Problem 72 - Algebra for College Students

Question

Solve each equation. \(|x+1|=|x-4|\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{3}{2}\).

Step 1: Understand the absolute values

The given equation \(|x+1| = |x-4|\) involves absolute values, which means that the values inside the absolute value bars can be equal or they can be opposites. We need to break this into cases to solve it properly.

Step 2: Separate into two case equations

When dealing with absolute values, we consider two cases: 1. \(x + 1 = x - 4\) (where the expressions inside the absolute values are equal)2. \(x + 1 = -(x - 4)\) (where the expressions inside the absolute values are opposites)

Step 3: Solve the first equation

For the first case \(x + 1 = x - 4\), subtract \(x\) from both sides to get:\[1 = -4\]This is a contradiction, which means there is no solution from this case.

Step 4: Solve the second equation

In the second case \(x + 1 = -(x - 4)\), simplify by distributing the negative sign on the right side:\[x + 1 = -x + 4\]Next, add \(x\) to both sides to obtain:\[2x + 1 = 4\]Subtract 1 from both sides:\[2x = 3\]Finally, divide both sides by 2:\[x = \frac{3}{2}\]

Step 5: Verify the solution

Substitute \(x = \frac{3}{2}\) back into the original equation to verify:Calculate \(|3/2 + 1| = |x + 1|\) which is \(|5/2|\) and \(|3/2 - 4| = |x - 4|\) which is also \(|5/2|\).Both sides equal, hence \(x = \frac{3}{2}\) is indeed a valid solution.

Key Concepts

Understanding Case Analysis

When faced with equations involving absolute values, such as \(|x+1| = |x-4|\), we must recognize that absolute values can produce two different scenarios. This is because the absolute value of a number represents the distance from zero, which means it disregards whether the number is positive or negative. Hence, we break the equation into two cases to ensure we capture all possible solutions.

• Case 1: The expressions inside the absolute values are equal, i.e., \(x + 1 = x - 4\).
• Case 2: The expressions inside the absolute values are opposites, i.e., \(x + 1 = -(x - 4)\).

This method of case analysis provides a comprehensive way to solve absolute value equations. By evaluating both scenarios, we can identify all potential solutions, ensuring none are overlooked.

Steps to Solving The Equations

Solving the equations derived from the absolute value equation involves handling each case separately. Let's walk through how to solve these equations using basic algebraic steps.

For the first case, \(x + 1 = x - 4\), attempt to isolate \(x\) by performing operations that simplify the equation. However, as we simplify:

1. Subtract \(x\) from both sides: - \(1 = -4\). 2. Notice this results in a contradiction, meaning this case yields no solution.

Now, consider the second case, \(x + 1 = -(x - 4)\), where you distribute the negative sign:

1. Simplify to get \(x + 1 = -x + 4\).2. Add \(x\) to both sides to maintain equality: - \(2x + 1 = 4\).3. Further simplify by subtracting \(1\) from both sides: - \(2x = 3\).4. Finally, divide each side by 2: - \(x = \frac{3}{2}\).

These systematic steps ensure clarity and accuracy in solving for \(x\).

Verification of Solutions

Verifying a solution is a crucial step in confirming the correctness of our result. After solving for \(x\), it’s important to substitute it back into the original equation to ensure both sides match. For the solution \(x = \frac{3}{2}\), checking involves substitution back into the absolute value equation:

• Calculate \(|x + 1|\): Substitute \(\frac{3}{2}\) into \(x\), yielding \(|\frac{3}{2} + 1| = |\frac{5}{2}|\).
• Calculate \(|x - 4|\): Substitute \(\frac{3}{2}\) into \(x\), yielding \(|\frac{3}{2} - 4| = |\frac{5}{2}|\).

Both expressions result in the value \(|\frac{5}{2}|\). Thus, the solution \(x = \frac{3}{2}\) is verified as correct by satisfying the original equation, ensuring our solution process was properly executed.