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Chapter 1: Problem 81

Solve each equation or inequality. $$ |x+4|+1=2 $$

Short Answer

Expert verified
The solutions are \(x = -3\) and \(x = -5\).

Step by step solution

01

Isolate the Absolute Value Expression

Subtract 1 from both sides of the equation to isolate the absolute value expression: \ \[ |x + 4| + 1 = 2 \] \ \[ |x + 4| = 1 \]
02

Set Up Two Separate Equations

The absolute value equation \(|x + 4| = 1\) can be split into two separate equations: \ \(x + 4 = 1\) and \(x + 4 = -1\).
03

Solve Each Equation

Solve each of the two equations separately: \ 1. For \(x + 4 = 1\): \ \(x = 1 - 4\) \ \(x = -3\) \ 2. For \(x + 4 = -1\): \ \(x = -1 - 4\) \ \(x = -5\)
04

Combine Solutions

The solutions to the equation are \(x = -3\) and \(x = -5\).

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Key Concepts

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

Absolute Value
To solve absolute value equations, it's important to understand what absolute value means.
The absolute value of a number is its distance from zero on a number line, regardless of direction.
It is always a non-negative number. For example, the absolute value of both -3 and 3 is 3.
Mathematically, it is represented as \(|x|\).
In the given equation \(|x+4| + 1 = 2\), the absolute value signs around \(x+4\) indicate we're looking at the distance from zero of the expression \(x+4\).
Equations
Equations are statements that assert the equality of two expressions.
The basic goal in solving an equation is to determine the value of the variable that makes the equation true.
In this problem, the equation given is an absolute value equation, \(|x + 4| + 1 = 2\).
This means we have to find the value(s) of \(x\) that satisfy this condition.
Isolation of Absolute Value
Isolating the absolute value expression is the key first step.
In our problem, the equation starts as \(|x+4| + 1 = 2\). To isolate \(|x+4|\), we subtract 1 from both sides:
\[|x + 4| + 1 - 1 = 2 - 1\]
Simplifying, we get: \[|x + 4| = 1\].
Now the absolute value is isolated, and we can move to the next step.
Solving Linear Equations
Once the absolute value is isolated, we can split it into two separate linear equations.
For the isolated form \(|x + 4| = 1\), this means:
\[x + 4 = 1 \ and \ x + 4 = -1\].
Solving these separately:
For \(x + 4 = 1\):
\[x = 1 - 4\],
so \(x = -3\).
For \(x + 4 = -1\):
\[x = -1 - 4\],
so \(x = -5\).
Combining these, the solutions are: \(x = -3\) and \(x = -5\).

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