Solution of ` | 3x- 2 |le (1)/(2)` is

Options

A. `[(1)/(2) , (5)/(6) ]`
B. ` ((1)/(2) , ( 5)/(6))`
C. ` ((5)/( 6),(1)/(2)) `
D. `[(5)/(6),(1)/(2) ]`

Answer

To solve the inequality \( |3x - 2| \leq \frac{1}{2} \), we can follow these steps:

### Step 1: Understand the absolute value inequality
The expression \( |t| \leq a \) implies that \( -a \leq t \leq a \). In our case, we have:
\[ - \frac{1}{2} \leq 3x - 2 \leq \frac{1}{2} \]

### Step 2: Break it into two inequalities
We can break this into two separate inequalities:
1. \( 3x - 2 \geq -\frac{1}{2} \)
2. \( 3x - 2 \leq \frac{1}{2} \)

### Step 3: Solve the first inequality
Starting with the first inequality:
\[ 3x - 2 \geq -\frac{1}{2} \]
Add 2 to both sides:
\[ 3x \geq -\frac{1}{2} + 2 \]
Convert 2 to a fraction:
\[ 2 = \frac{4}{2} \]
Now combine:
\[ 3x \geq \frac{4}{2} - \frac{1}{2} = \frac{3}{2} \]
Now divide by 3:
\[ x \geq \frac{3}{2} \times \frac{1}{3} = \frac{1}{2} \]

### Step 4: Solve the second inequality
Now for the second inequality:
\[ 3x - 2 \leq \frac{1}{2} \]
Add 2 to both sides:
\[ 3x \leq \frac{1}{2} + 2 \]
Convert 2 to a fraction:
\[ 2 = \frac{4}{2} \]
Now combine:
\[ 3x \leq \frac{1}{2} + \frac{4}{2} = \frac{5}{2} \]
Now divide by 3:
\[ x \leq \frac{5}{2} \times \frac{1}{3} = \frac{5}{6} \]

### Step 5: Combine the results
Now we have:
\[ \frac{1}{2} \leq x \leq \frac{5}{6} \]
This can be written in interval notation as:
\[ x \in \left[\frac{1}{2}, \frac{5}{6}\right] \]

### Final Answer
Thus, the solution of the inequality \( |3x - 2| \leq \frac{1}{2} \) is:
\[ x \in \left[\frac{1}{2}, \frac{5}{6}\right] \]