

So:

\[
4x + 6 = 9 + 3x
\]

✔ Step 3: Subtract 3x from both sides

\[
x + 6 = 9
\]

✔ Step 4: Subtract 6

\[
x = 3
\]

✔ Check for extraneous solution:
Denominator in original equation: \( 3 + x = 3 + 3 = 6 ≠ 0 \) → Valid!

✔ Answer: \( \boxed{3} \)

*(Note: The “C” icon next to this problem might indicate it's a multiple-choice question or has special marking — but since no choices are given, we solved it directly.)*

---

5.
\[
\frac{x}{9} + x - 1 = 19
\]

✔ Step 1: Combine like terms

Write \( x \) as \( \frac{9x}{9} \) to combine with \( \frac{x}{9} \):

\[
\frac{x}{9} + \frac{9x}{9} - 1 = 19
\]

\[
\frac{10x}{9} - 1 = 19
\]

✔ Step 2: Add 1 to both sides

\[
\frac{10x}{9} = 20
\]

✔ Step 3: Multiply both sides by 9

\[
10x = 180
\]

✔ Step 4: Divide by 10

\[
x = 18
\]

✔ Answer: \( \boxed{18} \)

---

## ✔ Final Answers:

1. \( \boxed{24} \)
2. \( \boxed{-2} \)
3. \( \boxed{40} \)
4. \( \boxed{3} \)
5. \( \boxed{18} \)

Let me know if you’d like to see any step explained further!