Problem 16 - Thomas Calculus

Question

Solve the equations in Exercises \(13-18\) $$ |1-t|=1 $$

Step-by-Step Solution

Answer

The solutions are \(t=0\) and \(t=2\).

Step 1: Understand the Absolute Value Equation

The given equation is \(|1-t| = 1\). The absolute value \(|x|\) represents the distance of \(x\) from 0 on the number line. So, \(|1-t| = 1\) implies that the expression \(1-t\) is 1 unit away from 0.

Step 2: Set Up Two Separate Equations

To solve the absolute value equation \(|1-t| = 1\), consider the two cases: one where the expression inside the absolute value is positive and equal to 1, and another where it is negative and equal to -1.1. Consider \(1 - t = 1\).2. Consider \(1 - t = -1\).

Step 3: Solve the First Equation

Solve the first equation, \(1 - t = 1\):Subtract 1 from both sides to isolate \(t\):\(1 - t - 1 = 1 - 1\)This simplifies to:\(-t = 0\)Multiplying through by -1, we find:\(t = 0\).

Step 4: Solve the Second Equation

Solve the second equation, \(1 - t = -1\):Add \(t\) to both sides to move \(t\) to the right:\(1 = t - 1\)Now, add 1 to both sides to isolate \(t\):\(1 + 1 = t\)This simplifies to:\(t = 2\).

Step 5: Verify the Solutions

Verify both solutions in the context of the original equation \(|1-t| = 1\).- Substitute \(t=0\): \(|1-0| = 1| = 1\), which is correct.- Substitute \(t=2\): \(|1-2| = |-1| = 1\), which is also correct.Therefore, both \(t=0\) and \(t=2\) are correct solutions.

Key Concepts

Equation Solving

Equation solving is a fundamental concept in algebra used to find values that satisfy a given mathematical statement. When dealing with equations, our goal is to determine the unknown variable that makes the equation true. In the given exercise, we have the equation \(|1-t| = 1\). This equation involves an absolute value expression.

Solving such an equation often requires setting up separate cases for different possible values resulting from the absolute value. An equation is a statement of equality, and solving it means finding the number(s) that can be plugged into the variable(s), making the equation hold true. With absolute value equations, the absolute value symbol alters how we solve compared to regular linear equations, requiring extra steps to handle the positive and negative scenarios.

Step by Step Solution

A step-by-step solution process allows us to systematically tackle a problem, ensuring we cover all possible scenarios and don't miss any solutions. Let's take a closer look at how this process works for the given absolute value equation \(|1-t| = 1\).

Understanding the problem is the first step. This equation tells us that the expression \(1-t\) is exactly one unit away from zero. In practical terms, the steps include:

• Formulating two separate cases to handle the absolute value, considering both positive and negative situations.
• For \(1-t = 1\), we solve it as a linear equation and simplify to find \(t = 0\).
• For \(1-t = -1\), we similarly solve to find \(t = 2\).
• Finally, confirming these results by substituting back into the original equation to check their validity.

By following each of these steps, we ensure a thorough solution is reached.

Absolute Value Properties

Understanding the properties of absolute values is crucial to solving equations that contain them. The absolute value of a number, denoted as \(|x|\), is its distance from zero on the number line, which ensures the value is always non-negative. This property creates two possible scenarios when an equation involves an absolute value:

• If \(|x| = a\) where \(a\) is non-negative, then \(x = a\) or \(x = -a\).
• This means both positive and negative values of the expression within the absolute value need to be considered as possible solutions.

In the context of our sample exercise \(|1-t| = 1\), it illustrates the need to check both \(1 - t = 1\) and \(1 - t = -1\) as potential solutions. By understanding these fundamental properties, we can confidently solve absolute value equations with accuracy.