Question

Sketch the graph of the inequality. $$x \leq 3.5$$

Short answer

The graph is a number line with a solid circle at 3.5 and shading to the left of and including 3.5.

Step-by-step solution

Identify the boundary

In this inequality \(x \leq 3.5\), 3.5 is the boundary on the number line. Mark a point on 3.5.

Draw the line

Since the inequality includes equal to, the point at 3.5 is included in the solution. To show this, draw a solid circle at 3.5.

Shade the area

The inequality \(x \leq 3.5\) indicates that the solution includes numbers that are less than or equal to 3.5. As a result, shade the area on the number line to the left of and including the point at 3.5.

Key concepts

Number Line

A number line is a visual representation that helps us understand the relative position of numbers. It's a straight line with numbers placed at equal intervals along its length. To use a number line for graphing inequalities, we usually start by identifying the important values mentioned in the inequality. For instance, with the inequality \(x \leq 3.5\), you would draw a horizontal line and mark points for integral values, and then include the specific value 3.5. The point for 3.5 is crucial because it acts as a reference for the inequality's boundary. Once the number line is established, it becomes a useful tool for illustrating not just single numbers but also entire ranges of values which is often required when dealing with inequalities.

Boundary of Inequality

When graphing inequalities on a number line, the boundary of the inequality signifies the pivotal point around which the graph is defined. In our example of \(x \leq 3.5\), the number 3.5 is the boundary. The way we represent this boundary visually depends on whether the inequality is 'strict' (like \(<\) or \(>\)), or 'non-strict' (like \(\leq\) or \(\geq\)). In the case of non-strict inequalities, such as the one we have, the boundary is included as a possible solution to the inequality and is typically marked with a solid circle or a closed dot. If the inequality were strict (\(x < 3.5\)), we would represent the boundary with an open circle, indicating that 3.5 itself is not part of the solution set but is still the point where the inequality applies.

Shading Number Line

Shading the number line is a critical step in graphically representing inequalities. This visual technique helps to easily identify the set of numbers that satisfy the inequality. For example, if we have \(x \leq 3.5\), the shading will occur on the side of the boundary that contains all numbers less than or equal to 3.5. Given that 3.5 is included (as indicated by the solid circle), you would darken the number line starting from 3.5 and going towards the left to indicate all the numbers that are less than or equal to 3.5. This shaded area on the number line tells anyone looking at the graph that any number found within this shaded zone is a solution to the inequality. Shading is a great way to create a clear visual distinction of the solutions and is particularly advantageous for those who prefer learning through visual methods.