Solve and graph the solution set of `|2x-1|le7`.

Answer

To solve the inequality \( |2x - 1| \leq 7 \), we will break it down into two cases based on the definition of absolute value.

### Step 1: Set up the inequality
The inequality \( |2x - 1| \leq 7 \) means that the expression inside the absolute value, \( 2x - 1 \), can be either less than or equal to 7 or greater than or equal to -7.

### Step 2: Break into two cases
This gives us two separate inequalities to solve:
1. \( 2x - 1 \leq 7 \)
2. \( 2x - 1 \geq -7 \)

### Step 3: Solve the first inequality
For the first inequality:
\[ 2x - 1 \leq 7 \]
Add 1 to both sides:
\[ 2x \leq 8 \]
Now, divide by 2:
\[ x \leq 4 \]

### Step 4: Solve the second inequality
For the second inequality:
\[ 2x - 1 \geq -7 \]
Add 1 to both sides:
\[ 2x \geq -6 \]
Now, divide by 2:
\[ x \geq -3 \]

### Step 5: Combine the results
From the two inequalities, we have:
\[ -3 \leq x \leq 4 \]
This means the solution set is:
\[ x \in [-3, 4] \]

### Step 6: Graph the solution set
On a number line, we will represent the solution set:
- Mark the points -3 and 4.
- Shade the region between -3 and 4, including both endpoints.

### Final Answer
The solution set is:
\[ x \in [-3, 4] \]

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