Graph each inequality. x ≤ 3

Verified step by step guidance

1. Identify the inequality given: \(x \leq 3\). This means we are looking for all values of \(x\) that are less than or equal to 3.
2. Draw a number line and locate the point corresponding to \(x = 3\) on it.
3. Since the inequality includes \(x \leq 3\) (less than or equal to), use a solid dot or closed circle at \(x = 3\) to indicate that 3 is included in the solution.
4. Shade the number line to the left of \(x = 3\) because all values less than 3 satisfy the inequality.
5. Label the shaded region clearly to show that it represents all \(x\) such that \(x \leq 3\).

Key Concepts

Inequalities on the Number Line

An inequality like x ≤ 3 represents all values of x that are less than or equal to 3. On a number line, this includes the point 3 and all points to its left. Understanding how to represent these values visually is essential for graphing inequalities.

Closed and Open Circles

When graphing inequalities, a closed circle is used to indicate that the endpoint is included (≤ or ≥), while an open circle shows the endpoint is excluded (< or >). For x ≤ 3, a closed circle is placed at 3 to show that 3 is part of the solution.

Shading the Solution Region

After marking the boundary point, the solution region is shaded to represent all values satisfying the inequality. For x ≤ 3, the shading extends to the left of 3, indicating all numbers less than or equal to 3 are solutions.