Problem 10 - Algebra and Trigonometry

Question

Solve the equation. $$ \left|\frac{1}{2} x-2\right|=1 $$

Step-by-Step Solution

Answer

The solutions are \( x = 6 \) and \( x = 2 \).

Step 1: Understanding Absolute Value Equations

An equation involving absolute values, such as \( \left|A\right| = B \), means that \( A \) can be either equal to \( B \) or \( -B \). This means we need to solve two separate equations.

Step 2: Set Up the Equations

Given the equation \( \left| \frac{1}{2} x - 2 \right| = 1 \), we can translate it into two separate equations: \( \frac{1}{2} x - 2 = 1 \) and \( \frac{1}{2} x - 2 = -1 \).

Step 3: Solve the First Equation \( \frac{1}{2} x - 2 = 1 \)

Add 2 to both sides of the equation to get \( \frac{1}{2} x = 3 \). Then multiply both sides by 2 to isolate \( x \), resulting in \( x = 6 \).

Step 4: Solve the Second Equation \( \frac{1}{2} x - 2 = -1 \)

Add 2 to both sides of the equation to get \( \frac{1}{2} x = 1 \). Then multiply both sides by 2 to isolate \( x \), resulting in \( x = 2 \).

Step 5: Verify the Solutions

For \( x = 6 \), substitute back into the original \( \left| \frac{1}{2} \cdot 6 - 2 \right| = \left| 3 - 2 \right| = 1 \). This holds true. For \( x = 2 \), substitute back to find \( \left| \frac{1}{2} \cdot 2 - 2 \right| = \left| 1 - 2 \right| = 1 \). This also holds true.

Key Concepts

Solving Equations

When solving equations involving absolute values, it's important to understand what absolute value signifies. The absolute value of a number, represented as \( |A| \), refers to its distance from zero on the number line.
The equation \( |A| = B \) implies that \( A \) can take two values: \( B \) or \( -B \). Therefore, you must always create two separate equations to solve such problems.For the equation given: \( \left| \frac{1}{2}x - 2 \right| = 1 \), this means you will solve:- \( \frac{1}{2}x - 2 = 1 \)- \( \frac{1}{2}x - 2 = -1 \)By approaching it through this method, we ensure that all possible solutions are accounted for, as absolute values often introduce multiple valid scenarios. This approach is crucial in accurately solving absolute value equations.

Algebraic Manipulation

Algebraic manipulation is essential in solving absolute value equations. Once you've set up your two separate equations, algebraic techniques help you isolate the variable to find solutions. Let's see the algebraic manipulation used in this context:

• **Solving \( \frac{1}{2}x - 2 = 1 \):** Begin by adding 2 to both sides, turning the equation into \( \frac{1}{2}x = 3 \). After that, multiply both sides by 2 to isolate \( x \), which gives \( x = 6 \).
• **Solving \( \frac{1}{2}x - 2 = -1 \):** Similarly, add 2 to both sides to get \( \frac{1}{2}x = 1 \). Then, multiply by 2 to find \( x = 2 \).

Effective algebraic manipulation begins with understanding operations that will simplify and solve the equation. As you adjust the equation step by step, ensure the operations maintain equality to preserve the integrity of the solution.

Equation Verification

Verifying solutions is a key step to ensure correctness. This involves plugging the solutions back into the original equation to confirm they satisfy the condition.For our problem, we derived two solutions: \( x = 6 \) and \( x = 2 \). Now let's verify these:

• **For \( x = 6 \):** Substitute into the original equation to check: \[ \left| \frac{1}{2} \, \times \, 6 - 2 \right| = \left| 3 - 2 \right| = 1 \]

This confirms that \( x = 6 \) is correct.

• **For \( x = 2 \):** Substitute back: \[ \left| \frac{1}{2} \, \times \, 2 - 2 \right| = \left| 1 - 2 \right| = 1 \]

This confirms that \( x = 2 \) is also correct.Verification is crucial, as it ensures no step in problem-solving was missed and that the values are truly solutions of the absolute value equation. Always perform this step to confirm your results are accurate.