Problem 5 - Algebra and Trigonometry

Question

Solve the equation. $$ |x-3|=2 $$

Step-by-Step Solution

Answer

The solutions are \(x = 5\) and \(x = 1\).

Step 1: Understand the Absolute Value Definition

The absolute value equation \(|x-3|=2\) implies two equations: one where the expression inside the absolute value is equal to 2 and one where it is equal to -2. This is because the absolute value of a number is its distance from zero on the number line, meaning it is always non-negative.

Step 2: Set Up the Positive Case Equation

For the case where \(x-3 = 2\), we simply remove the absolute value and set up the equation as: \[ x - 3 = 2 \] This represents the scenario where the expression inside the absolute value equals its non-negative counterpart.

Step 3: Solve the Positive Case Equation

Add 3 to both sides of the equation to solve for \(x\): \[ x - 3 + 3 = 2 + 3 \] Which simplifies to \[ x = 5 \] This is one potential solution for the equation.

Step 4: Set Up the Negative Case Equation

For the case where \(x-3 = -2\), we consider the expression inside the absolute value to be negative:\[ x - 3 = -2 \] This represents the scenario where the expression inside the absolute value results in the opposite number before taking the absolute value (i.e., the negative counterpart).

Step 5: Solve the Negative Case Equation

Add 3 to both sides of the equation to solve for \(x\): \[ x - 3 + 3 = -2 + 3 \] Which simplifies to \[ x = 1 \] This is the second potential solution for the equation.

Step 6: Verify Solutions

Substitute the solutions \(x = 5\) and \(x = 1\) back into the original absolute value equation \(|x-3|=2\) to verify:- For \(x = 5\): \[ |5 - 3| = |2| = 2 \] - For \(x = 1\): \[ |1 - 3| = |-2| = 2 \] Both solutions satisfy the initial equation.

Key Concepts

Solving Absolute Value Equations

When solving absolute value equations like \(|x-3| = 2\), the key is to remember what absolute value signifies. The absolute value of a number represents its distance from zero on the number line. Thus, it can never be negative. This characteristic allows you to split the absolute value equation into two separate equations.

In our example, we consider the expression inside the absolute value, \(x-3\). To solve \(|x-3| = 2\), we assume two scenarios:

• The expression \(x-3\) equals the positive value, which results in the equation \(x-3 = 2\).
• The expression \(x-3\) equals the negative value, leading to the equation \(x-3 = -2\).

Both cases derive from the definition of absolute value: a non-negative number equal to 2 could either be positive or negative before applying the absolute value. This dual nature is the fundamental strategy for solving absolute value equations.

Algebraic Solutions for Absolute Value Equations

After establishing the two equations \(x-3 = 2\) and \(x-3 = -2\), we proceed to solve each equation using basic algebraic techniques.


For the equation \(x-3 = 2\), we solve for \(x\) by isolating it on one side:

• Add 3 to both sides to get \(x = 5\). This is the first solution.

Similarly, for the equation \(x-3 = -2\):

• Add 3 to both sides to get \(x = 1\). This is the second solution.

These algebraic steps involve simple addition and subtraction, reflecting the core operations needed to rearrange the equation and solve for \(x\). Understanding these steps solidifies one's ability to approach and solve equations systematically.

Verification of Solutions

Verifying solutions is a crucial step to ensure that they satisfy the original absolute value equation. In our problem, we have potential solutions \(x = 5\) and \(x = 1\).


To verify, substitute each solution back into the equation \(|x-3| = 2\) and check:

• Substitute \(x = 5\) into the equation: \(|5 - 3| = |2| = 2\). The equation holds true, so \(x = 5\) is a valid solution.
• Substitute \(x = 1\) into the equation: \(|1 - 3| = |-2| = 2\). The equation holds true again, making \(x = 1\) another valid solution.

This verification confirms that the solutions meet all conditions of the original equation, guaranteeing their correctness. Always remember to verify solutions to prevent errors and ensure complete accuracy in solving equations.