Problem 18 - Thomas Calculus

Question

Solve the equations in Exercises \(13-18\) $$ \left|\frac{s}{2}-1\right|=1 $$

Step-by-Step Solution

Answer

The solutions are \( s = 4 \) and \( s = 0 \).

Step 1: Understanding the Absolute Value

The equation given is an absolute value equation: \( \left|\frac{s}{2}-1\right|=1 \). This means that the expression inside the absolute value, \( \frac{s}{2}-1 \), can be either 1 or -1.

Step 2: Setting Up Two Equations

Write two separate equations based on the absolute value definition: (1) \( \frac{s}{2} - 1 = 1 \) and (2) \( \frac{s}{2} - 1 = -1 \). These come from the fact that if \( \left|x\right| = a \), then \( x = a \) or \( x = -a \).

Step 3: Solving the First Equation

Solve \( \frac{s}{2} - 1 = 1 \). First, add 1 to both sides: \( \frac{s}{2} = 2 \). Then, multiply both sides by 2 to solve for \( s \): \( s = 4 \).

Step 4: Solving the Second Equation

Solve \( \frac{s}{2} - 1 = -1 \). First, add 1 to both sides: \( \frac{s}{2} = 0 \). Then, multiply both sides by 2 to solve for \( s \): \( s = 0 \).

Step 5: Conclusion of Solutions

Combine the solutions from both equations to express the final answer. The solutions to the equation \( \left|\frac{s}{2}-1\right|=1 \) are \( s = 4 \) and \( s = 0 \).

Key Concepts

Solving Equations

When tackling equations that involve absolute values, it is crucial to understand the fundamental process for reaching a solution. Absolute value equations like \( \left|\frac{s}{2}-1\right|=1 \) typically split into two separate scenarios that reflect the properties of absolute value. An important first step is to convert a single equation into two linear equations. This arises from how absolute values function: observing that the expression inside the absolute value can equal either the positive or negative of the given value (in this case, 1 or -1).
Solving these types of equations efficiently involves:

• Identifying that absolute value expressions \( \left|A\right| = b \) lead to two scenarios: \( A = b \) and \( A = -b \).
• Writing these conditions into separate linear equations.
• Simplifying and solving for the unknown variable (in our example \( s \)).

By methodically setting up and solving each equation, you ensure that all possible values of the variable are considered, leading to a complete solution.

Equation Solutions

The solutions to an equation provide the values that satisfy the equation when substituted back into it. For an absolute value equation like \( \left|\frac{s}{2}-1\right|=1 \), the solutions represent the values of \( s \) that render the equation true.
In the example used, we found two solutions for \( s \):

• Solving \( \frac{s}{2} - 1 = 1 \) yields \( s = 4 \).
• Solving \( \frac{s}{2} - 1 = -1 \) yields \( s = 0 \).

Both solutions \( s = 4 \) and \( s = 0 \) are valid, as they satisfy one of the two scenarios for the absolute value condition. When considering solutions, always plug them back into the original equation to verify correctness. This ensures that no steps were mistaken throughout the process.
Additionally, it is crucial to consider the domain and range in relation to the context of the problem to eliminate any extraneous solutions that may arise.

Mathematical Concepts

The concept of absolute value is a foundational principle in mathematics, particularly reflected in absolute value equations. The absolute value of a number represents its distance from zero on a number line, regardless of direction. This mathematical idea is crucial in understanding how absolute value influences equations.
In absolute value equations, the focus is often on breaking down the expression within the absolute value into positive and negative components. For example, if \( |x| = a \), it indicates two possibilities: \( x = a \) or \( x = -a \).
Key mathematical concepts to remember when dealing with these equations include:

• Recognizing absolute values indicate magnitude only and always yield non-negative outcomes.
• When solving a split outcome, it is vital to check if the solutions fit other given conditions of the problem, such as signs or contexts provided.
• Graphing solutions on a number line can provide a visual representation of these concepts, helping to reinforce understanding.

The inherent symmetry of absolute values often simplifies mathematical reasoning, making them a powerful tool in algebra and mathematical analysis.