Problem 13 - Elementary Algebra

Question

Solve the inequalities by graphing. $$ x \leq 0 $$

Step-by-Step Solution

Answer

Question: Graph the inequality \(x \leq 0\) and describe the solution on the number line. Answer: The graph of the inequality \(x \leq 0\) has a closed circle at x=0 and shading to the left of the circle. This means that all the values of x less than or equal to 0 are included in the solution.

Step 1: Draw the number line

Begin by drawing a number line, which is a horizontal line with tick marks indicating integers. Make sure to include 0 and both negative and positive integers on either side of it, for example, -3, -2, -1, 0, 1, 2, and 3.

Step 2: Indicate the boundary point

On the number line, the boundary point is 0, representing the value at which the inequality changes its direction. Since the inequality is less than or equal to zero, we want to include the boundary point in our solution. You can indicate this by placing a closed circle on top of the tick mark representing zero.

Step 3: Shade the solution region

As x must be less than or equal to zero, the solution region is to the left of the boundary point. Shade this region by drawing an arrow pointing to the left, starting at the closed circle on 0. The graph of the inequality \(x \leq 0\) consists of a closed circle at x=0 and shading to the left of the circle, indicating that all the values of x less than or equal to 0 are included in the solution.

Key Concepts

Algebra and Inequalities

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. Inequalities are relations that compare two expressions and can be visualized on a number line. They reveal the range of possible values that satisfy the inequality's condition.

In the context of inequalities, algebra's primary role is to enable us to manipulate and solve equations and inequations, which in this case is graphing the inequality. A simple inequality equation like the one given, \(x \leq 0\), can tell us a great deal about the value of \(x\) — for instance, \(x\) can be zero or any negative number. The act of graphing this inequality translates that information into a visual form, making it easier to grasp and interpret.

Number Line

A number line is a straight, horizontal line that serves as a visual representation of numbers, where each point corresponds to a real number. It's a fundamental tool in mathematics, particularly useful for illustrating concepts in algebra such as inequalities.

To graph an inequality on a number line, it's important to plot a whole range of numbers, including both positive and negative integers, as well as zero. In the exercise, the presence of negative and positive integers gives us the context within which the inequality will be represented.

Boundary Point

In graphing inequalities, a boundary point is a number that represents the threshold or limit of the region where the inequality holds true. This point is demarcated on a number line to show where the conditions of the inequality begin or end.

When an inequality includes the boundary point, such as \(x \leq 0\), it is represented by a solid or closed circle on that number. Conversely, an open circle would be used if the inequality did not include the boundary point, such as \(x < 0\). The closed circle or dot on the boundary emphasizes that the number zero itself is a solution to the inequality.

Solution Region

The solution region refers to the set of all possible values that satisfy the inequality. It's the area on the number line where the truth of the statement is maintained. In the exercise, since we are looking at \(x \leq 0\), the solution region encompasses all numbers equal to zero and to the left of zero, which are the negative numbers.

To indicate the solution region on a number line after the boundary point has been established, one typically shades or draws an arrow in the direction where the inequality holds true. By doing so, the viewer can immediately identify the range of acceptable values for \(x\), which in this case, consists of zero and any number to its left.