Problem 2 - College Algebra with Modeling and Visualization

Question

Exercises \(1-8:\) Let \(a \neq 0\) $$ \text { Solve }|x| \leq 3 $$

Step-by-Step Solution

Answer

\(-3 \leq x \leq 3\)

Step 1: Understand the Absolute Value Inequality

The inequality \(|x| \leq 3\) implies that the distance between \(x\) and 0 on the number line is less than or equal to 3. This means \(x\) can be any real number between -3 and 3, inclusive.

Step 2: Rewrite the Absolute Value Inequality

Rewrite the inequality \(|x| \leq 3\) as a compound inequality: \[-3 \leq x \leq 3\] This represents all the values of \(x\) that satisfy the inequality.

Step 3: Interpret the Inequality Solution

The inequality \(-3 \leq x \leq 3\) tells us that \(x\) can assume any value starting from -3 to 3, including the endpoints. Therefore, the solution to the inequality is all numbers between -3 and 3, inclusive.

Key Concepts

Compound Inequalities

In algebra, compound inequalities are expressions involving two separate inequalities joined by the words "and" or "or." They are used to describe ranges of values for a variable. When looking at the absolute value inequality \(|x| \leq 3\), we can understand this as a compound inequality because it describes the values that \(x\) can take on.
The absolute value \(|x|\) represents the distance of \(x\) from 0 on a number line. Therefore, \(|x| \leq 3\) translates into two inequalities: \(-3 \leq x\) and \(x \leq 3\). They are connected by the word "and" since both must be true simultaneously.

• These inequalities together describe the range of x as being between -3 and 3.
• The solution to the compound inequality is the intersection of values that satisfy both conditions: \(-3 \leq x \leq 3\).

This logical combination helps in understanding how values interact in algebra, offering a way to communicate multiple conditions in one statement.

Number Line

A number line is a visual tool that helps to understand positions and distances between numbers. When solving absolute value inequalities, such as \(|x| \leq 3\), a number line is especially useful.
Number lines allow you to visually represent the solution set of inequalities. In this case:

• Plot the points \(-3\) and \(3\).
• Shade the area between them to indicate all numbers in this range.
• Include the endpoints by using closed circles (or dots) on \(-3\) and \(3\), since they are included in the inequality due to the "less than or equal to" (\( \leq \)) sign.

This shading visually demonstrates the solution set of the inequality and helps make the concept intuitive. Using number lines aids in understanding the nature of boundaries and solutions in mathematical expressions.

Algebra

Algebra is the branch of mathematics that deals with symbols and rules for manipulating those symbols. It provides the foundation for working with absolute values and inequalities. Let's break down how it applies to solving \(|x| \leq 3\).
In algebra:

• Absolute values are denoted by vertical bars (\\(| \cdot |\\)). They are used to express the non-negative magnitude of a number.
• When dealing with absolute inequalities, the \(\leq\) or \(\geq\) sign indicates a boundary within which the values can exist.

For \(|x| \leq 3\), we transform this inequality into an algebraic compound inequality: \(-3 \leq x \leq 3\). This linear transformation helps illustrate that the distance from zero is limited by 3 units in either direction.
By practicing with inequalities and absolute values, you develop skills to manipulate and interpret algebraic expressions more proficiently, laying the groundwork for more complex mathematical concepts.