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).