Problem 62 - Algebra and Trigonometry

Question

In Exercises 61–78, solve each absolute value equation or indicate that the equation has no solution. $$|x|=6$$

Step-by-Step Solution

Answer

The solutions to the equation \(|x| = 6\) are \(x = 6\) and \(x = -6\).

Step 1: Understanding Absolute Value

An absolute value is the positive magnitude of a particular number or variable and is indicated by two vertical lines: \(|x|\). So, \(|x|=x\) if \(x>=0\) and \(|x|=-x\) if \(x<0\). So, \(|x|=6\) denotes that \(x\) could be either 6 or -6.

Step 2: Splitting the equation

Accordingly, the above equation can be split into two separate equations: \(x = 6\) and \(x = -6\).

Step 3: Solving the equation

Both equations already have \(x\) isolated, therefore the solution to both equations is \(x = 6\) and \(x = -6\) respectively.

Key Concepts

Understanding Absolute Value

The absolute value of a number represents its distance from zero on the number line, irrespective of direction. In mathematics, we denote the absolute value of a variable, say x, with the symbol |x|. For example, the absolute value of both 5 and -5 is 5, as they are both five units away from zero.

Formally, the definition of absolute value is:

• For x >= 0, |x| = x
• For x < 0, |x| = -x

This means that the absolute value of x is x itself if x is positive or zero, and it is the negation of x if x is negative. Understanding this concept is crucial as it sets the foundation for solving absolute value equations.

Solving Absolute Value Equations

Solving an absolute value equation involves finding all the values of the variable that make the equation true. Since absolute value signifies how far a number is from zero without considering direction, we often end up with two potential solutions — one positive and one negative. For the equation |x| = 6, the variable x could be either 6 or -6 because both of these numbers are 6 units away from zero on the number line.

To solve an equation like |x| = 6, we can split it into two separate cases:

1. If x is non-negative, then |x| = x and the equation becomes x = 6.
2. If x is negative, then |x| = -x, and the equation becomes -x = 6 or x = -6.

After removing the absolute value symbol by considering both scenarios, we simply solve the resulting linear equations to find the values of x.

Isolation of Variable

Isolating the variable is a fundamental step in solving any algebraic equation. It means to manipulate the equation so that the variable we are trying to find is on one side of the equation by itself. In the context of absolute value equations, after removing the absolute value signs, we often get a straightforward equation where the variable is already isolated. For example, from our exercise, once we have determined that our equations are x = 6 and x = -6, the variable x is isolated in both cases.

To isolate a variable in more complex equations, we may need to perform operations such as adding or subtracting terms from both sides, multiplying or dividing by coefficients, and applying inverse operations. The goal is always to get the variable x by itself so that we can easily see what it equals. With practice, isolation of the variable becomes a more intuitive and straightforward process.