Solution set for inequality `|x-1|le5`is.

Options

A. [-6,4]
B. [-4,0]
C. [-4,6]
D. [0,6]

Answer

To solve the inequality \( |x - 1| \leq 5 \), we will follow these steps:

### Step 1: Understand the Absolute Value Inequality
The expression \( |x - 1| \leq 5 \) means that the distance between \( x \) and \( 1 \) on the number line is at most \( 5 \). This leads to two inequalities:
\[ -5 \leq x - 1 \leq 5 \]

### Step 2: Break it into Two Inequalities
From the absolute value inequality, we can split it into two separate inequalities:
1. \( x - 1 \leq 5 \)
2. \( x - 1 \geq -5 \)

### Step 3: Solve the First Inequality
Now, we solve the first inequality:
\[ x - 1 \leq 5 \]
Adding \( 1 \) to both sides gives:
\[ x \leq 6 \]

### Step 4: Solve the Second Inequality
Next, we solve the second inequality:
\[ x - 1 \geq -5 \]
Adding \( 1 \) to both sides gives:
\[ x \geq -4 \]

### Step 5: Combine the Results
Now we combine the results from both inequalities:
\[ -4 \leq x \leq 6 \]
This can be written in interval notation as:
\[ [-4, 6] \]

### Final Solution
Thus, the solution set for the inequality \( |x - 1| \leq 5 \) is:
\[ [-4, 6] \]

---