Solve : `|3x-5|=1`

Answer

To solve the equation \( |3x - 5| = 1 \), we need to consider the definition of absolute value. The absolute value of a number is its distance from zero on the number line, which means it can be expressed in two cases:

1. The expression inside the absolute value is equal to the positive value.
2. The expression inside the absolute value is equal to the negative value.

### Step 1: Set up the two cases

From the equation \( |3x - 5| = 1 \), we can create two separate equations:

**Case 1:**
\[ 3x - 5 = 1 \]

**Case 2:**
\[ 3x - 5 = -1 \]

### Step 2: Solve Case 1

For the first case, we solve the equation:

\[ 3x - 5 = 1 \]

Add 5 to both sides:

\[ 3x = 1 + 5 \]
\[ 3x = 6 \]

Now, divide both sides by 3:

\[ x = \frac{6}{3} = 2 \]

### Step 3: Solve Case 2

Now, we solve the second case:

\[ 3x - 5 = -1 \]

Add 5 to both sides:

\[ 3x = -1 + 5 \]
\[ 3x = 4 \]

Now, divide both sides by 3:

\[ x = \frac{4}{3} \]

### Step 4: Summary of Solutions

The solutions to the equation \( |3x - 5| = 1 \) are:

\[ x = 2 \quad \text{and} \quad x = \frac{4}{3} \]