The inequality you provided is:

$-\frac{1}{5}(x + 9) + \frac{1}{2}x \leq -3$

Let's solve it step by step:

Step 1: Distribute the $-\frac{1}{5}$ across $(x + 9)$

$-\frac{1}{5}(x + 9) = -\frac{1}{5}x - \frac{9}{5}$ So the inequality becomes: $-\frac{1}{5}x - \frac{9}{5} + \frac{1}{2}x \leq -3$

Step 2: Combine the like terms involving $x$

The terms involving $x$ are $-\frac{1}{5}x$ and $\frac{1}{2}x$. To combine these, find a common denominator.

$-\frac{1}{5}x + \frac{1}{2}x = \left(-\frac{2}{10} + \frac{5}{10}\right)x = \frac{3}{10}x$ So the inequality becomes: $\frac{3}{10}x - \frac{9}{5} \leq -3$

Step 3: Eliminate the fractions

To eliminate the fractions, multiply every term by 10 (the least common denominator).

$10\left(\frac{3}{10}x - \frac{9}{5}\right) \leq 10(-3)$ $3x - 18 \leq -30$

Step 4: Solve for $x$

First, add 18 to both sides: $3x \leq -30 + 18$ $3x \leq -12$

Finally, divide by 3: $x \leq -4$

Final Answer:

The solution to the inequality is: $x \leq -4$

Would you like more details or further clarification?

Here are 5 related questions you can explore:

1. How do you solve inequalities with variables on both sides?
2. What are the rules for solving inequalities when multiplying or dividing by a negative number?
3. How can we verify if a solution to an inequality is correct?
4. What is the graphical representation of $x \leq -4$ on a number line?
5. How does solving linear inequalities differ from solving linear equations?

Tip: Always remember to flip the inequality sign when multiplying or dividing by a negative number!