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 or coordinate plane with the \(x\)-axis clearly labeled. Mark the point \(x = 3\) on the axis.
3. Since the inequality includes \(\leq\) (less than or equal to), use a solid vertical line at \(x = 3\) to indicate that points on this line are included in the solution.
4. Shade the region to the left of the line \(x = 3\) on the number line or coordinate plane, because \(x\) 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

Inequalities like x ≤ 3 represent 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 interpret and represent these sets visually is essential for graphing inequalities.

Graphing Inequalities

Graphing inequalities involves shading the region of the number line or coordinate plane that satisfies the inequality. For x ≤ 3, you shade all points at 3 and to the left, often using a solid dot at 3 to indicate that 3 is included in the solution.

Boundary Points and Solid vs. Open Dots

When graphing inequalities, the boundary point (here, x = 3) is marked with a solid dot if the inequality includes equality (≤ or ≥). If the inequality is strict (< or >), an open dot is used. This distinction shows whether the boundary value is part of the solution set.