Problem 57 - Calculus with Concepts in Calculus

Question

Solve the equation. $$ |x+4|=|x-4| $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0\).

Step 1: Understand the Absolute Value

The absolute value of a number describes its distance from zero on a number line, regardless of direction. Therefore, for the equation \(|x+4|=|x-4|\), the expressions inside the absolute value signs determine the possible cases to consider.

Step 2: Set Up Cases Based on Absolute Values

Since absolute value expressions can affect the sign, we consider two main cases: 1) \(x+4\) and \(x-4\) are equal, and 2) \(x+4\) and \(x-4\) have opposite signs.

Step 3: Case 1 - Equal Expressions

Assume that both expressions are equal (i.e., \(x+4 = x-4\)). Solve for \(x\) to find one such solution: \[ x + 4 = x - 4 \] Subtract \(x\) from both sides: \[ 4 = -4 \] This case results in a contradiction, so it does not provide a valid solution.

Step 4: Case 2 - Opposite Signs

Consider the case where one expression is the negative of the other: \(x+4 = -(x-4)\). Solve for \(x\): \[ x + 4 = -x + 4 \] Add \(x\) to both sides: \[ 2x + 4 = 4 \] Subtract 4 from both sides: \[ 2x = 0 \] Divide by 2: \[ x = 0 \] This is a valid solution.

Step 5: Verify the Solution

Substitute \(x = 0\) back into the original equation to confirm:\( |0 + 4| = |0 - 4| \). Simplify both sides: \( 4 = 4 \). The solution holds true, confirming \( x = 0\) as correct.

Key Concepts

Distance on a Number Line

Understanding distance on a number line is essential for grasping absolute value equations. The number line is a visual representation where numbers increase to the right and decrease to the left. Absolute value refers to the distance a number is from zero, regardless of its direction on the number line. For example, the absolute value of both -3 and 3 is 3, because each is three units away from zero. This measurement is always non-negative, which is why absolute values are frequently encountered in real-world applications that involve distance, as actual distance cannot be negative.

When solving the equation \(|x+4| = |x-4|\), we rely on the principle that absolute values relate to distances. It poses the question: at what values of \(x\) do both of these expressions have the same distance from zero? Such equations invite us to consider different scenarios or cases, as the situation depends on whether and how the numbers related to \(x\) are positive or negative.

Solving Equations

Solving equations involves finding values for variables that make the equation true. In the context of absolute value equations like \(|x+4| = |x-4|\), this process can be slightly tricky, as it requires managing equations within the absolute value. The primary steps involve simplifying equations and isolating the variable to one side. We begin this process by identifying the expressions inside the absolute value symbols, focusing on solving them separately.

Achieving this includes performing algebraic operations such as addition, subtraction, and sometimes division or multiplication, to isolate the variable \(x\). It's important to monitor both sides of the equation, maintaining balance and equality through every step. In standard equations, once simplified, substitutions into the original equation are made to verify the accuracy of proposed solutions. Keep in mind, potential solutions must satisfy the equation when replaced back, thus ensuring they are truly valid.

Case Analysis in Equations

Case analysis is vital when dealing with equations containing absolute values, as these can imply multiple scenarios. For instance, \(|x+4| = |x-4|\) can be broken into different cases based on how absolute values affect the expressions inside.

• First case: Both expressions are equal. This assumes that the distances or absolute values are naturally identical without sign changes. However, in our example, this led to a false result, revealing no valid solution under this circumstance.
• Second case: The expressions have opposite signs. Here, one expression could be the negative counterpart of the other. Solving for this scenario, \(x + 4 = -(x-4)\), guided us to discover that \(x = 0\) satisfies the equation.

Analyzing each case helps in understanding possibilities and ruling out non-viable solutions. Case analysis ensures all scenarios are accounted for, thereby offering a comprehensive understanding of the problem and identifying all possible solutions efficiently. This analytical approach is crucial for thoroughly solving and interpreting absolute value equations.