Problem 3 - Algebra for College Students

Question

Solve each inequality and graph the solutions. \(|x| \leq 2\)

Step-by-Step Solution

Answer

The solution is \(-2 \leq x \leq 2\), and it is represented by a shaded interval between \(-2\) and \(2\) on the number line with solid dots.

Step 1: Understand Absolute Inequality

The inequality \(|x| \leq 2\) represents all values of \(x\) for which the absolute value of \(x\) is less than or equal to 2. Absolute value inequalities often describe intervals on a number line.

Step 2: Split the Absolute Value Inequality

An inequality of the form \(|x| \leq c\) can be rewritten as two inequalities: \(-c \leq x \leq c\). Thus, for \(|x| \leq 2\), we have the double inequality \(-2 \leq x \leq 2\).

Step 3: Graph the Solution

To graph \(-2 \leq x \leq 2\), draw a number line. Mark the points \(-2\) and \(2\) with solid dots to include them in the solution. Shade the region between \(-2\) and \(2\) to indicate all the values that satisfy the inequality.

Key Concepts

Solving Absolute Value Inequalities

Inequalities can be a bit tricky, especially when they involve absolute values. But don't worry, let's break it down together. An absolute value inequality such as \(|x| \leq 2\) asks us to find all numbers within a specific range. The absolute value, \(|x|\), represents the distance of a number from zero on a number line, without considering the direction. When we solve \(|x| \leq 2\), we seek all numbers whose distance from zero is no more than 2. Here’s how to tackle it:

• Start by splitting the inequality into two separate inequalities: \(-c \leq x \leq c\). Here \(c\) is 2, so you will get \(-2 \leq x \leq 2\).
• This actually represents a range of numbers including \(-2\) and \(2\) themselves. This is known as a double inequality, and it’s your solution set!

By understanding this concept, you can solve any basic absolute value inequality.

Graphing Inequalities

After solving the inequality, it’s time to visualize the solution. We achieve this by graphing the inequality on a number line. Graphing helps us see the solution set as a part of the number line, making it much easier to understand and communicate the range of values that work. To graph \(-2 \leq x \leq 2\):

• Draw a straight line representing the number line. It doesn’t have to be long, but make sure to include zero, negative values, and positive values.
• Locate and mark the numbers \(-2\) and \(2\) on this line.
• Since our inequality includes \(-2\) and \(2\) (indicated by the \(\leq\) symbol which means "less than or equal to"), place solid dots on the \(-2\) and \(2\).

This sketch will help you verify which numbers are solutions and whether the boundary points are included.

Number Line Representation of Inequality Solutions

Representing solutions on a number line is a powerful way to illustrate how values relate to each other. The number line shows you the continuum of possible values in a simple, visual manner. For our double inequality, \(-2 \leq x \leq 2\):

• After placing solid dots on \(-2\) and \(2\), shade the entire portion between these two points. This shaded section signifies all the numbers that make the inequality true.
• In this case, any number between \(-2\) and \(2\) can replace \(x\) in the inequality and maintain the truth of the statement.
• The solid dots signify that both \(-2\) and \(2\) are included as possible solutions. If we were dealing with a \(<\) or \(>\) scenarios, you would use open circles instead, indicating those points are not part of the solution set.

Understanding this form of representation aids in seeing solutions quickly and is especially useful for more complex inequalities.