Problem 67 - Algebra for College Students

Question

Explain how you would solve the equation \(|2 x-3|=0\).

Step-by-Step Solution

Answer

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

Step 1: Understanding Absolute Value

The equation given is \(|2x - 3| = 0\). Recall that the absolute value \(|a|\) represents the distance of \(a\) from zero on the number line. It is always non-negative. Therefore, for an absolute value to be equal to zero, the expression inside must be zero.

Step 2: Set the Expression Inside Absolute Value to Zero

Since \(|2x - 3| = 0\), we set the expression inside the absolute value to zero: \(2x - 3 = 0\). This equation simply expresses when the contents are equal to their absolute value (since they are non-negative).

Step 3: Solve the Simple Equation

To solve \(2x - 3 = 0\), add 3 to both sides of the equation to get \(2x = 3\).

Step 4: Isolate the Variable

To isolate \(x\), divide both sides by 2 to get \(x = \frac{3}{2}\). This is the solution to the equation.

Key Concepts

Absolute Value

When dealing with absolute value equations like \(|2x - 3| = 0\), it's important to understand that absolute value represents the distance a number is from zero on the number line. This distance is always non-negative, meaning it cannot be a negative number. Think of it as ignoring any negative sign that might appear before a number.

• If a number is at -5 on the number line, its absolute value is 5.
• Similarly, if it is at 5, the absolute value is also 5.

In our problem, the equation's absolute value being zero suggests that the content inside must equal zero itself because zero is the only non-negative number at zero distance. Hence, the first step is to set the expression inside the absolute value, \(2x - 3\), to zero and solve for \(x\).

Isolating Variable

Isolating a variable is an essential part of solving equations and is done to find its specific value. After setting the absolute value equation \(|2x - 3| = 0\) to \(2x - 3 = 0\), we need to isolate \(x\) to find its value. Start by moving all terms, except with \(x\), to the other side of the equation. We begin with:- The given equation: \(2x - 3 = 0\)- Add 3 to both sides to simplify: \(2x = 3\)This process effectively removes terms not containing \(x\) from one side to the other, allowing us to focus solely on \(x\). Here we look to solely express \(x\) by itself on one side of the equation.

Simple Equations

Simple equations are the foundation of algebra and typically involve arithmetic operations such as addition, subtraction, multiplication, and division. Solving \(2x - 3 = 0\) is a straightforward example. To tackle this equation, we perform the following:- Addition: Add 3 to both sides, leading to \(2x = 3\).- Division: Divide both sides by 2, resulting in \(x = \frac{3}{2}\).These operations combined help us to unearth the solution by adhering to the fundamental balance principle: whatever you do to one side of the equation, you must do to the other. This keeps the equation balanced, allowing us to solve it correctly.

Number Line

The number line is a visual tool that can help better understand concepts of absolute value and equations. Visualizing \(2x - 3\) on a number line can illuminate how its absolute value relates to zero. - On a number line, zero is the midpoint, with negative numbers to the left and positive numbers to the right.- Absolute value, being non-negative, shows the span from any point to zero.For instance, when the expression \(2x - 3\) is zero, it translates to a single point on the number line where \(2x\) equals 3. This visual representation might simplify understanding how operations performed in equations move points around on this line, leading to solutions like \(x = \frac{3}{2}\).