https://file.aiquickdraw.com/mathbot/file/1728314191137obrhm3uj.jpg

The inequality in the image is:

\[
\frac{3x - 1}{-4} < 8 - \frac{x + 3}{-6}
\]

### Step-by-step solution:

#### Step 1: Simplify both sides

First, simplify the right-hand side:
\[
\frac{x + 3}{-6} = -\frac{x + 3}{6} = -\frac{x}{6} - \frac{3}{6} = -\frac{x}{6} - \frac{1}{2}
\]
So, the inequality becomes:
\[
\frac{3x - 1}{-4} < 8 + \frac{x}{6} + \frac{1}{2}
\]

#### Step 2: Eliminate fractions by finding a common denominator
The least common denominator between \( -4 \), 6, and 2 is 12. Multiply every term by 12 to eliminate the fractions:
\[
12 \times \left(\frac{3x - 1}{-4}\right) < 12 \times \left(8 + \frac{x}{6} + \frac{1}{2}\right)
\]
This simplifies to:
\[
3 \times (3x - 1) < 12 \times 8 + 2x + 6
\]

Simplifying both sides:
\[
9x - 3 < 96 + 2x + 6
\]
\[
9x - 3 < 102 + 2x
\]

#### Step 3: Isolate the variable
Move all terms involving \(x\) to one side and constants to the other:
\[
9x - 2x < 102 + 3
\]
\[
7x < 105
\]

#### Step 4: Solve for \(x\)
Divide both sides by 7:
\[
x < 15
\]

### Final Solution:
The solution to the inequality is:
\[
x < 15
\]

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. How would the inequality change if the signs of the fractions were reversed?
2. What happens if the coefficient of \(x\) on one side is negative?
3. Can you graph the solution \(x < 15\) on a number line?
4. How would the solution change if the inequality was \( \leq \) instead of \( < \)?
5. What happens if there are no fractions involved?

**Tip:** Always be careful when multiplying or dividing an inequality by a negative number, as this will reverse the inequality sign.

Solve the inequality (3x - 1) / -4 < 8 - (x + 3) / -6.

Learn how to solve the linear inequality (3x - 1) / -4 < 8 - (x + 3) / -6. This detailed solution covers key algebraic concepts, including fraction simplification, solving inequalities, and isolating variables. Suitable for students in Grades 9-11.

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