Which inequality expresses all numbers that are less than 3 units away from 10 on a number line?

To find the inequality that expresses all numbers that are less than 3 units away from 10 on a number line, we can use the concept of absolute value. The absolute value of the difference between a number \( x \) and 10 should be less than 3. This can be expressed as:

\[ |x - 10| < 3 \]

This inequality states that the distance between \( x \) and 10 is less than 3 units.

To solve this inequality, we can rewrite it without the absolute value:

\[ -3 < x - 10 < 3 \]

Next, we solve this compound inequality for \( x \) by adding 10 to all parts of the inequality:

\[ -3 + 10 < x - 10 + 10 < 3 + 10 \]

Simplifying this gives:

\[ 7 < x < 13 \]

Thus, the inequality that expresses all numbers less than 3 units away from 10 on a number line is:

\[ 7 < x < 13 \]

### Summary:
\[ |x - 10| < 3 \quad \text{is equivalent to} \quad 7 < x < 13 \]

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### Related Questions:
1. What is the inequality for numbers more than 5 units away from 15?
2. How do you represent the inequality \( |x - a| \leq b \) on a number line?
3. Can you solve \( |2x - 4| < 6 \)?
4. How do you handle absolute value inequalities involving a variable on both sides?
5. What is the solution set for \( |x + 5| > 2 \)?
6. How do you graph \( |x - 10| \leq 3 \) on a number line?
7. What are some applications of absolute value inequalities in real life?
8. How do you convert compound inequalities into interval notation?

### Tip:
When solving absolute value inequalities, always consider the two cases that arise from the definition of absolute value to convert it into a compound inequality.

Learn how to express numbers less than 3 units away from 10 on a number line using absolute value inequalities. This page provides a detailed explanation and solution to the problem, covering key mathematical concepts and formulas. Suitable for high school students.

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