Problem 74 - Algebra and Trigonometry

Question

In Exercises 61–78, solve each absolute value equation or indicate that the equation has no solution. $$|x+1|+6=2$$

Step-by-Step Solution

Answer

The equation has no solutions because an absolute value cannot be negative

Step 1: Isolate the Absolute Value

To isolate the absolute value \(|x+1|\), the first step is to subtract 6 from both sides of the equation \(|x+1| + 6 = 2\), which would result in \(|x+1| = -4\)

Step 2: Analyze the equation within the absolute values

The result from step 1 gives \(|x+1| = -4\). This indicates that the equation has no solution because the absolute value of any number can't be negative.

Key Concepts

Solving Absolute Value Equations

When tackling absolute value equations, the goal is to express them in a form where we can clearly analyze and solve. Absolute value represents the distance a number is from zero on the number line, typically leading to two possible solutions. However, not all equations follow this pattern. To begin solving these equations, you need to first work towards isolating the absolute value expression, allowing for a clearer understanding of the possibilities of solutions or lack thereof. This foundational step is crucial for further analysis and problem-solving.

No Solution Absolute Value

The concept of an absolute value equation having no solution might seem strange at first. But it's simpler than it sounds: if you end up with an equation where the absolute value equals a negative number, there is no solution. Absolute values can only be zero or positive, as they measure distance. So, if your equation results in an expression like \(|x| = -4\), this means there’s no possible value for \(x\) that satisfies the equation. Knowing when an absolute value equation has no solution can save time and clarify misunderstandings.

Isolating Absolute Value

Isolating the absolute value in an equation is a critical first step that sets the groundwork for solving it. To isolate means to get the absolute value expression on one side of the equation alone. This can involve either adding, subtracting, multiplying, or dividing terms on both sides to achieve this. Consider the equation \(|x+1| + 6 = 2\). You first subtract 6 from both sides to find that \(|x+1| = -4\). Isolating makes it clear what values or solutions, if any, you can proceed to evaluate. It simplifies the equation and allows you to focus on solving the absolute value itself.

Negative Absolute Value Analysis

Analyzing an absolute value equation, once isolated, involves checking if a solution is possible. If like in our exercise, you find an equation such as \(|x+1| = -4\), a negative result indicates impossibility. Negative outcomes in absolute value equations mean the absolute value cannot be achieved, as distances are always positive. This kind of analysis checks the logic and feasibility before taking further steps. Understanding this principle can prevent unnecessary calculations and guide you accurately to conclude that some equations have no solution.