Problem 51 - A Graphical Approach to College Algebra

Question

Solve each equation or inequality. $$\frac{1}{2}\left|-2 x+\frac{1}{2}\right|=\frac{3}{4}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 1\).

Step 1: Clear the Fraction

Multiply both sides of the equation by 2 to eliminate the fraction. This gives:\[ |-2x + \frac{1}{2}| = \frac{3}{2} \]

Step 2: Set up Absolute Value Equations

The absolute value equation \(|a| = b\) can be written as two separate equations: \(a = b\) or \(a = -b\). Thus, set up:1. \(-2x + \frac{1}{2} = \frac{3}{2}\)2. \(-2x + \frac{1}{2} = -\frac{3}{2}\)

Step 3: Solve the First Equation

Solve the equation \(-2x + \frac{1}{2} = \frac{3}{2}\):1. Subtract \(\frac{1}{2}\) from both sides: \(-2x = \frac{3}{2} - \frac{1}{2}\)2. Simplify the right side: \(-2x = 1\)3. Divide by -2: \(x = -\frac{1}{2}\)

Step 4: Solve the Second Equation

Solve the equation \(-2x + \frac{1}{2} = -\frac{3}{2}\):1. Subtract \(\frac{1}{2}\) from both sides: \(-2x = -\frac{3}{2} - \frac{1}{2}\)2. Simplify the right side: \(-2x = -2\)3. Divide by -2: \(x = 1\)

Step 5: Verify Solutions

Check each solution in the original equation:1. For \(x = -\frac{1}{2}\), substituting gives \(-2(-\frac{1}{2}) + \frac{1}{2} = 1.\) The equation becomes \(\frac{1}{2} \times 1 = \frac{1}{2},\) which does not equal \(\frac{3}{4}.\)2. For \(x = 1\), substituting gives \(-2(1) + \frac{1}{2} = -\frac{3}{2}.\) The equation becomes \(\frac{1}{2} \times \frac{3}{2} = \frac{3}{4},\) which is correct.Therefore, \(x = 1\) is the only valid solution.

Key Concepts

Solving Equations

When solving equations, we are looking for the value of the variable that makes the equation true. The process involves using different techniques to simplify and eventually solve for the variable. In this exercise, we have an absolute value equation, where the focus is on understanding how the absolute value works in equations.

Absolute value equations like \(|-2x + \frac{1}{2}| = \frac{3}{2}\) need to be broken down into two separate cases:

• One where the expression inside the absolute value equals a positive value
• Another where it equals a negative value

This stems from the definition of absolute value: \(|a| = b\) implies \(a = b\) or \(a = -b\). By approaching the solution with these two perspectives, we handle the positive and negative sides of the expression separately.

Once the equations are set up, the following steps typically involve simplifying using basic arithmetic operations—like subtraction and division—to isolate the variable, achieving a straightforward expression that represents the possible solutions to the equation.

Fractions and Decimals

Fractions and decimals are often encountered in algebraic problems, including in equations. They are different ways of representing numbers and, when present in equations, can sometimes make them look more complex at first.

When encountering fractions, a common first step in simplifying equations is to clear them by multiplying through by a common denominator. This can turn fractions into whole numbers, making the equation easier to work with. For example, in the given equation \(\frac{1}{2}\left|-2x+\frac{1}{2}\right|=\frac{3}{4}\), multiplying the entire equation by 2 eliminates the fraction. This is a strategic way to get rid of fractions without changing the equality of the equation.

Remember that operations like these do not change the balance of the equation because they are applied equally to both sides. Practicing this technique can help in reducing complexity, allowing you to focus on solving the algebraic expression without the distraction of fractions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can consist of numbers, variables, and operators. They are an essential component in equations and require manipulation to solve problems.

In our exercise, the expression \(-2x + \frac{1}{2}\) appears within the absolute value. To handle this, we first stabilize the equation by clearing fractions as mentioned, and then deal with the core of the problem—solving the expression within the absolute value symbol.

• Start simplifying the expression by removing any constants (e.g., add or subtract fractions or whole numbers)
• Isolate the term involving the variable (e.g., getting \(-2x\) by itself)
• Finally, solve for the variable by performing operations such as division or multiplication

The complexity of algebraic expressions is managed by carefully applying operations to isolate variables and simplify the equation. The goal is to transform the expression into a form that explicitly clarifies the solution.