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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!

Learn how to solve the linear inequality -(1/5)(x + 9) + (1/2)x ≤ -3 with a step-by-step solution. This problem covers essential algebraic concepts, including distributing, combining like terms, and solving inequalities. Suitable for high school students in Grades 9-11.

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