Problem 39 - Intermediate Algebra

Question

Solve each equation. See Example 3. $$ \left|\frac{1}{5} x+2\right|-8=-8 $$

Step-by-Step Solution

Answer

The solution is \( x = -10 \).

Step 1: Isolate the Absolute Value Expression

We begin by isolating the absolute value expression. The equation given is \( \left|\frac{1}{5}x + 2\right| - 8 = -8 \). To isolate the absolute value, add 8 to both sides:\[ \left| \frac{1}{5}x + 2 \right| = 0 \].

Step 2: Remove the Absolute Value

If the absolute value of a number is 0, the number itself must be 0. Hence, we set up the equation without the absolute value:\[ \frac{1}{5}x + 2 = 0 \]

Step 3: Solve for x

To solve \( \frac{1}{5}x + 2 = 0 \), first subtract 2 from both sides:\[ \frac{1}{5}x = -2 \]Then multiply both sides by 5 to solve for \( x \):\[ x = -10 \].

Key Concepts

Understanding Algebra

Algebra is like a magical language that helps us express and solve problems involving unknown numbers. In our example, the unknown is represented by the variable \( x \). The main goal in algebra is to find out the value of this unknown by following a series of logical steps. Variables are like placeholders; they represent numbers that we are trying to find. We often see equations in algebra, where we have expressions on either side of the equal sign. By manipulating these expressions using various algebraic rules, we strive to isolate the variable and find its value. It's important to remember that whatever operation you apply to one side of the equation, you must apply the same to the other side to maintain balance.

Equation Solving Process

Equation solving is a process where we find the value of unknowns in mathematical expressions that are set equal to each other. It often involves several steps, similar to peeling layers off an onion, until the unknown variable is isolated.

Here’s a straightforward procedure to solve equations:

• Identify the equation: Recognize the type of equation you're dealing with, such as linear, quadratic, or involving absolute values.
• Simplify the equation: Use algebraic operations like addition, subtraction, multiplication, or division to simplify the equation if necessary.
• Isolate the variable: Your goal is to have the variable alone on one side of the equation. This often involves moving terms around and ensuring the operations performed are inverse to what was originally done.
• Solve for the variable: Once isolated, simplify further to find the exact value of the variable.

In the exercise provided, we applied these steps by first isolating the absolute value term, then removing the absolute value to simplify the equation, and finally solving for \( x \) by performing arithmetic operations like addition and multiplication.

Absolute Value Basics

The absolute value of a number is essentially its distance from zero on the number line, regardless of the direction. It is always non-negative. Think of it as a measure of magnitude without considering direction.

For instance:

• The absolute value of both -3 and 3 is 3, because both are three units away from zero.
• This property is crucial when solving equations with absolute values, as it converts any negative result into a positive one.

When faced with an absolute value equation, like in the exercise, you often set the expression inside the absolute value equal to both the positive and negative of the value on the other side of the equation. However, when the absolute value equals zero, it simplifies because the expression inside must also be zero. Understanding this helps simplify the problem-solving process and provides a unique solution when dealing with an absolute value of zero.