https://file.aiquickdraw.com/mathbot/file/1726379877448pqdn3c1l.jpg

The equation in the image is:

\[
\frac{-2}{3}(x + 12) + \frac{2}{3}x = \frac{-5}{4}x + 2
\]

Let's solve this step-by-step:

### Step 1: Distribute the \(\frac{-2}{3}\) over \((x + 12)\)
\[
\frac{-2}{3}(x + 12) = \frac{-2}{3}x + \frac{-2}{3} \times 12 = \frac{-2}{3}x - 8
\]

Now the equation becomes:
\[
\left(\frac{-2}{3}x - 8\right) + \frac{2}{3}x = \frac{-5}{4}x + 2
\]

### Step 2: Combine like terms on the left side
\[
\frac{-2}{3}x + \frac{2}{3}x - 8 = \frac{-5}{4}x + 2
\]

The \(\frac{-2}{3}x + \frac{2}{3}x\) cancels out, leaving:
\[
-8 = \frac{-5}{4}x + 2
\]

### Step 3: Move the constant 2 to the left side
\[
-8 - 2 = \frac{-5}{4}x
\]
\[
-10 = \frac{-5}{4}x
\]

### Step 4: Solve for \(x\)
To isolate \(x\), multiply both sides by the reciprocal of \(\frac{-5}{4}\), which is \(\frac{4}{-5}\):
\[
x = -10 \times \frac{4}{-5} = \frac{-10 \times 4}{-5} = \frac{-40}{-5} = 8
\]

### Final Answer:
\[
x = 8
\]

Let me know if you need any more details!

### 5 Related Questions:
1. What happens if you change the constant on the right side of the equation?
2. How would the solution change if the equation had a different fraction for the coefficient of \(x\)?
3. What are some alternative methods for solving this equation?
4. How can you verify that \(x = 8\) is a correct solution?
5. What if the distributive property was applied incorrectly—what kind of errors would appear?

**Tip:** Always simplify the equation by combining like terms early in the process to avoid unnecessary complexity later.

Learn how to solve the linear equation -(2/3)(x + 12) + (2/3)x = -(5/4)x + 2 with a detailed step-by-step solution. This problem involves key algebraic concepts such as distributing, combining like terms, and solving for x. Suitable for students in Grades 8-10.

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