Problem 59 - Algebra 2

Question

Solve each equation or inequality. Check your solutions. $$ |x-1|=3 $$

Step-by-Step Solution

Answer

The solutions are \(x = 4\) and \(x = -2\).

Step 1: Understanding the Absolute Value Equation

The expression \(|x-1| = 3\) represents an absolute value equation. Absolute value indicates the distance of a number from zero without considering direction, meaning it results in a non-negative value.

Step 2: Splitting the Absolute Value into Two Equations

For the equation \(|x-1| = 3\), the absolute value can result from two scenarios:1. \(x-1 = 3\)2. \(x-1 = -3\). We solve these separately.

Step 3: Solving the First Equation: \(x-1 = 3\)

Solve the first scenario by adding 1 to each side of the equation:\[x - 1 + 1 = 3 + 1\]\[x = 4\].

Step 4: Solving the Second Equation: \(x-1 = -3\)

Solve the second scenario by adding 1 to each side of the equation:\[x - 1 + 1 = -3 + 1\]\[x = -2\].

Step 5: Checking the Solutions

Substitute the solutions back into the original equation to confirm:For \(x = 4\):\(|4 - 1| = |3| = 3\) which is correct.For \(x = -2\):\(|-2 - 1| = |-3| = 3\) which is also correct. Thus, both solutions satisfy the original equation.

Key Concepts

Solving Equations

Solving equations is a foundational skill in mathematics. It involves finding the value of a variable that makes an equation true. For the equation \(|x-1|=3\), you're tasked with finding values of \(x\) that satisfy this condition.

The process starts by understanding the rule that governs absolute values: they measure the distance a number is from zero. Because distance is always non-negative, this impacts how we solve equations involving absolute values. Here, the distance from zero is 3.

To solve \(|x-1|=3\), we recognize two possible scenarios that satisfy the equation: when \(x-1\) equals 3, and when \(x-1\) equals -3. Each scenario is an equation itself. Solving these requires simple algebra. By breaking the absolute value equation into these two separate equations, the problem becomes easier to manage.

Checking Solutions

Once you find potential solutions, it's important to ensure they are correct. This involves checking, or verifying, each solution by substituting them back into the original equation.

For our example, we determined the possible solutions: \(x = 4\) and \(x = -2\). To check, substitute each into \(|x-1|=3\). First, try \(x = 4\): substitute and simplify. You get \(|4-1| = |3| = 3\) which confirms \(x = 4\) as a solution. Now, try \(x = -2\): substitute and simplify. You get \(|-2-1| = |-3| = 3\), confirming \(x = -2\) as also correct.

This step is crucial because it affirms your solutions fit the original equation. Verifying helps catch any mistakes made during calculation, ensuring accuracy in your problem-solving.

Absolute Value

Understanding absolute value is key to solving related equations. Absolute value denotes the magnitude of a number relative to zero, disregarding negative or positive signs.

The absolute value of a number is always non-negative. For example, \(|3|\) and \(|-3|\) both equal 3, since they are both three units away from zero. This property is crucial when solving equations like \(|x-1|=3\).

With absolute value affecting equations, it creates two possible scenarios: the expression inside the absolute value can be equal to the given number (\(x-1=3\)) or its negative (\(x-1=-3\)). Always remember this duality in solutions when dealing with absolute values in equations.

Mathematical Problem Solving

Mathematical problem solving involves a structured approach to finding solutions. It is about understanding the problem, breaking it into manageable parts, developing strategies, and evaluating your answers.

For absolute value equations, begin by interpreting the problem. Understand what an absolute value conveys, and what it means for the equation at hand. Next, dissect the equation into simpler parts—here, into two linear equations.

Apply algebraic techniques to solve each part individually, then combine and check the solutions. Finally, reflect on the process to ensure accuracy and deepen understanding. This methodical approach enhances your ability to tackle more complex mathematical challenges effectively.