Solve $\frac{5-2 x}{3} \leq \frac{x}{6}-5$.

Step 1: Clear the denominators

To eliminate the denominators and simplify the inequality, we multiply both sides by the least common multiple (LCM) of the denominators, which are 3 and 6. The LCM of 3 and 6 is 6.

Step 2: Simplify the inequality

After multiplying by 6, we simplify each term. On the left side, $6/3 = 2$, so we have $2(5-2x)$. On the right side, $6/6 = 1$, so we have $x$, and $6 \times 5 = 30$.

Step 3: Distribute and combine like terms

Next, we distribute the 2 on the left side: $2 \times 5 = 10$ and $2 \times (-2x) = -4x$. Now, we have a linear inequality with no fractions.

Step 4: Isolate the variable term

To isolate the variable $x$, we move all terms containing $x$ to one side and constant terms to the other. We add $4x$ to both sides and add $30$ to both sides.

Step 5: Simplify and solve for x

Combine the terms on both sides: $10 + 30 = 40$ and $x + 4x = 5x$. Finally, divide both sides by 5 to solve for $x$. Since we are dividing by a positive number, the inequality sign remains the same.

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