Problem 29 - Algebra 2

Question

Solve each equation. Check your solutions. \(0=|2 z-3|\)

Step-by-Step Solution

Answer

The solution is \(z = \frac{3}{2}\) and checks out.

Step 1: Understand the Absolute Value Equation

The absolute value equation given is \(0 = |2z - 3|\). Since the absolute value represents the distance from zero, the expression inside the absolute value, \(2z - 3\), must equal zero for the absolute value itself to be zero.

Step 2: Set Up the Equation Without Absolute Value

Remove the absolute value by setting the expression inside it equal to zero. Thus, solve the equation: \(2z - 3 = 0\).

Step 3: Solve for z

To solve \(2z - 3 = 0\), add 3 to both sides to get \(2z = 3\). Next, divide both sides by 2 to isolate \(z\), resulting in \(z = \frac{3}{2}\).

Step 4: Check Your Solution

Substitute \(z = \frac{3}{2}\) back into the original equation to verify it. Calculate \(|2(\frac{3}{2}) - 3| = |3 - 3| = |0| = 0\). Since this satisfies the equation \(0 = |2z - 3|\), the solution is correct.

Key Concepts

Solving Absolute Value Equations

When tackling an absolute value equation, it's important to understand that the absolute value represents the distance a number is from zero on a number line. In the case of our exercise, this means we are looking at the equation \(0 = |2z - 3|\). Here, the absolute value is zero, which implies that the expression inside the absolute value, \(2z - 3\), must also be zero. This is because zero is a unique case where there is no positive or negative scope, only the neutral middle point.

To approach such equations, follow these steps:

• Identify the expression inside the absolute value.
• Set the expression equal to zero since the outer value is zero.
• Solve the equation as you would any linear equation to find the variable's value.

Recognizing that the absolute value is zero streamlines our solution process by reducing it to solving the internal expression for zero.

Algebraic Manipulation

Algebraic manipulation is a crucial skill in solving equations, and it involves rearranging and altering equations to isolate the variable of interest—in this case, \(z\). For the equation \(2z - 3 = 0\), we need to get \(z\) by itself. We can achieve this through two main steps:

• Add or subtract terms to both sides of the equation to simplify it. Here, adding 3 makes the equation \(2z = 3\).
• Divide both sides of the equation by the coefficient of the variable to isolate it. Dividing by 2 results in \(z = \frac{3}{2}\).

This straightforward manipulation ensures that the variable is isolated, providing a clear solution that can later be verified.

Checking Solutions

After calculating a solution for an equation, it's vital to verify that the solution is correct and satisfies the original equation. Checking solutions helps confirm that calculations were accurate and that the equation was properly manipulated.

To check the solution of \(z = \frac{3}{2}\), substitute this value back into the original equation:\

• Replace \(z\) with \(\frac{3}{2}\) in the expression \(2z - 3\).
• Calculate \(|2(\frac{3}{2}) - 3| = |3 - 3| = |0| = 0\).

Since the absolute value simplifies back down to zero, which matches the left side of the original equation \(0 = |2z - 3|\), we can be confident that \(z = \frac{3}{2}\) is indeed the correct solution. This step reinforces the understanding of both the algebraic process and the properties of absolute value.