Solve the equation:
(i) `x+10gt4x-5`
(ii) `8x-2gt5x`.
(iii) `3x-10gt5x+1`.

Answer

Let's solve each of the given inequalities step by step.

### (i) Solve the inequality: \( x + 10 > 4x - 5 \)

**Step 1:** Rearrange the inequality.
\[ x + 10 > 4x - 5 \]
Subtract \( x \) from both sides:
\[ 10 > 4x - x - 5 \]
This simplifies to:
\[ 10 + 5 > 3x \]
So,
\[ 15 > 3x \]

**Step 2:** Divide both sides by 3.
\[ \frac{15}{3} > x \]
This simplifies to:
\[ 5 > x \]
or
\[ x < 5 \]

**Step 3:** Write the solution in interval notation.
The solution is:
\[ (-\infty, 5) \]

### (ii) Solve the inequality: \( 8x - 2 > 5x \)

**Step 1:** Rearrange the inequality.
\[ 8x - 2 > 5x \]
Subtract \( 5x \) from both sides:
\[ 8x - 5x - 2 > 0 \]
This simplifies to:
\[ 3x - 2 > 0 \]

**Step 2:** Add 2 to both sides.
\[ 3x > 2 \]

**Step 3:** Divide both sides by 3.
\[ x > \frac{2}{3} \]

**Step 4:** Write the solution in interval notation.
The solution is:
\[ \left( \frac{2}{3}, \infty \right) \]

### (iii) Solve the inequality: \( 3x - 10 > 5x + 1 \)

**Step 1:** Rearrange the inequality.
\[ 3x - 10 > 5x + 1 \]
Subtract \( 5x \) from both sides:
\[ 3x - 5x - 10 > 1 \]
This simplifies to:
\[ -2x - 10 > 1 \]

**Step 2:** Add 10 to both sides.
\[ -2x > 11 \]

**Step 3:** Divide both sides by -2 (remember to reverse the inequality).
\[ x < -\frac{11}{2} \]

**Step 4:** Write the solution in interval notation.
The solution is:
\[ (-\infty, -\frac{11}{2}) \]

### Summary of Solutions:
1. \( (-\infty, 5) \)
2. \( \left( \frac{2}{3}, \infty \right) \)
3. \( (-\infty, -\frac{11}{2}) \)