If |x - 5| < 1, what is the answer in intervals

The inequality \( |x - 5| < 1 \) represents the absolute value of \(x - 5\) being less than 1. This means the distance between \(x\) and 5 must be less than 1.

We can break this into two inequalities:

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

Now, solve for \(x\):

1. Add 5 to all parts of the inequality:
   \[
   -1 + 5 < x - 5 + 5 < 1 + 5
   \]
   \[
   4 < x < 6
   \]

So, the solution in interval notation is:

\[
(4, 6)
\]

This means that \(x\) lies strictly between 4 and 6.

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Would you like a more detailed explanation of absolute value inequalities or have any questions about this? Here are some related questions you could explore:

1. How would you solve an absolute value equation like \( |x - 5| = 1 \)?
2. What does \( |x + 3| > 2 \) represent in interval form?
3. How do you graph the inequality \( |x - 5| < 1 \)?
4. Can absolute value inequalities have solutions for all \(x\)?
5. What happens if the inequality is \( |x - 5| \geq 1 \)?

**Tip**: To solve absolute value inequalities like this, always rewrite the inequality without the absolute value by considering both the positive and negative cases.

If |x - 5| < 1, what is the answer in intervals?

Learn how to solve the absolute value inequality |x - 5| < 1, and understand how to represent it in interval notation. This solution demonstrates the process of breaking down the inequality into two parts and solving for x. Suitable for students in Grades 8-10.

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