`-(5)/(3)lt(1)/(2)-(1)/(3)xlt-(3)/(2)`
For the inequality above, what is a possible values of `x-3`?

Answer

To solve the inequality \(-\frac{5}{3} < \frac{1}{2} - \frac{1}{3}x < -\frac{3}{2}\), we will break it down step by step.

### Step 1: Rewrite the inequality
The given inequality is:
\[ -\frac{5}{3} < \frac{1}{2} - \frac{1}{3}x < -\frac{3}{2} \]

### Step 2: Multiply the entire inequality by 6
To eliminate the fractions, we can multiply the entire inequality by 6 (the least common multiple of the denominators 2 and 3):
\[ 6 \cdot \left(-\frac{5}{3}\right) < 6 \cdot \left(\frac{1}{2} - \frac{1}{3}x\right) < 6 \cdot \left(-\frac{3}{2}\right) \]

This simplifies to:
\[ -10 < 3 - 2x < -9 \]

### Step 3: Split the inequality
Now we can split the inequality into two parts:
1. \(-10 < 3 - 2x\)
2. \(3 - 2x < -9\)

### Step 4: Solve the first part
For the first part, \(-10 < 3 - 2x\):
1. Subtract 3 from both sides:
\[ -10 - 3 < -2x \implies -13 < -2x \]
2. Divide by -2 (remember to reverse the inequality):
\[ \frac{-13}{-2} > x \implies x < 6.5 \]

### Step 5: Solve the second part
For the second part, \(3 - 2x < -9\):
1. Subtract 3 from both sides:
\[ -2x < -9 - 3 \implies -2x < -12 \]
2. Divide by -2 (again, reverse the inequality):
\[ x > 6 \]

### Step 6: Combine the results
From the two parts, we have:
\[ 6 < x < 6.5 \]

### Step 7: Find possible values for \(x - 3\)
Now, we need to find possible values for \(x - 3\):
1. Subtract 3 from all parts of the inequality:
\[ 6 - 3 < x - 3 < 6.5 - 3 \]
This simplifies to:
\[ 3 < x - 3 < 3.5 \]

### Conclusion
Thus, a possible value for \(x - 3\) could be any number between 3 and 3.5. For example, \(x - 3 = 3.1\) is a valid choice.

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