You've uploaded an image containing several algebraic equations. Let's go through them one by one and solve them. I'll provide detailed solutions and explanations for each.

(i) $\frac{2}{3} - \frac{1}{2} x = x + \frac{1}{6}$

Solution:

Step 1: Find the least common denominator (LCD) for the fractions, which is 6.

• Multiply both sides by 6 to eliminate the fractions.

$6 \times \left( \frac{2}{3} - \frac{1}{2} x \right) = 6 \times \left( x + \frac{1}{6} \right)$

$4 - 3x = 6x + 1$

Step 2: Move all the terms with $x$ on one side and constants on the other:

$4 - 1 = 6x + 3x$

$3 = 9x$

Step 3: Solve for $x$:

$x = \frac{3}{9} = \frac{1}{3}$

(ii) $\frac{x - 3}{3} - \frac{x - 2}{2} = -1$

Solution:

Step 1: Find the least common denominator (LCD), which is 6, and multiply both sides by 6.

$6 \times \left( \frac{x - 3}{3} - \frac{x - 2}{2} \right) = 6 \times (-1)$

$2(x - 3) - 3(x - 2) = -6$

Step 2: Expand both sides:

$2x - 6 - 3x + 6 = -6$

$-x = -6$

Step 3: Solve for $x$:

$x = 6$

(iii) $\frac{1}{2} \left( x - \frac{1}{6} \right) + \frac{2}{3} = \frac{5}{6} + \frac{1}{3} \left( \frac{1}{2} - 3x \right)$

Solution:

Step 1: Distribute and simplify both sides:

Left side:

$\frac{1}{2} \left( x - \frac{1}{6} \right) + \frac{2}{3} = \frac{x}{2} - \frac{1}{12} + \frac{2}{3}$

Right side:

$\frac{5}{6} + \frac{1}{3} \left( \frac{1}{2} - 3x \right) = \frac{5}{6} + \frac{1}{6} - x$

Step 2: Combine terms on both sides.

$\frac{x}{2} + \frac{7}{12} = \frac{2}{3} - x$

Step 3: Multiply through by 12 to eliminate fractions.

$6x + 7 = 8 - 12x$

Step 4: Collect like terms:

$18x = 1$

Step 5: Solve for $x$:

$x = \frac{1}{18}$

I can continue solving the remaining problems (iv) through (x) if you want. Do you need more detailed explanations for the steps or prefer to focus on any specific equation?

Here are 5 related questions to explore further:

1. How can you find the least common denominator for fractions quickly?
2. What is the importance of distributing terms when solving algebraic equations?
3. How do you handle equations where variables appear on both sides?
4. How do you solve equations with conditions like $x \neq 2$?
5. What are the general steps to check your solutions?

Tip: Always verify your solution by substituting it back into the original equation to see if it satisfies the equation.