Problem 10 - Algebra for College Students

Question

Solve each equation. $$ |x-5|=13 $$

Step-by-Step Solution

Answer

The solutions are \( x = 18 \) and \( x = -8 \).

Step 1: Understand the Absolute Value Equation

An absolute value equation like \( |x - 5| = 13 \) means the expression inside the absolute value can be either positive or negative, resulting in two separate equations.

Step 2: Set Up the Two Equations

Since \( |x - 5| = 13 \), we can write this as two equations: \( x - 5 = 13 \) and \( x - 5 = -13 \).

Step 3: Solve the First Equation

Solve \( x - 5 = 13 \) by adding 5 to both sides. \( x - 5 + 5 = 13 + 5 \), which simplifies to \( x = 18 \).

Step 4: Solve the Second Equation

Solve \( x - 5 = -13 \) by adding 5 to both sides. \( x - 5 + 5 = -13 + 5 \), which simplifies to \( x = -8 \).

Step 5: Combine the Solutions

The solutions to the equation \( |x - 5| = 13 \) are \( x = 18 \) and \( x = -8 \).

Key Concepts

Understanding Absolute Value

The absolute value of a number is its distance from zero on a number line, regardless of direction. This is always a non-negative value. For example, the absolute value of both -3 and 3 is 3, represented as \(|-3| = 3\) and \(|3| = 3\). In an absolute value equation like \(|x - 5| = 13\), the expression inside the absolute value can be either positive or negative, leading to two possible cases.

Solving Absolute Value Equations

When solving an absolute value equation, separate the expression inside the absolute value into two different equations. One equation considers the positive scenario and the other considers the negative scenario. For example, if given \(|x - 5| = 13\), convert it into two equations: \x - 5 = 13\ (positive case) and \x - 5 = -13\ (negative case). This way, we can account for both possible values inside the absolute value.

Working with Two-Variable Solutions

After setting up the two equations from the absolute value equation, solve them individually to find two possible solutions. For \(|x - 5| = 13\), solve the equations \x - 5 = 13\ and \x - 5 = -13\:

• Add 5 to both sides to solve each equation:

- Solve \x - 5 + 5 = 13 + 5\ to get \x = 18\.
- Solve \x - 5 + 5 = -13 + 5\ to get \x = -8\.
Thus, there are two solutions for \x\: 18 and -8.

Step-by-Step Solution for Clarity

Following a clear step-by-step approach can help break down the process of solving absolute value equations:

• Step 1: Start by understanding the nature of the absolute value equation.
• Step 2: Form two equations by considering both positive and negative scenarios.
• Step 3: Solve the first equation by isolating the variable.
• Step 4: Solve the second equation similarly.
• Step 5: Combine the solutions from both equations.

This method provides clear and manageable steps to ensure a thorough understanding and successful solution of the problem.