Express the following inequality​ a) using a number​ line, b) in interval​ notation, and​ c) as a solution set​ (use set builder​ notation).
x less than or equals 6

To express the inequality \( x \leq 6 \) in various forms, here's how it is done:

### a) Number line representation:
On a number line, the inequality \( x \leq 6 \) means that all numbers less than or equal to 6 are included. On the number line:
- A **solid dot** is placed at \( 6 \) to indicate that \( 6 \) is included.
- An arrow extends to the left from \( 6 \) to represent all numbers less than 6.

### b) Interval notation:
In interval notation, we express the range of values that satisfy the inequality:
\[
(-\infty, 6]
\]
Here, \( -\infty \) represents that there is no lower bound (all negative numbers), and the square bracket \( ] \) at \( 6 \) indicates that \( 6 \) is included.

### c) Set builder notation:
In set builder notation, the inequality is written as:
\[
\{ x \mid x \leq 6 \}
\]
This reads as "the set of all \( x \) such that \( x \) is less than or equal to 6."

Would you like a more detailed explanation of any part of this solution?

### Related Questions:
1. How would you express \( x > 6 \) on a number line?
2. What would the interval notation be for \( x > 6 \)?
3. How would you represent \( x \geq -2 \) using set-builder notation?
4. What is the difference between open and closed intervals?
5. How do you represent compound inequalities using interval notation?

**Tip:** In interval notation, use a square bracket [ ] when the endpoint is included (closed), and a parenthesis ( ) when it’s not included (open).

Express the following inequality​ a) using a number​ line, b) in interval​ notation, and​ c) as a solution set​ (use set builder​ notation). x less than or equals 6

Learn how to express the inequality x ≤ 6 using a number line, interval notation, and set builder notation. This explanation covers core concepts of inequalities and representations on a number line, suitable for students in Grades 6-8.

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