Problem 21 - Elementary and Intermediate Algebra

Question

Solve each equation. \(|x-5|=8\)

Step-by-Step Solution

Answer

The solutions are \(x = 13\) and \(x = -3\).

Step 1: Understanding Absolute Value

The equation given is \(|x-5|=8\). An absolute value equation like this represents two cases, because absolute value measures the distance of a number from zero on the number line. Therefore, \(|x-5|=8\) can be split into two separate equations: \(x-5=8\) and \(x-5=-8\).

Step 2: Solving First Equation

For the first case, \(x-5=8\), add 5 to both sides to isolate \(x\). This results in \(x=8+5\), or \(x=13\).

Step 3: Solving Second Equation

For the second case, \(x-5=-8\), again add 5 to both sides. This gives \(x=-8+5\), which simplifies to \(x=-3\).

Step 4: Conclusion

The solutions to the equation \(|x-5|=8\) are \(x=13\) and \(x=-3\). These two values for \(x\) satisfy the original equation as they maintain either a positive or negative difference from 5 equal to 8.

Key Concepts

Solving Absolute Value Equations

When tackling absolute value equations, it's important to understand that absolute value measures how far a number is from zero on a number line. This means it could be a positive or negative distance. For the equation \(|x-5|=8\), the absolute value could equal 8 by having the expression inside the absolute value be 8 or -8.

To solve absolute value equations like this, always set up two separate equations:

• One with the expression equal to the positive value: \(x-5=8\)
• And one with the expression equal to the negative value: \(x-5=-8\)

This approach ensures you've considered both scenarios where the absolute value could hold true. After setting up the two equations, solve each one independently to find all potential solutions for \(x\).

It's a straightforward process once you get the hang of splitting the equations up and solving them independently!

Algebra: Manipulating Equations

In algebra, solving equations involves manipulating them using basic arithmetic operations. For equations like \(x-5=8\) or \(x-5=-8\), you'll typically use addition or subtraction to isolate the variable.

Let's take \(x-5=8\) as an example. To get \(x\) by itself, you add 5 to both sides of the equation. This balances the equation and gives you \(x=13\). For \(x-5=-8\), you'd also add 5 to both sides, resulting in \(x=-3\). This step is essential in solving equations because it helps you isolate the variable, allowing you to find its specific value.

The key is always to perform the same operation on both sides of the equation. This keeps the equation balanced, much like keeping a scale even. Practicing these techniques enhances your algebra skills and builds a strong foundation for more complex math problems.

The Importance of Understanding Math Concepts

In math education, grasping the underlying concepts is crucial for problem-solving and comprehension. Understanding how to solve absolute value equations is a perfect example of this idea. Once you understand why an absolute value equation splits into two different possibilities, it facilitates your approach to the problem.

By grasping these concepts:

• Students develop confidence in solving various types of problems.
• They become better at recognizing patterns and relationships in math.
• They are more prepared for advanced topics that build on these principles.

Encouraging students to explore these "why" questions not only enhances their learning but also fosters a deeper appreciation of math. When you understand the reasoning behind algebraic processes, you don’t just memorize steps—you gain a toolkit for solving future equations and facing new challenges.