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Chapter 1: Problem 82

Solve each equation or inequality. $$ |x+5|-2=12 $$

Short Answer

Expert verified
\( x = 9 \) or \( x = -19 \)

Step by step solution

01

Understand the Absolute Value Property

Recall that for any absolute value equation of the form \(|A| = B\), it implies two possible scenarios: \(A = B\) or \(A = -B\).
02

Isolate the Absolute Value Expression

Start by isolating the absolute value expression on one side of the equation. Given \(|x+5|-2=12\), add 2 to both sides to isolate \(|x+5|\): \[|x+5| = 14\]
03

Set Up Two Separate Equations

Now, set up two separate equations based on the absolute value equality: \[x+5 = 14\] and \[x+5 = -14\]
04

Solve Each Equation

Solve each of these equations separately: \[x+5=14 \Rightarrow x=9\] \[x+5=-14 \Rightarrow x=-19\]
05

Combine Solutions

Combine the solutions from both equations to get the final answer: \[x = 9\] or \[x = -19\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolate Absolute Value
Start solving any absolute value equation by isolating the absolute value expression. This helps to simplify the equation. For example, in the equation \( |x+5|-2=12 \), you need to first get rid of the \(-2\) by adding 2 to both sides. This leaves you with \[ |x+5| = 14 \].

Removing constants and other terms outside the absolute value symbol is a necessary first step. Always perform basic arithmetic operations like addition or subtraction to keep the absolute value expression alone on one side of the equation.
Solving Linear Equations
Once you isolate the absolute value, the next step is to handle the linear equations that arise. Absolute value equations like \[ |A| = B \] translate into two separate linear equations: A = B and A = -B.

Using our example \[ |x+5| = 14 \], the two linear equations are: \[ x+5 = 14 \] and \[ x+5 = -14 \]. From here, solve each of these equations separately.
- For \( x+5 = 14 \), subtract 5 from both sides to get \( x = 9 \).
- For \( x+5 = -14 \), subtract 5 from both sides to get \( x = -19 \).

Simplifying each linear equation gives us two possible values for x.
Absolute Value Properties
Understanding the properties of absolute value is crucial for solving these types of equations. Absolute value represents the distance from zero, so it is always non-negative. For any number A, \[ |A| \] is equal to A if A is positive or zero, and equal to -A if A is negative.

In our case, when we found \[ |x+5| = 14 \], this meant \[ x+5 \] could either be 14 or -14. Hence, we developed two equations: \[ x+5=14 \] and \[ x+5=-14 \].

This dual nature of absolute value is why each absolute value equation can lead to two distinct solutions, highlighting its fundamental property of distance in mathematics.

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