Problem 77 - College Algebra Essentials

Question

Absolute value expressions are equal when the expressions inside the absolute value bars are equal to or opposites of each other. $$ |3 x-1|=|x+5| $$

Step-by-Step Solution

Answer

The solution for the absolute value equation \(|3x - 1| = |x + 5|\) is \(x = 3\) and \(x = -1\).

Step 1: Separate the absolute value equation into two scenarios

First scenario is when the values inside the absolute signs are equal. That's when \(3x - 1 = x + 5\). The second scenario is when the values inside the absolute signs are opposite of each other, i.e., \(3x - 1 = - (x + 5)\)

Step 2: Solve for x in both scenarios

For equation \(3x - 1 = x + 5\), solving for \(x\) gives \(x = 3\). For equation \(3x - 1 = - (x + 5)\), after solving for \(x\), we get \(x = -1\)

Step 3: Confirm the solutions

Substituting the values of \(x\) in the initial equation verifies both solutions. For \(x = 3\), both sides equal to 8 and for \(x = -1\), both sides equal to 4.

Key Concepts

Solving Absolute Value Equations

To understand the beauty of absolute value equations, let's delve into what they are. Absolute value captures the distance of a number from zero on the number line. This means it's always a non-negative number. When we solve absolute value equations, we're looking at two potential scenarios: the expressions inside the absolute value bars are either equal or they are opposites.
In a typical problem, like \(|3x - 1| = |x + 5|\), we want to determine the values of \(x\) that make this true. This boils down to considering two cases:

• Case 1: The expressions inside the absolute value are equal. This gives us the first equation: \(3x - 1 = x + 5\).
• Case 2: The expressions are opposites, leading to the second equation: \(3x - 1 = -(x + 5)\).

By solving each of these simpler equations, we'll find all possible solutions for \(x\).
Remember, these scenarios arise from the definition of absolute values and these methods help catch all potential solutions.

Algebraic Equations

Algebraic equations are all about finding the unknowns, and they come in handy when solving absolute value problems. When faced with the equations derived from absolute values, it's crucial to solve them using basic algebra techniques like combining like terms and isolating the variable.
Let's break this down:

• For the equation \(3x - 1 = x + 5\), rearrange it to isolate \(x\) by first subtracting \(x\) from both sides. This simplifies to \(2x - 1 = 5\). Adding 1 to both sides gives \(2x = 6\). Then, divide by 2 to find \(x = 3\).
• In the second equation \(3x - 1 = -(x + 5)\), you'll distribute the negative sign, leading to \(3x - 1 = -x - 5\). Adding \(x\) to both sides gives \(4x - 1 = -5\). Then add 1 to both sides to get \(4x = -4\), and dividing by 4 results in \(x = -1\).

Solving these algebraic equations step-by-step will unveil the potential values for \(x\) that satisfy the original absolute value equation.

Step-by-Step Solution

Going through each step systematically ensures a clear path to the solution. Here's how we can follow a structured approach to such problems.
First, identify the problem and separate it into real-world scenarios based on what absolute values mean. As mentioned before, recognize that you're dealing with either the expressions being the same or totally opposite.
Second, solve each derived algebraic equation without skipping steps:

• For the first case equation \(3x - 1 = x + 5\), methodically bring like terms together and simplify to find \(x\).
• For the opposite case equation \(3x - 1 = - (x + 5)\), distribute the negative, then isolate \(x\) step-by-step to get your solution.

Last and importantly, check the solutions by plugging them back into the original absolute value equation. This verification step is crucial as sometimes you may find extraneous solutions that don’t actually work. Checking helps confirm the validity of each solution.
This structured method not only hones your problem-solving skills but also builds a robust understanding of algebra concepts.