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Chapter 3: Problem 16

Solve. $$|x+5|=1$$

Short Answer

Expert verified
x = -4 or x = -6

Step by step solution

01

Understand the Absolute Value Equation

An absolute value equation can be split into two separate equations. For the given equation \(|x+5| = 1\), this means we need to consider both \(x + 5 = 1\) and \(x + 5 = -1\).
02

Solve the First Equation

Start by solving \(x + 5 = 1\). Subtract 5 from both sides: \[x + 5 - 5 = 1 - 5 \] which simplifies to \[x = -4\].
03

Solve the Second Equation

Next, solve \(x + 5 = -1\). Again, subtract 5 from both sides: \[x + 5 - 5 = -1 - 5 \] which simplifies to \[x = -6\].
04

Combine the Solutions

The solutions to the original equation \(|x+5|=1\) are the values we found: \(x = -4\) and \(x = -6\).

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Key Concepts

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

solving absolute value equations
To solve absolute value equations, you need to understand that the absolute value of a number represents its distance from zero on a number line, regardless of direction. This means that an equation like \(|x+5|=1\) can be split into two scenarios: one where the expression is equal to 1 and one where it is equal to -1. For our equation, these two scenarios are: \(x+5=1\) and \(x+5=-1\).
absolute value properties
Absolute values have unique properties that simplify solving equations. Key properties include:
  • Non-negativity: The absolute value of any number is always non-negative. \(|a| \geq 0\) for all real numbers \(a\).
  • Piecewise Definition: The absolute value of \(x\) can be defined piecewise: \(|x| = x\) if \(x \geq 0\), and \(|x| = -x\) if \(x < 0\).
  • Equation Splitting: For an equation \(|x|=a\), where \(a\geq 0\), it translates to \((x=a \text{ or } x=-a)\).
Using these properties in solving \(|x+5|=1\), it becomes two linear equations: \(x+5=1\) and \(x+5=-1\). Solve these individually to find possible values for \(x\).
step-by-step math solutions
Here's a step-by-step approach to solving the absolute value equation \(|x+5|=1\):

Step 1: Split the equation. \(|x+5|=1\) means we set up two equations: \(x+5=1\) and \(x+5=-1\).
Step 2: Solve the first equation. For \(x+5=1\), subtract 5 from both sides: \(x=1-5=-4\).
Step 3: Solve the second equation. For \(x+5=-1\), subtract 5 from both sides: \(x=-1-5=-6\).
Step 4: Combine solutions. The solutions to \(|x+5|=1\) are \(x=-4\) and \(x=-6\).

By writing each step out clearly, you ensure a thorough understanding of how to approach and solve absolute value equations.

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