solve the inequality of |x-4|<7

Answer 1

To solve the inequality \( |x - 4| < 7 \), we can use the definition of absolute value. The inequality \( |A| < B \) means that \( -B < A < B \).

For our inequality:

\[ |x - 4| < 7 \]

This can be expressed as:

\[ -7 < x - 4 < 7 \]

Now, we will solve the compound inequality step by step.

1. Add 4 to all parts of the inequality:

\[ -7 + 4 < x - 4 + 4 < 7 + 4 \]

This simplifies to:

\[ -3 < x < 11 \]

So, the solution to the inequality \( |x - 4| < 7 \) is:

\[ x \in (-3, 11) \]

In interval notation, the solution is \( (-3, 11) \).