If `|(30-x)/(3)|lt4`, then find x

Options

A. `-6ltxlt6`
B. `10ltxlt50`
C. `xgt10`
D. none of these

Answer

To solve the inequality \( \left| \frac{30 - x}{3} \right| < 4 \), we will follow these steps:

### Step 1: Remove the absolute value
The expression \( \left| \frac{30 - x}{3} \right| < 4 \) implies that:
\[ -4 < \frac{30 - x}{3} < 4 \]

### Step 2: Multiply through by 3
Since 3 is positive, we can multiply the entire inequality by 3 without changing the direction of the inequalities:
\[ -12 < 30 - x < 12 \]

### Step 3: Split into two inequalities
Now, we can split this into two separate inequalities:
1. \( -12 < 30 - x \)
2. \( 30 - x < 12 \)

### Step 4: Solve the first inequality
For the first inequality \( -12 < 30 - x \):
\[ -12 - 30 < -x \\ -42 < -x \\ x < 42 \]

### Step 5: Solve the second inequality
For the second inequality \( 30 - x < 12 \):
\[ 30 - 12 < x \\ 18 < x \\ x > 18 \]

### Step 6: Combine the results
Now, we combine the results from the two inequalities:
\[ 18 < x < 42 \]

### Final Answer
Thus, the solution for \( x \) is:
\[ x \in (18, 42) \]

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