Problem 35 - Introductory Algebra for College Students

Question

Graph each inequality. $$x \geq 0$$

Step-by-Step Solution

Answer

The graph of \(x \geq 0\) is a number line with a closed circle at 0, and the portion of the line extending to the right from 0 is shaded.

Step 1: Draw a Number Line

Draw a number line. Place an arrowhead at the end indicating the direction of positive infinity. The center of the number line represents zero.

Step 2: Plot the Inequality Boundary

Mark zero, the boundary of the inequality, on the number line. Since the inequality includes an equals sign (\(\geq\)), the boundary itself is part of the solution. Therefore, a closed circle needs to be drawn at \(x = 0\).

Step 3: Shade the Solution Region

Because the inequality is \(x \geq 0\), the solution includes all values greater than or equal to zero. To represent this visually, shade the portion of the number line that extends to the right from zero (including zero itself).

Key Concepts

Number Line

A number line is a visual representation of numbers placed in order along a straight line. It's a simple yet powerful tool that helps in understanding the relationship between numbers. You can visualize positive and negative numbers, zero, fractions, and even inequalities on a number line.

When drawing a number line for an inequality, begin by drawing a horizontal line. On this line, you mark the critical points that are relevant to the inequality you are working with. An arrowhead should be placed at each end of the line to indicate that the numbers extend infinitely in both directions.

• The center is usually marked as zero.
• The direction to the right of zero represents positive numbers, while to the left are negative numbers.

Using a number line effectively can help to fully grasp the concept of inequalities and how they are visually represented.

Inequality Boundary

The inequality boundary is a crucial concept when graphing inequalities. It's a point or value that sets the limit where the inequality begins or ends. For the inequality \(x \geq 0\), the boundary is at zero.

Mark this boundary on the number line to identify the starting point of the solution for the inequality. The type of boundary depends on whether the inequality includes equality:

• If the inequality includes 'greater than or equal to' (\(\geq\)) or 'less than or equal to' (\(\leq\)), use a closed circle to denote that the boundary value itself is included in the solution set.
• If the inequality is strict, such as 'greater than' (\(>\)) or 'less than' (\(<\)), use an open circle to show that the boundary value is not part of the solution set.

For our inequality, \(x \geq 0\), draw a closed circle at zero to show that zero is included in the solution set.

Solution Region

The solution region is the part of the number line that satisfies the inequality. It is the visual representation of all possible solutions to the inequality. Once you have identified and marked the inequality boundary, you need to determine which part of the number line to shade.

For the inequality \(x \geq 0\), the solution region includes zero and all the numbers to the right of zero. This is because any number greater than or equal to zero satisfies the inequality. To graph this:

• Begin shading at the boundary point, zero in this case.
• Continue shading to the right, extending the shade arrow to signify all numbers infinitely greater than zero.

Shading helps to visually capture the set of all values that are solutions, making it clear which values satisfy the inequality on the number line.