Problem 33 - College Algebra

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ -2 \leq x<5 $$

Step-by-Step Solution

Answer

The solution set of the inequality \(-2 \leq x < 5\) is the set of all real numbers that are greater than or equal to \(-2\) and less than \(5\). This solution set is represented on the number line by a line segment that starts at \(-2\) and goes up to but does not include \(5\), with a closed circle at \(-2\) and an open circle at \(5\).

Step 1: Understand the inequality

This is a compound inequality. It's saying that \(x\) is greater than or equal to \(-2\) and less than \(5\). This means that all numbers that are greater than or equal to \(-2\) and less than \(5\) are solutions to the inequality.

Step 2: Graph the inequality

To graph the inequality, first draw a number line. Place two marks on the number line at \(-2\) and \(5\). Since the inequality includes \(-2\) (i.e. \(x\) is greater than \(\underline{or equal to}\) \(-2\)), the mark on \(-2\) should be a closed circle or a filled in dot. However, as the inequality does not include \(5\) (i.e. \(x\) is just less \(5\), not less than or equal to), the mark at \(5\) should be an open circle or an empty dot. Draw a line or a bar from \(-2\) to \(5\), including \(-2\) but excluding \(5\). The segment between the points \(-2\) and \(5\) on the number line is the solution set of the inequality.

Key Concepts

Compound Inequalities

Compound inequalities combine two simple inequalities. These can either be joined by "and" or "or." In this exercise, we have a compound inequality that combines conditions using "and." This sets the range for the values of \(x\). The expression \(-2 \leq x < 5\) stipulates that \(x\) must be greater than or equal to \(-2\) and less than \(5\). This means both conditions must be satisfied simultaneously. Compound inequalities create solution sets that are often continuous, forming a segment on the number line where each point is part of the solution. Recognizing compound inequalities is critical for correctly interpreting and graphing them, ensuring each part of the inequality is correctly expressed in the graph.

Number Line

The number line is a powerful visual tool for representing inequalities. It helps us to clearly see the range of solutions and the relationships between the numbers involved. To accurately represent the inequality \(-2 \leq x < 5\), a number line is drawn first.

• Position the key points: in this case,
  • -2, because \(-2\) is the minimum value \(x\) can take.
  • 5, because \(5\) is the maximum value \(x\) is approaching but not reaching.
• Include both negative and positive values if needed for broader context.

The line should capture all numbers within the specified interval, providing a clear overview of what values \(x\) can actually assume.

Solution Set

The solution set of an inequality consists of all possible values for a variable that satisfy the inequality. For the compound inequality \(-2 \leq x < 5\), the solution set includes every number from \(-2\) to just below \(5\). To express this set graphically, we draw a line segment on a number line that starts at \(-2\) and ends just before \(5\). The line segment visually represents all values \(x\) could be, marking a continuous range of solutions.

• The arrows do not extend indefinitely because the inequality does not express "greater than" or "less than" in a broad sense but defines specific boundaries.
• The critical aspect of a solution set is showing which endpoints are included or excluded.

In essence, the solution set is the part of the number line that satisfies all parts of the compound inequality.

Closed and Open Circles

Closed and open circles are symbols used in graphing inequalities on a number line to denote inclusion and exclusion of boundary points.

• A closed circle indicates that a number is included in the solution set. This corresponds to "greater than or equal to" or "less than or equal to," as in \(-2 \leq x\), marking a closed circle at \(-2\).
• An open circle signifies that a number is not part of the solution set, representing strict inequalities like "greater than" or "less than," such as \(x < 5\) in the example, leading to an open circle at \(5\).

These visual tools help differentiate between what values within a set are eligible solutions. The precise usage of closed and open circles removes ambiguity, ensuring that anyone interpreting the graph understands which boundary points are part of the inequality's solution.