Problem 75 - College Algebra Essentials

Question

Solve each absolute value equation or indicate that the equation has no solution. $$ |2 x-1|+3=3 $$

Step-by-Step Solution

Answer

The solution to the equation \(|2x-1|+3=3\) is \(x = \frac{1}{2}\).

Step 1: Isolate the absolute value equation

The first task is to isolate the absolute value equation on one side of the equation. This can be done by subtracting 3 from both sides of the absolute value equation. As such, the equation becomes \( |2x - 1| = 3 - 3 = 0 \).

Step 2: Consider the positive and negative possibilities

When working with absolute value equations, the value inside the absolute value can either be positive or negative. This means that \(2x - 1\) can be either 0 or -0. However, -0 is the same as 0, so we only have one possibility in this case.

Step 3: Solve the equation

Now we can solve the equation \(2x - 1 = 0\) for \(x\). This results in \(2x = 1\) and then dividing both sides by 2 gives \(x = \frac{1}{2}\).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra, and it's all about finding the value of the variable that makes the equation true. When dealing with absolute value equations, the challenge lies in understanding that the absolute value refers to the distance from zero on the number line. This means that whatever is inside the absolute value brackets can be either positive or negative, but never less than zero.
In the exercise we are looking at, the equation is \(|2x-1|+3=3\). To solve it, we need to make sure the absolute value component is isolated. Once isolated, we can go forward by determining the potential scenarios for what could be inside the brackets - positive or negative. Always remember, the key here is recognizing the two scenarios \((2x - 1 = 0)\) and taking action accordingly to determine the value of \(x\).

Algebraic Solutions

Algebraic solutions involve manipulating the equation with operations such as addition, subtraction, multiplication, or division to reach an answer. For the given problem, once we isolate the absolute value, we must consider only realistic mathematical scenarios.
Absolute value can often lead to two potential solutions due to its dual nature with positive and negative possibilities. However, in this case, since the value inside the brackets is set equal to zero, we only have one scenario to attend to. Here, we directly solve the equation \(2x - 1 = 0\) algebraically.

• Add or subtract numbers to isolate terms with the variable.
• Use inverse operations to simplify the equation step-by-step.
• Solve for \(x\) by dividing to isolate the variable completely.

Following these steps carefully, as done here by setting \(2x - 1 = 0\) and solving to find \(x = \frac{1}{2}\), ensures that all necessary algebraic manipulations are correctly applied.

Isolation of Variables

Isolation of variables is an essential process when solving equations as it allows you to find the solution for a specific variable. In our equation, isolation means getting \(|2x-1|\) all by itself on one side of the equation initially. Here’s how you can successfully isolate variables:

• First, ensure any additional numbers on the same side of the absolute value are moved across to the other side of the equation. For \(|2x-1|+3 = 3\), subtracting 3 from both sides isolates \(|2x-1|\).
• Once isolated, focus on what the equation represents. For absolute value, recognize that the expression inside can approach zero but won't be negative.
• Identify the range of values the variable can take, considering the essence of absolute value.

The goal of isolation is to simplify and condense the equation so you can directly solve for the variable \(x\), leading to a clean and definite solution, such as \(x = \frac{1}{2}\) in this example.