What is the additive inverse of `(-5)/(9) + (1)/(3)` ?

Options

A. `(-2)/(9)`
B. `(2)/(9)`
C. `(-4)/(9)`
D. `(4)/(9)`

Answer

To find the additive inverse of the expression \(-\frac{5}{9} + \frac{1}{3}\), we can follow these steps:

### Step 1: Simplify the expression
First, we need to simplify the expression \(-\frac{5}{9} + \frac{1}{3}\). To do this, we need a common denominator for the fractions.

**Hint:** Identify the least common multiple (LCM) of the denominators to find a common denominator.

### Step 2: Find a common denominator
The denominators are 9 and 3. The least common multiple of 9 and 3 is 9. We can rewrite \(\frac{1}{3}\) with a denominator of 9.

\[ \frac{1}{3} = \frac{1 \times 3}{3 \times 3} = \frac{3}{9} \]

### Step 3: Rewrite the expression
Now we can rewrite the original expression using the common denominator:

\[ -\frac{5}{9} + \frac{3}{9} \]

### Step 4: Combine the fractions
Now that the fractions have the same denominator, we can combine them:

\[ -\frac{5}{9} + \frac{3}{9} = \frac{-5 + 3}{9} = \frac{-2}{9} \]

### Step 5: Find the additive inverse
The additive inverse of a number \(x\) is the number \(y\) such that \(x + y = 0\). Therefore, to find the additive inverse of \(-\frac{2}{9}\), we need to find a number that, when added to \(-\frac{2}{9}\), gives us 0.

This number is simply the opposite of \(-\frac{2}{9}\):

\[ y = \frac{2}{9} \]

### Conclusion
Thus, the additive inverse of \(-\frac{5}{9} + \frac{1}{3}\) is \(\frac{2}{9}\).

---