Problem 62 - College Algebra Essentials

Question

Solve each absolute value equation or indicate that the equation has no solution. $$ |x|=6 $$

Step-by-Step Solution

Answer

Therefore, the solutions for the equation |x|=6 are x=6 and x=-6.

Step 1: Understand absolute value

To start off, understand what absolute value means. The absolute value of number is its distance from zero. Therefore, it is always a positive number or zero.

Step 2: Create two equations

The equation |x|=6 can be rewritten as two equations: x=6 and x=-6. This is because both 6 and -6 are 6 units away from zero.

Step 3: Solve the equations

Both equations x=6 and x=-6 are already solved. The solutions are x=6 and x=-6 respectively. This means that this absolute value equation has two solutions.

Key Concepts

Solving Equations

When solving equations, especially absolute value equations, we aim to find the values of the variable that make the equation true. An absolute value equation like \(|x| = 6\) involves understanding the concept of distance in mathematical terms. Since the absolute value represents a number's distance from zero on a number line, it is always positive or zero. The equation \(|x| = 6\) implies the variable \(x\) is 6 units away from zero. To solve it, we split it into two separate equations: \(x = 6\) and \(x = -6\). Thus, solving the equation involves identifying all possible values that the variable can take to satisfy the given condition.

Algebraic Solutions

In the realm of algebraic solutions, rewriting the equation to simplify or isolate the variable is key. For illustrating purposes, consider the absolute value equation \(|x| = 6\). We rewrite it as two equations: \(x = 6\) and \(x = -6\). This takes into account both the positive and negative solutions that arise due to the nature of absolute values.
The step of creating two separate equations is crucial. It leverages the definition that absolute values reflect a number's non-negative distance from zero. Each solution — \(x = 6\) and \(x = -6\) — corresponds to separate scenarios where the variable is either 6 units to the right or left of zero on a number line.

Mathematical Concepts

Grasping the mathematical concepts behind absolute value equations is fundamental to mastering them. The absolute value, denoted \(|x|\), fundamentally asks: "How far is \(x\) from zero?" This understanding helps when considering why there are potentially two answers. An equation like \(|x|=6\) recognizes that 6 and -6 both lie six units from zero, making them valid solutions.
Absolute value equations often appear simple, but they teach valuable lessons in symmetry and properties of integers. Recognizing that absolute values ignore direction (positive or negative) but only consider magnitude reinforces the concept that solutions can come from divergent scenarios. Embracing these dual possibilities enhances one's problem-solving toolkit in broader algebraic contexts.