Of course! Let's solve each equation step-by-step.
### Part (i): Solve [tex]\( 25x + 2 = 10(x - 5) \)[/tex]
1. Begin by expanding the right-hand side:
[tex]\[
25x + 2 = 10x - 50
\][/tex]
2. Next, we want to isolate the terms involving [tex]\( x \)[/tex] on one side of the equation. Subtract [tex]\( 10x \)[/tex] from both sides:
[tex]\[
25x - 10x + 2 = -50
\][/tex]
3. Simplify the left side:
[tex]\[
15x + 2 = -50
\][/tex]
4. Now, subtract 2 from both sides to isolate the [tex]\( x \)[/tex]-term:
[tex]\[
15x = -52
\][/tex]
5. Finally, divide by 15 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = -\frac{52}{15}
\][/tex]
### Part (ii): Solve [tex]\( \frac{3x}{11} - \frac{x}{4} = 2 \)[/tex]
1. To combine the fractions, find a common denominator. The common denominator for 11 and 4 is 44:
[tex]\[
\frac{3x \cdot 4}{11 \cdot 4} - \frac{x \cdot 11}{4 \cdot 11} = 2
\][/tex]
2. This simplifies to:
[tex]\[
\frac{12x}{44} - \frac{11x}{44} = 2
\][/tex]
3. Combine the fractions on the left-hand side:
[tex]\[
\frac{12x - 11x}{44} = 2
\][/tex]
4. Simplify within the numerator:
[tex]\[
\frac{x}{44} = 2
\][/tex]
5. To isolate [tex]\( x \)[/tex], multiply both sides by 44:
[tex]\[
x = 88
\][/tex]
### Part (iii): Solve [tex]\( \frac{x + 5}{x - 8} = \frac{3}{2} \)[/tex]
1. Start by cross-multiplying to eliminate the fraction:
[tex]\[
2(x + 5) = 3(x - 8)
\][/tex]
2. Distribute both sides:
[tex]\[
2x + 10 = 3x - 24
\][/tex]
3. Get all [tex]\( x \)[/tex]-terms on one side by subtracting [tex]\( 2x \)[/tex] from both sides:
[tex]\[
10 = x - 24
\][/tex]
4. Then add 24 to both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
x = 34
\][/tex]
### Summary
The solutions are:
- For [tex]\( 25x + 2 = 10(x - 5) \)[/tex], [tex]\( x = -\frac{52}{15} \)[/tex]
- For [tex]\( \frac{3x}{11} - \frac{x}{4} = 2 \)[/tex], [tex]\( x = 88 \)[/tex]
- For [tex]\( \frac{x + 5}{x - 8} = \frac{3}{2} \)[/tex], [tex]\( x = 34 \)[/tex]
