Problem 28 - College Algebra with Modeling and Visualization

Question

Solve the absolute value equation. $$|-x-4|=0$$

Step-by-Step Solution

Answer

The solution is \(x = -4\).

Step 1: Set the Expression Inside the Absolute Value Equal to Zero

Since the absolute value of any expression is zero only when the expression itself is zero, we set \(-x-4=0\).

Step 2: Solve for x

To solve for \(x\), add 4 to both sides of the equation:\(-x = 4\). Next, multiply both sides by -1 to get \(x = -4\).

Key Concepts

Algebra

Algebra is a branch of mathematics representing numbers through variables and represents relationships between things. When you're dealing with algebra, you're working with equations that involve unknown variables and constants.

The key purpose of algebra is to solve for these unknown variables. A simple algebra equation could look like: x + 3 = 5, which can be solved by subtracting 3 from 5, resulting in x = 2.

A critical concept in algebra, relevant to our exercise, is the absolute value. Absolute value represents the distance of a number from zero on the number line, regardless of direction, so |-4| = 4.Algebra often progresses to more complex problems, where equations and inequalities may involve absolute values, like in our original exercise, where we set up and solve the equation \(|-x-4|=0\). Understanding the properties of numbers and how they interact through different algebraic operations is crucial for solving these equations.

Solving Equations

When solving equations, the goal is to find the value of the unknown variable that makes the equation true. This involves manipulating the equation using algebraic rules to isolate the variable on one side.

For the absolute value equation \(|-x-4|=0\), we need to consider what the absolute value represents. Since absolute value measures distance from zero and is only zero when the expression inside the absolute value is zero itself, we must first set the expression inside to zero: \(-x-4=0\).

Next, solve for the variable by undoing the operations using inverse operations. Here's how we do it:

• Add 4 to both sides: \(-x = 4\).
• Multiply both sides by -1 to isolate x, resulting in \(x = -4\).

This step-by-step process demonstrates the logical sequence in solving algebraic equations.

Mathematical Modeling

Mathematical modeling is the process of translating real-world scenarios into mathematical language. This involves using equations to represent patterns and express relationships between variables and constants.

For instance, consider a simple model where the absolute value equation \(|-x-4|=0\) could represent a physical process where a measurement should be exactly zero at a certain point, like balancing a scale.

By setting up an equation to depict the balance, we identify constraints and limits within the scenario. So, solving the equation \(|-x-4|=0\) helps us comprehend and predict outcomes under these specific conditions.

Mathematical modeling bridges the gap by helping us quantify and solve practical problems using mathematical methods. In the realm of algebra and solving equations, it's a powerful tool to simulate and refine understanding of complex systems in the real world.