Problem 27 - Algebra for College Students

Question

Graph \(|x|<2\). [Hint: Remember that \(|x|<2\) is equivalent to \(-2

Step-by-Step Solution

Answer

Shade the interval between -2 and 2 on a number line, excluding -2 and 2.

Step 1: Understand the inequality

The problem involves graphing the inequality \(|x|<2\). By the absolute value property, this inequality can be split into a compound inequality as suggested by the hint: \(-2 < x < 2\). This means that we want the values of \(x\) such that \(x\) lies between \(-2\) and \(2\) exclusive.

Step 2: Plot the critical points

Begin by marking the critical points on a number line. These points are \(-2\) and \(2\), which represent the boundaries of the inequality.

Step 3: Indicate open intervals

Since the inequality is strict (i.e., it uses '<' rather than '≤'), use open circles at the critical points \(-2\) and \(2\) on the number line. Open circles demonstrate that these endpoints are not included in the set of solutions.

Step 4: Shade the interval

Shade the region of the number line that falls between \(-2\) and \(2\). This shaded interval represents all the numbers \(x\) that satisfy the inequality \(|x|<2\).

Key Concepts

Number Line Graphing

Graphing on a number line is a simple and effective way to visualize the solution of inequalities, including absolute value inequalities. To graph an inequality like \(|x|<2\), we begin by understanding the type of inequality we are dealing with. Since it is an absolute value inequality, we translate it into a compound inequality: \(-2 < x < 2\). To plot this on a number line:

• Identify the critical points, \(-2\) and \(2\) in this case, as the boundaries of the inequality.
• Place these points on the number line.
• Draw open circles at \(-2\) and \(2\) because these values are not part of the solution set due to the strict inequality sign "<".
• Shade the area between these points representing all values of \(x\) that satisfy \(-2 < x < 2\).

This visual representation helps you quickly see the range of values that fulfill the inequality.

Compound Inequalities

Compound inequalities express two sets of conditions that a solution must satisfy simultaneously. In essence, they are two simple inequalities combined into one. Take \( |x| < 2 \) for example, which becomes \(-2 < x < 2\). This compound inequality states that \(x\) must be greater than \(-2\) and simultaneously less than \(2\).

• The solution is a continuous range of numbers between these two points.
• Both conditions must be true at the same time for any number in the set.

Compound inequalities are critical in narrowing down exact values or ranges that solve an absolute value inequality problem. When graphing such inequalities, we consider the conjunction of both simple inequalities, meaning values of \(x\) plotted satisfy both conditions simultaneously.

Inequality Notation

Inequality notation is a concise way to express that one value is greater than, less than, equal to, or not equal to another. In inequalities like \( |x| < 2 \), the use of a symbol such as "<" specifies a strict inequality meaning we're considering numbers less than 2 and greater than -2 without including these endpoints.Open circles are used in graphing to indicate that endpoints are not part of the solution (as opposed to closed circles, which would indicate inclusion, used for "≤" or "≥" inequalities). This notation is essential not just for representing solutions but also for understanding the critical differences between strict and non-strict inequalities. The absence of an equality part ("≤" or "≥") in the inequality \( |x| < 2 \) is why we use open circles on the number line.

Critical Points

Critical points are key values that define the boundaries of a solution set for an inequality. For the inequality \( |x| < 2 \), the critical points are \(-2\) and \(2\). These determine the interval within which all potential solutions lie.

• Mark these critical points on the number line.
• Check if they should be included in the solution set - in this case, they are not because of the strict inequality, hence open circles are used.
• They help identify the region on the number line to shade, banning any \(|x|=2\) endpoint values from the solution.

When dealing with inequalities, these points often indicate transitions between different solution types, helping visualize and grasp the impact of the inequality on the solution set.