Problem 60 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding Absolute value

The absolute value of a number signifies its distance from zero on the number line. Therefore, \(|x|=n\), where \(n\) is a positive number, typically has two solutions: \(x=n\) and \(x=-n\).

Step 2: Apply absolute value rules

Given the equation \(|x|=6\), based on the absolute value definition, \(x\) can be either 6 (since absolute value of 6 is 6) or -6 (since absolute value of -6 is also 6).

Key Concepts

Distance on the Number Line

Distances are a crucial aspect of understanding absolute values. Imagine you are looking at a straight line with numbers marked along it. This is your number line, a simple visual tool to represent numbers and their distances from each other. The number line extends infinitely in both directions, with zero typically placed at the center. Numbers to the right of zero are positive, while those to the left are negative.

When we talk about the distance of a number from zero, we are asking, "How far is this number from zero?" on the number line. The absolute value of a number is this distance, always taken as a non-negative value. So, if you stand at -3 on the number line, you are 3 units away from zero, just as you are 3 units away if you are at 3. This distance concept helps you understand why both -6 and 6 have the same absolute value of 6; they are both 6 units away from zero.

Absolute Value Definition

The absolute value of a number tells us about its magnitude, ignoring its direction. It is symbolized as \(|x|\), which can be read as "the absolute value of x". The absolute value function transforms negative numbers into their positive counterparts, while leaving positive numbers unchanged.

Here's how it works:

• If \(x\) is a positive number or zero, \(|x|\) simply equals \(x\).
• If \(x\) is a negative number, \(|x|\) equals \(-x\), reversing the negative sign.

For example, \(|3| = 3\) and \(|-3| = 3\). The absolute value is always zero or a positive number, never negative. This property makes it very handy in both mathematics and real-world problems where only magnitudes are of concern, such as distance, regardless of direction.

Positive Numbers in Equations

When solving equations involving absolute values, it’s crucial to remember that the result of \(|x|\) is always a positive number or zero. This is because of its definition as the distance from zero, which can never be negative.

In our example equation \(|x| = 6\), the number 6 indicates a distance from zero. As per the properties of absolute values:

• The solution set includes a positive number 6, because \(x = 6\) satisfies the equation.
• It also includes \(-6\), because \(|-6| = 6\), which is also a valid distance from zero.

Thus, recognizing that solutions must be positive numbers or zero helps simplify solving these equations, as negative results cannot be solutions unless they are converted to positive distances.