Solve `0 < (-x)/(3)<1`,`xinR`.

Answer

To solve the inequality \( 0 < \frac{-x}{3} < 1 \), we will break it down step by step.

### Step 1: Split the Compound Inequality
We start with the compound inequality:
\[ 0 < \frac{-x}{3} < 1 \]

This can be split into two separate inequalities:
1. \( 0 < \frac{-x}{3} \)
2. \( \frac{-x}{3} < 1 \)

### Step 2: Solve the First Inequality
Let's solve the first inequality:
\[ 0 < \frac{-x}{3} \]

To eliminate the fraction, multiply both sides by 3:
\[ 0 < -x \]

Now, multiply both sides by -1 (remember to reverse the inequality sign):
\[ 0 > x \]
or
\[ x < 0 \]

### Step 3: Solve the Second Inequality
Now, let's solve the second inequality:
\[ \frac{-x}{3} < 1 \]

Again, multiply both sides by 3:
\[ -x < 3 \]

Now, multiply both sides by -1 (again, remember to reverse the inequality sign):
\[ x > -3 \]

### Step 4: Combine the Results
Now we have two inequalities:
1. \( x < 0 \)
2. \( x > -3 \)

Combining these gives us:
\[ -3 < x < 0 \]

### Step 5: Write the Final Answer
In interval notation, this can be expressed as:
\[ x \in (-3, 0) \]