Linear Inequalities in the Number Line - Justin Skycak - Free Printable

The image shows four number lines, each representing a different inequality involving the variable \( x \). Let's analyze each one step by step:

1. First Number Line: \( x > 3 \)

- Inequality: \( x > 3 \)
- Explanation: This means \( x \) is greater than 3. On the number line, we shade all the values to the right of 3.
- Circle at 3: Since \( x \) is strictly greater than 3 (not equal to 3), we use an open circle at 3 to indicate that 3 is not included in the solution set.
- Shading: The shading extends to the right of 3, indicating all values greater than 3.

2. Second Number Line: \( x \geq 3 \)

- Inequality: \( x \geq 3 \)
- Explanation: This means \( x \) is greater than or equal to 3. On the number line, we shade all the values to the right of 3, including 3.
- Circle at 3: Since \( x \) can be equal to 3, we use a closed (filled) circle at 3 to indicate that 3 is included in the solution set.
- Shading: The shading extends to the right of 3, including 3, indicating all values greater than or equal to 3.

3. Third Number Line: \( x < 3 \)

- Inequality: \( x < 3 \)
- Explanation: This means \( x \) is less than 3. On the number line, we shade all the values to the left of 3.
- Circle at 3: Since \( x \) is strictly less than 3 (not equal to 3), we use an open circle at 3 to indicate that 3 is not included in the solution set.
- Shading: The shading extends to the left of 3, indicating all values less than 3.

4. Fourth Number Line: \( x \leq 3 \)

- Inequality: \( x \leq 3 \)
- Explanation: This means \( x \) is less than or equal to 3. On the number line, we shade all the values to the left of 3, including 3.
- Circle at 3: Since \( x \) can be equal to 3, we use a closed (filled) circle at 3 to indicate that 3 is included in the solution set.
- Shading: The shading extends to the left of 3, including 3, indicating all values less than or equal to 3.

Summary of the Solutions:

1. \( x > 3 \): Open circle at 3, shading to the right.
2. \( x \geq 3 \): Closed circle at 3, shading to the right.
3. \( x < 3 \): Open circle at 3, shading to the left.
4. \( x \leq 3 \): Closed circle at 3, shading to the left.

The final answer is:

\[
\boxed{x > 3, \, x \geq 3, \, x < 3, \, x \leq 3}
\]