Solve for $$ x $$ and write your answer in simplest form. $$ 3\left(-\frac{1}{3} x+\frac{1}{2}\right)+2 x=-1-\left(6 x-\frac{1}{2}\right) $$

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UpStudy AI · Accepted Answer

To solve for $$ x $$ in the equation:

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

**Step 1: Distribute and Simplify Both Sides**

**Left Side:**
$$
3\left(-\frac{1}{3}x + \frac{1}{2}\right) + 2x = 3 \cdot \left(-\frac{1}{3}x\right) + 3 \cdot \frac{1}{2} + 2x = -x + \frac{3}{2} + 2x = x + \frac{3}{2}
$$

**Right Side:**
$$
-1 - \left(6x - \frac{1}{2}\right) = -1 - 6x + \frac{1}{2} = -\frac{1}{2} - 6x
$$

**Step 2: Set the Simplified Expressions Equal and Solve for $$ x $$**
$$
x + \frac{3}{2} = -\frac{1}{2} - 6x
$$

Add $$ 6x $$ to both sides:
$$
7x + \frac{3}{2} = -\frac{1}{2}
$$

Subtract $$ \frac{3}{2} $$ from both sides:
$$
7x = -\frac{1}{2} - \frac{3}{2} = -2
$$

Divide both sides by 7:
$$
x = \frac{-2}{7} = -\frac{2}{7}
$$

**Answer:**
$$
\boxed{-\dfrac{2}{7}}
$$
$$ x = -\frac{2}{7} $$

Solve for and write your answer in simplest form.

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