Problem 59 - College Algebra

Question

Solve each absolute value equation or indicate the equation has no solution. $$ |x|=8 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 8\) and \(x = -8\).

Step 1: Solve for positive solution

Since the absolute value of \(x\) is equal to 8, this means that \(x\) could be a positive 8. So, set \(x = 8\). This is one possible solution.

Step 2: Solve for negative solution

Again, because the absolute value of \(x\) is 8, \(x\) could also be a negative 8, since the absolute value of -8 is also 8. So, set \(x = -8\). This is the second possible solution.

Step 3: Combine solutions

In the end, we have two possible values for \(x\) that would make the absolute value equation true. These are \(x = 8\) and \(x = -8\).

Key Concepts

Solving Equations

Solving equations is the process of finding the values of variables that make the equation true. In the case of equations that involve an absolute value, the process can seem a bit different at first. When you encounter an equation like \(|x| = 8\), it's important to understand what this represents. The absolute value, denoted by vertical bars \(|\cdot|\), measures the distance of a number from zero on a number line, regardless of direction.

To solve the equation \(|x| = 8\), we need to find all possible values of \(x\) that satisfy this condition. We approach it by considering that any number could be either positive or negative. Thus, both \(x = 8\) and \(x = -8\) are solutions, because both have an absolute value of 8.

• Identify the absolute value expression and isolate it on one side.
• Set up two separate equations to consider both possible cases (positive and negative).
• Solve each equation individually.

This method ensures you find all possible solutions to the equation.

Absolute Value Property

The absolute value property is a fundamental concept in solving equations that include absolute values. Absolute value tells us how far a number is from zero, whether on the positive side or the negative side of a number line. This is why absolute value expressions can result in two potential solutions.

When dealing with absolute values in equations, remember:

• The absolute value expression \(|x|\) indicates distance from zero, not direction.
• If \(|x| = 8\), then \(x\) could be either 8 or -8.
• Consider both the positive and negative possibilities arising from this distance.

By understanding the absolute value property, you'll be able to tackle even more complex absolute value equations with confidence.

Positive and Negative Solutions

In absolute value equations, identifying the positive and negative solutions is key. For the equation \(|x| = 8\), the absolute value property simplifies the task. Essentially, you recognize that the result could come from positive 8 or negative 8. This means:

• For the positive solution: assume the number is on the positive side of the number line (\(x = 8\)).
• For the negative solution: assume the number is on the negative side (\(x = -8\)).

The beauty of this concept lies in its simplicity. The absolute value makes negative numbers positive by measuring only the distance. In practice, this results in multiple potential solutions, ready to be checked against the original equation.
By always looking for both the positive and negative solutions, you ensure a complete and thorough understanding of every equation you solve. This approach not only helps in simple equations but builds a solid foundation for solving more advanced mathematical problems.