The equation in the image is:

$-\frac{1}{3}x + 1 = \frac{3}{2}x - 1$

Let's solve for $x$:

Step 1: Eliminate the fractions

To make the equation easier to work with, multiply through by the least common denominator (LCD) of 3 and 2, which is 6:

$6 \left(-\frac{1}{3}x + 1 \right) = 6 \left(\frac{3}{2}x - 1 \right)$

Distribute the 6:

$-2x + 6 = 9x - 6$

Step 2: Move all terms involving $x$ to one side

Add $2x$ to both sides:

$6 = 11x - 6$

Step 3: Move constant terms to the other side

Add 6 to both sides:

$12 = 11x$

Step 4: Solve for $x$

Divide both sides by 11:

$x = \frac{12}{11}$

So, the solution is:

$x = \frac{12}{11}$

Would you like more details, or do you have any questions?

Related Questions:

1. How do you handle equations with variables on both sides?
2. What is the least common denominator and how is it used?
3. Why do we eliminate fractions in equations?
4. How would the steps change if one side had no fractions?
5. What is the meaning of isolating the variable in an equation?

Tip:

Always check your solution by substituting it back into the original equation to verify that both sides are equal.