Problem 1 - College Algebra with Modeling and Visualization

Question

Exercises \(1-8:\) Let \(a \neq 0\) Solve \(|x|=3\)

Step-by-Step Solution

Answer

Solutions are \(x = 3\) and \(x = -3\).

Step 1: Understand the Absolute Value Equation

The absolute value equation \(|x| = 3\) means that the expression inside the absolute value, which is \(x\) in this case, is at a distance of 3 units from zero on the number line. Thus, \(x\) can either be \(3\) or \(-3\).

Step 2: Solve for Possible Values of x

Since \(|x| = 3\) implies that the distance from zero is 3, we set up two separate equations to remove the absolute value: 1. \(x = 3\)2. \(x = -3\)

Step 3: Check Solutions for Each Case

Verify both solutions satisfy the original equation:- For \(x = 3\),\(|3| = 3\) which is true.- For \(x = -3\), \(|-3| = 3\) which is also true.

Key Concepts

Solving Equations with Absolute Values

When dealing with absolute value equations like \(|x| = 3\), we're essentially figuring out which numbers are exactly 3 units away from zero on the number line. Absolute value measures distance, not direction, which means both \(x = 3\) and \(x = -3\) satisfy this equation. Here's how to approach solving these equations:

• Identify the Absolute Equation: Recognize that \(|x| = 3\) means your goal is to find the values of \(x\) where the distance to zero is 3 units.
• Set Up Two Equations: Because the absolute value of a number is its distance from zero, you create two separate equations: \(x = 3\) and \(x = -3\).
• Evaluate Each Equation: Solve \(x = 3\) and \(x = -3\) to get the possible solutions.
• Verify Solutions: Plug each solution back into the original equation to ensure they satisfy \(|x| = 3\).

This systematic approach ensures that no possible solution is overlooked, and confirms the accuracy of your findings. Every time you solve such an equation, remember to analyze and declare all potential solutions as demonstrated.

Understanding the Number Line

To fully understand absolute value equations, it's helpful to think about the number line. A number line is a straight line where every point on it corresponds to a real number.Key aspects:

• Zero is the midpoint: The zero on a number line is the reference point for measuring distance. In the equation \(|x| = 3\), the values are measured as 3 units away from this point.
• Positive and negative values: On a number line, moving to the right of zero gives positive numbers, and moving to the left gives negative numbers.
• Distance is always positive: Distances on the number line are absolute, meaning they don't consider direction, only magnitude.

Visualizing this, if you move three spaces right of zero, you land on 3, and moving three spaces left lands on -3. Both of these points are equidistant from zero, thus fulfilling the requirement of the absolute value equation.

Essential Algebra Concepts

Algebra often involves manipulating equations to find unknown values. In absolute value equations, the algebraic concepts revolve around understanding and simplifying expressions.Let's break this down:

• Expression Simplification: Absolute values require you to consider both the positive and negative scenarios that make an equation true.
• Equation Solving: Once the absolute value is removed (as two separate cases), solving these linear equations becomes straightforward.
• Validation: After finding solutions, always substitute them back into the original equation to ensure they satisfy the equality—\(|3| = 3\) and \(|-3| = 3\) hold true in this context.

Understanding these fundamental algebra concepts is crucial, as they form the backbone of many problem-solving techniques used throughout mathematics. Embrace each step of the process to develop solid algebraic skills.