Solve the equation:
(i) `(5x)/(2)+(3x)/(4)ge(39)/(4)`
(ii) `(5-2x)/(3)le(x)/(6)-5`.

Answer

Let's solve the given inequalities step by step.

### Part (i): Solve the inequality \(\frac{5x}{2} + \frac{3x}{4} \geq \frac{39}{4}\)

**Step 1: Find a common denominator.**
The denominators are 2 and 4. The least common multiple (LCM) of 2 and 4 is 4.

**Step 2: Rewrite each term with the common denominator.**
\[ \frac{5x}{2} = \frac{5x \cdot 2}{2 \cdot 2} = \frac{10x}{4} \]
So the inequality becomes:
\[ \frac{10x}{4} + \frac{3x}{4} \geq \frac{39}{4} \]

**Step 3: Combine the fractions on the left side.**
\[ \frac{10x + 3x}{4} \geq \frac{39}{4} \]
This simplifies to:
\[ \frac{13x}{4} \geq \frac{39}{4} \]

**Step 4: Eliminate the denominator by multiplying both sides by 4.**
\[ 13x \geq 39 \]

**Step 5: Solve for \(x\) by dividing both sides by 13.**
\[ x \geq 3 \]

**Final Answer for Part (i):**
\[ x \in [3, \infty) \]

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### Part (ii): Solve the inequality \(\frac{5 - 2x}{3} \leq \frac{x}{6} - 5\)

**Step 1: Eliminate the fractions by finding a common denominator.**
The denominators are 3 and 6. The least common multiple (LCM) of 3 and 6 is 6.

**Step 2: Rewrite each term with the common denominator.**
\[ \frac{5 - 2x}{3} = \frac{2(5 - 2x)}{6} = \frac{10 - 4x}{6} \]
So the inequality becomes:
\[ \frac{10 - 4x}{6} \leq \frac{x}{6} - 5 \]

**Step 3: Rewrite the right side with a common denominator.**
\[ -5 = \frac{-30}{6} \]
Thus, the inequality now reads:
\[ \frac{10 - 4x}{6} \leq \frac{x - 30}{6} \]

**Step 4: Since the denominators are the same, we can eliminate them.**
\[ 10 - 4x \leq x - 30 \]

**Step 5: Rearrange the inequality.**
Add \(4x\) to both sides:
\[ 10 \leq x + 4x - 30 \]
This simplifies to:
\[ 10 + 30 \leq 5x \]
\[ 40 \leq 5x \]

**Step 6: Divide both sides by 5.**
\[ 8 \leq x \]
or
\[ x \geq 8 \]

**Final Answer for Part (ii):**
\[ x \in [8, \infty) \]

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