Questions: Which of the following is equivalent to x-1<5 ? x-1<5 5<x-1<5 -5<x-1<5 -5>x-1<5

Solution Steps

To solve the absolute value inequality \( |x-1| < 5 \), we need to rewrite it as a compound inequality. The absolute value inequality \( |A| < B \) can be rewritten as \( -B < A < B \).

1. Recognize that \( |x-1| < 5 \) can be rewritten as a compound inequality.
2. Apply the rule \( -B < A < B \) where \( A = x-1 \) and \( B = 5 \).
3. This results in \( -5 < x-1 < 5 \).

We start with the absolute value inequality:

\[ |x-1| < 5 \]

The definition of absolute value states that \( |A| < B \) can be rewritten as:

\[ -B < A < B \]

In our case, \( A = x-1 \) and \( B = 5 \). Therefore, we can rewrite the inequality as:

\[ -5 < x-1 < 5 \]

Now, we can solve the compound inequality by isolating \( x \):

1. From the left side: \[ -5 < x - 1 \implies -5 + 1 < x \implies -4 < x \]

2. From the right side: \[ x - 1 < 5 \implies x < 5 + 1 \implies x < 6 \]

Combining these results, we have:

\[ -4 < x < 6 \]

Final Answer