Problem 19 - College Algebra with Modeling and Visualization

Question

Solve the absolute value equation. $$ |-2 x|=4 $$

Step-by-Step Solution

Answer

The solutions are \(x = -2\) and \(x = 2\).

Step 1: Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering negative signs. So, if \(|-2x| = 4\), this means that \(-2x\) is either 4 or -4.

Step 2: Set Up Equations

Based on the property of absolute values, we set up two equations from \(|-2x| = 4\): 1. \[-2x = 4\]2. \[-2x = -4\]

Step 3: Solve First Equation

Taking the first equation, \(-2x = 4\):1. Divide both sides by -2:\[-2x \div -2 = 4 \div -2\] This simplifies to:\[x = -2\]

Step 4: Solve Second Equation

Taking the second equation, \(-2x = -4\):1. Divide both sides by -2:\[-2x \div -2 = -4 \div -2\]This simplifies to:\[x = 2\]

Step 5: Verify Solutions

Substitute \(x = -2\) and \(x = 2\) back into original equation:1. For \(x = -2\): \[ |-2(-2)| = |4| = 4 \] (True)2. For \(x = 2\): \[ |-2(2)| = |-4| = 4 \] (True)Both solutions satisfy \(|-2x| = 4\).

Key Concepts

Solving Equations

Solving equations is one of the fundamental skills in algebra and mathematics in general. When dealing with an equation, the main goal is to find the value of the variable that makes the equation true. Equations can be linear, quadratic, absolute value, and more.
For absolute value equations like \(|-2x| = 4\), we start by understanding the nature of absolute value. Absolute value represents the distance from zero, meaning it is always non-negative.

To solve \(|-2x| = 4\), we set up two separate equations because absolute values can be equal to a positive or negative number that has the same absolute magnitude:

• \(-2x = 4\)
• \(-2x = -4\)

Each of these individual equations has to be solved to find potential solutions for the variable \(x\). By solving these, we find that \(x\) can be -2 or 2, showing there can be more than one solution.

Algebraic Properties

Algebraic properties are rules that govern operations on numbers and variables. They are essential tools for simplifying and solving equations. One of the key algebraic properties used when solving equations is the division property of equality.
When we have an equation such as \(-2x = 4\), we use the division property of equality to solve for \(x\). This property states that if we divide both sides of the equation by the same non-zero number, the equality remains true. So, dividing both sides by -2 yields \(x = -2\).

This principle applies to the second equation \(-2x = -4\) as well, leading to the solution \(x = 2\) when divided by -2.

Remember, these properties ensure that as long as we perform the same operations on both sides of an equation, the equation remains balanced and valid. Understanding and applying these fundamental properties correctly is crucial to manipulating and solving any algebraic equation.

Real Numbers

Real numbers encompass both rational numbers (like fractions and integers) and irrational numbers (like \(\pi\) or \(\sqrt{2}\)). They make up the continuous number line, which represents the set of all possible numbers that can be used in mathematical equations.
When solving equations such as absolute value equations, the solutions we find are typically real numbers, as with this exercise resulting in \(x = -2\) and \(x = 2\).

These solutions mean that \(x\) can take on these values to satisfy the original equation \(|-2x| = 4\). Both -2 and 2 are examples of integers, a subset of real numbers. They demonstrate the concept of distance in real terms on the number line:

• -2 is two units to the left of 0
• 2 is two units to the right of 0

This wide coverage by real numbers allows for a full representation of possible solutions in contexts involving a wide range of mathematical and applied problems.