Solve : ` (5 - 2x)/(3) + 5le (x)/(6) `

Explanation

To solve the inequality \( \frac{5 - 2x}{3} + 5 \leq \frac{x}{6} \), we will follow these steps:

### Step 1: Rewrite the inequality
Start with the given inequality:
\[ \frac{5 - 2x}{3} + 5 \leq \frac{x}{6} \]

### Step 2: Find a common denominator for the left-hand side
The left-hand side consists of two terms. To combine them, we need a common denominator. The common denominator here is 3:
\[ \frac{5 - 2x}{3} + \frac{15}{3} \leq \frac{x}{6} \]
This simplifies to:
\[ \frac{(5 - 2x) + 15}{3} \leq \frac{x}{6} \]
Which further simplifies to:
\[ \frac{20 - 2x}{3} \leq \frac{x}{6} \]

### Step 3: Eliminate the fractions
To eliminate the fractions, we can multiply both sides by 6 (the least common multiple of the denominators 3 and 6). Remember to maintain the direction of the inequality:
\[ 6 \cdot \frac{20 - 2x}{3} \leq x \]
This simplifies to:
\[ 2(20 - 2x) \leq x \]
Which expands to:
\[ 40 - 4x \leq x \]

### Step 4: Rearrange the inequality
Now, we will rearrange the inequality to isolate the variable \(x\):
\[ 40 \leq x + 4x \]
This simplifies to:
\[ 40 \leq 5x \]

### Step 5: Solve for \(x\)
Now divide both sides by 5:
\[ 8 \leq x \]
This can also be written as:
\[ x \geq 8 \]

### Step 6: Write the solution in interval notation
The solution in interval notation is:
\[ [8, \infty) \]

### Summary of the solution
The solution to the inequality \( \frac{5 - 2x}{3} + 5 \leq \frac{x}{6} \) is:
\[ x \geq 8 \quad \text{or} \quad [8, \infty) \]

---