Problem 26 - College Algebra with Modeling and Visualization

Question

Solve the absolute value equation. $$|6 x-9|=0$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{3}{2}\) or \(x = 1.5\).

Step 1: Understand the absolute value equation

The equation \(|6x - 9| = 0\) states that the expression inside the absolute value, \(6x - 9\), must equal 0. This is because the absolute value of zero is zero.

Step 2: Remove the absolute value

Since the absolute value is set to 0, we can directly set the inside of the absolute value to 0. Therefore, the equation becomes \(6x - 9 = 0\).

Step 3: Solve for x

To solve the equation \(6x - 9 = 0\), first add 9 to both sides to isolate the term with \(x\). This gives us \(6x = 9\). Then, divide both sides by 6 to solve for \(x\). Thus, \(x = \frac{9}{6}\), which simplifies to \(x = \frac{3}{2}\) or \(x = 1.5\).

Key Concepts

Solving Equations

Equations are like puzzles that we need to solve to find out the unknown value, usually represented by a letter like \(x\). When we solve an equation, we aim to find out what value \(x\) must take for the equation to be true.
Specifically, when working with absolute value equations like \(|6x - 9| = 0\), understanding that the expression inside the absolute value must equal zero is key advice. This step simplifies solving the equation because it subtracts the complexity of the absolute function.
Start by setting the inside expression equal to zero, which turns the absolute value equation into a simpler linear format. Here, \(6x - 9 = 0\) shows this transition. Solving it involves straightforward algebraic manipulations such as adding or subtracting numbers to both sides and then dividing if needed to isolate \(x\). This step-by-step approach will lead you to the answer efficiently.

Mathematical Modeling

Mathematical modeling involves using mathematics to solve real-world problems or to represent real-life scenarios. In an absolute value equation like \(|6x - 9| = 0\), the setup can model situations where perfect balance or a starting point is required.
Absolute values often symbolize distances, regardless of direction, which can be crucial in various fields such as physics or engineering where equality and thresholds must be determined.
Translating real-world problems into mathematical terms helps in analyzing and deriving solutions that are both practical and efficient. This process simplifies complex problems, making them more manageable and solvable through algebra and other mathematical tools.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like + or -) without an equality sign. They form the building blocks of equations. In our equation \(|6x - 9| = 0\), \(6x - 9\) is the algebraic expression.
The first step is often rewriting or rethinking this expression, especially in the context of absolute values. Understanding how expressions behave when equal to zero gives more insight into the possible solutions of the equation.
Simplifying algebraic expressions usually involves combining like terms or breaking them down into more digestible parts, which enables easier manipulation and problem-solving. When handling expressions within absolute value terms, eliminating the absolute value by setting the inside equal to zero can drastically streamline the process.