Problem 12 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph the solutions of each inequality on a number line. $$-2 \leq x \leq 0$$

Step-by-Step Solution

Answer

The solution for the given inequality \(-2 \leq x \leq 0\) is graphed on a number line with closed circles at -2 and 0 and a solid line marking all the values between -2 and 0.

Step 1: Identify the Boundaries

Study the inequality \(-2 \leq x \leq 0\). Here, -2 and 0 are the lower and upper bounds of the inequality, respectively.

Step 2: Draw a Number Line

Draw a number line and mark these two points, -2 and 0, on it. Make sure to label these points clearly for easy reference.

Step 3: Graph the Inequality

Plot the inequality on the number line. Since \(-2 \leq x\), we will mark the point -2 with a closed circle. Similarly, because \(x \leq 0\), the point 0 is also marked with a closed circle. Draw a solid line (or shade) connecting these two points to show all the possible solutions to the inequality.

Key Concepts

Graphing Inequalities

Graphing inequalities involves visually representing the range of x-values that satisfy an inequality statement. To do this, we rely on a number line, which is a straight line depicting numbers in sequential order. By effectively graphing, you can see all the possible solutions that meet the condition set by the inequality. In our exercise, the inequality \[ -2 \leq x \leq 0 \]can be represented on a number line. This indicates that x can be any number from -2 to 0, inclusive of both endpoints. To represent it on the graph, we follow specific rules using symbols like closed circles to denote which points are part of the solution set.

Number Line

A number line is crucial in graphing inequalities, as it gives a clear visual context to numerical concepts. It is simply a horizontal line with evenly spaced markings indicating numbers. Typically, these markings have zero at the center, with negative numbers extending to the left and positive numbers stretching to the right. Number lines help in:

• Visualizing where numbers lie in relation to each other.
• Identifying boundaries when plotting inequalities.

In the given exercise, we begin by marking key points, -2 and 0, on the number line. These will serve as boundaries of our solution set for the inequality. It’s important to accurately label these points for clarity in understanding which areas of the number line represent solutions.

Closed Circle

When graphing inequalities, a closed circle is used to indicate that a particular value is included in the solution set. A closed circle appears as a solid dot on the number line. This visual marker tells anyone looking at the graph that the endpoint, where the dot is placed, is part of the solution.

In the inequality \(-2 \leq x\), the closed circle is used at -2, signaling that -2 is included in the range of possible solutions. Similarly, in \(x \leq 0\), the closed circle appears at 0, indicating that 0 is also included. This contrasts with an open circle, which would imply that an endpoint is not included in the solution.

Solutions of Inequalities

Solutions of inequalities represent all possible values of a variable that satisfy the inequality condition. In other words, they are the set of numbers that you can substitute into the variable place without making the inequality false.

For the problem at hand, the inequality \[-2 \leq x \leq 0\]means that x can take any value from -2 to 0, including both -2 and 0. This range of values is the set of solutions. Graphically, this is depicted by shading or drawing a solid line between these endpoints on the number line. Thus, the segment or the shaded area between -2 and 0 on the number line visually confirms the solutions to the inequality.