Question

Sketch the graph of the inequality. $$x \leq 5$$

Short answer

The solution is understood by drawing a vertical line through \(x = 5\) and shade the region to the left, including the line (since the inequality is less than or equal to 5).

Step-by-step solution

Understanding the equation

The inequality presented is \(x \leq 5\), this is a simple one-dimensional inequality. The solution to this inequality is every real number that is less than or equal to 5.

Sketch the line

Sketch a vertical line through the point where \(x = 5\). This line represents all the points where \(x\) equals 5. Note: since the inequality include 'equal to', make the line solid.

Shade the area

The inequality is \(x \leq 5\), which means all the values less than or equal to 5. So, you shade the region on the left side of the line since that represents all the values where \(x < 5\).

Validate the solution

To validate this solution, choose any point from the shaded region and substitute it for \(x\) in the inequality. The inequality should hold true.

Key concepts

Understanding One-Dimensional Inequalities

One-dimensional inequalities are mathematical expressions that bear a lot of similarities to equations. However, instead of indicating equality, they show a relationship where one quantity is smaller or larger than another, or within a certain range. When you come across an inequality like \(x \leq 5\), imagine you are looking at a scale, where all the points to the left are less than 5 and all the points to the right are more than 5.

Understanding these inequalities is crucial for graphically representing them on a number line. In one-dimensional space, it's essentially a linear representation, indicating a range of values along a single axis. For the inequality \(x \leq 5\), the solution includes every number that is 5 or less, extending infinitely to the left. When illustrating this concept, think of the temperature on a thermometer. If the temperature must not exceed a certain point, we're interested in the area up to and including that point, similar to the 'less than or equal to' scenario in our inequality.

Sketching Vertical Lines for Inequalities

To visually convey inequalities on a graph, sketching lines is the first step. Specifically, when dealing with an inequality like \(x \leq 5\), we will sketch a vertical line. Why vertical, you might ask? Because in a two-dimensional coordinate system, a vertical line indicates all the points where x has a specific value.

In this case, the line is at \(x = 5\). Remember to use a solid line for this inequality since it includes the 'equal to' condition, which means the value at exactly x equals 5 is included in the solution set. A dashed line, on the other hand, would indicate that the value where x equals 5 is not included. When visualizing this, imagine a fence that demarcates boundaries for a property—a solid fence includes the boundary line, whereas a dashed line would exclude it.

Shading Number Lines to Represent Solutions

After sketching the vertical line, the next step is to represent the solution set of the inequality by shading the appropriate side of the number line. For \(x \leq 5\), we shade to the left of the line to indicate all the values of x that are less than or equal to 5. This shaded region expresses the range of solutions in a way that is immediately understandable.

Shading is like painting over the part of a piece of paper where a particular condition is met—everything within your colored area is a solution. To ensure you've shaded the right side, you can test a point. Pick any value less than 5 and see if it satisfies the inequality (it should). Practice with different types of inequalities will make shading feel more intuitive, helping students to instantly visualize solutions to these kinds of problems.