Question

\(|2 x-6|=|x|\)

Short answer

The solutions to the absolute value equation are \(x = 6\) and \(x = 2\).

Step-by-step solution

Isolate the Absolute Value Equations

To begin, split the equation into its two possible cases forming: \(2x - 6 = x\) and \(2x - 6 = -x\).

Solving for \(x\) in the first equation

The first equation \(2x - 6 = x\) simplifies to \(x = 6\).

Solving for \(x\) in the second equation

The second equation \(2x - 6 = -x\) simplifies to \(x = 2\).

Key concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. These symbols represent numbers and quantities in equations and expressions. For example, in our given problem, both sides of the equation

• \(|2x-6| = |x|\)

involve algebraic expressions. The vertical bars represent absolute value, which is a fundamental concept in algebra. Absolute value refers to the distance of a number from zero on the number line, regardless of direction. This means that \(|a| = a\) if \(a\) is positive, and \(|a| = -a\) if \(a\) is negative. Understanding absolute values allows us to manipulate equations in a way that accounts for both positive and negative solutions.

Equation Solving

Equation solving involves finding the value of variables that make the equation true. In this particular problem, we are solving for the variable \(x\). This involves isolating \(x\) within two distinct equations derived from the absolute value setup:

• \(2x - 6 = x\)
• \(2x - 6 = -x\)

For the first equation, we rearrange and simplify the equation to find \(x = 6\), by subtracting \(x\) from both sides, leading to \(2x - x = 6\).
For the second equation, we follow similar steps and simplify it to get \(x = 2\), which involves adding \(x\) to both sides, resulting in \(2x + x = 6\). This process shows that two different values of \(x\) satisfy the original absolute value equation.

Mathematical Solutions

Mathematical solutions are the answers or resolutions to problems expressed in mathematical terms. To reach these solutions, we often have to apply structured logic and processes, such as solving equations. For absolute value equations, like

• \(|2x-6| = |x|\)

the process of finding solutions involves handling both potential cases that arise from the absolute value properties.
This results in two separate linear equations to solve. Each gives a potential solution to the original problem. The solutions \(x = 6\) and \(x = 2\) were found by treating the absolute value equation distinctly for its positive and negative implications. By applying these procedures, one can determine all possible solutions to the equation, verifying each within the original equation framework.