https://file.aiquickdraw.com/mathbot/file/17248484548414h6pdvfl.jpg

The inequality given in the image is:

\[
-1 + \frac{1}{2}x < -\left(-\frac{3}{2}x + 1\right) - 9
\]

Let's solve it step by step.

### Step 1: Distribute the negative sign on the right-hand side

Distribute the negative sign inside the parentheses:

\[
-1 + \frac{1}{2}x < \frac{3}{2}x - 1 - 9
\]

### Step 2: Simplify the right-hand side

Combine the constant terms on the right-hand side:

\[
-1 + \frac{1}{2}x < \frac{3}{2}x - 10
\]

### Step 3: Move the \(x\)-terms to one side

Subtract \(\frac{1}{2}x\) from both sides to get all \(x\)-terms on one side:

\[
-1 < \frac{3}{2}x - \frac{1}{2}x - 10
\]

Simplify the \(x\)-terms on the right-hand side:

\[
-1 < \frac{2}{2}x - 10
\]

\[
-1 < x - 10
\]

### Step 4: Isolate \(x\)

Add 10 to both sides to isolate \(x\):

\[
9 < x
\]

Or equivalently:

\[
x > 9
\]

### Final Answer:
\[
x > 9
\]

Would you like more details or have any questions? Here are a few related questions you might consider:

1. How does the solution change if the inequality sign is reversed?
2. What would the inequality look like if there were an additional constant term added to both sides?
3. How can you verify the solution graphically on a number line?
4. What impact does the coefficient of \(x\) have on the inequality?
5. How do the properties of inequalities change if both sides are multiplied by a negative number?

**Tip:** After solving an inequality, it’s always good practice to test the solution by substituting a number from the solution set back into the original inequality.

Learn how to solve the inequality -1 + (1/2)x < -(-3/2x + 1) - 9 with a detailed step-by-step solution. This problem covers solving inequalities, distributing negative signs, combining like terms, and isolating variables. Suitable for students in Grades 9-12.

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