Problem 73 - College Algebra

Question

Solve each absolute value equation or indicate the equation has no solution. $$ |3 x-1|=|x+5| $$

Step-by-Step Solution

Answer

The solutions to the absolute equation \(|3x - 1| = |x + 5|\) are \(x = 3\) and \(x = -1\).

Step 1: Break Down into Two Cases

Since the absolute value is the non-negative value of the expression inside, we need to account for two possibilities: \((3x - 1) = (x + 5)\) or \((3x - 1) = -(x + 5)\). This makes sense if we recall that the absolute value of a number is its distance to zero.

Step 2: Solve the first equation

First, solve \((3x - 1) = (x + 5)\) for \(x\). This simplifies to \(2x = 6\) and thus, \(x = 3\). However, remember to check the solution in the original equation.

Step 3: Validate Solution for first equation

Substituting \(x = 3\) into the original absolute equation, we get \(|3*3 - 1| = |3 + 5|\), which is \(|8| = |8|\). Indeed, both sides of the equation yield 8, supporting our solution that \(x = 3\).

Step 4: Solve the second equation

Now, solve \((3x - 1) = -(x + 5)\) for \(x\). This simplifies to \(4x = -4\) and thus, \(x = -1\). Again, it's crucial to confirm that this solution also satisfies the original equation.

Step 5: Validate Solution for second equation

Substitute \(x = -1\) into the original absolute equation and get \(|3*(-1) - 1| = |-1 + 5|\), which is \(|4| = |4|\). Both sides of the equation yield 4, which proves our second solution that \(x = -1\).

Key Concepts

Absolute Value

When we talk about the absolute value, we refer to the distance a number lies from zero on the number line, regardless of direction. It is always a non-negative value. In mathematical terms, the absolute value of a given number 'a' is denoted by \(|a|\) and it reflects how far 'a' is away from zero.

For example, the absolute value of both 3 and -3 is 3. In the context of the exercise \(|3x-1|=|x+5|\), the absolute values on each side of the equation force us to consider two scenarios: one where the expressions inside the absolute value bars are equal and another where they are opposite in sign. This effectively means that we're looking for the points where these linear expressions have the same distance to zero on the number line, either in the positive or negative direction.

Understanding this concept is crucial when solving absolute value equations as it helps us set up the correct algebraic expressions needed to find a solution.

Linear Equations

Linear equations are algebraic expressions that represent a straight line when graphed on a coordinate plane. These equations take on the general form \(Ax + By = C\), where 'A', 'B', and 'C' are constants. In the case of one variable, the equation simplifies to \(Ax = C\), and the solution is simply \(x = C/A\).

In our exercise, the initial absolute value equation breaks down into two simpler linear equations, \((3x - 1) = (x + 5)\) and \((3x - 1) = -(x + 5)\). Solving these equations requires isolating 'x' on one side, which involves combining like terms and then dividing both sides by the coefficient before the 'x' term. It's straightforward arithmetic, but accuracy is essential to avoid errors.

Breaking down absolute value equations into linear equations allows us to apply foundational algebraic skills to find solutions. It also enables us to graph these equations and visually understand the relationship between the solutions.

Algebraic Solution Validation

Algebraic solution validation is a critical step in the problem-solving process. It ensures the solutions we find satisfy the original equation. To validate a solution, we substitute it back into the initial equation and check if the equation holds true.

For the exercise provided, validation takes place in two steps. First, the potential solution \(x = 3\) is tested by substituting '3' into the original absolute value equation, yielding \(|8| = |8|\). Since the equation is balanced, the solution \(x = 3\) is confirmed. Secondly, using the other potential solution \(x = -1\), we again substitute into the original equation to confirm \(|-4| = |-4|\), validating \(x = -1\) as another solution.

It's worthwhile to note that every potential solution must be validated this way. Without validation, there's a risk of including extraneous solutions—solutions that emerge from the algebraic process but do not satisfy the original equation. Always remember: validation is a key part of the algebraic process that confirms the credibility of our solutions.