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Chapter 3: Problem 15

Solve. $$|x+2|=6$$

Short Answer

Expert verified
x = 4 or x = -8

Step by step solution

01

- Understand Absolute Value Equation

The equation \(|x+2| = 6\) means that the expression \(x+2\) can either be 6 or -6 because the absolute value symbol \(| \) makes any number positive.
02

- Set Up Two Separate Equations

Since \(|x+2| = 6\), you can split it into two separate equations: \(x+2 = 6\) and \(x+2 = -6\).
03

- Solve for x

Now solve each equation separately. For \(x+2 = 6\), subtract 2 from both sides to get \(x = 4\). For \(x+2 = -6\), subtract 2 from both sides to get \(x = -8\).
04

- Write the Solution

The solutions to the equation \(|x+2| = 6\) are \(x = 4\) and \(x = -8\).

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Key Concepts

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

solving absolute value equations
Absolute value equations might seem tricky at first, but they follow a consistent method, making them straightforward once you understand the steps. The absolute value of a number is its distance from zero on the number line, regardless of direction. For example, the absolute value of both 6 and -6 is 6.
An absolute value equation has the format \(| expression | = value\), which represents the distance of the expression from zero being equal to the specified value. To solve it, you set up two separate equations.
This is because \(|a| = b\) implies that \(a = b\) or \(a = -b\).
splitting absolute value into two equations
Once you understand the absolute value concept, the next step is to split the equation into two simpler equations. This follows directly from the definition of absolute value. Taking our example \(|x+2| = 6\), the expression inside the absolute value signs (in this case \(x + 2\)) can be equal to the positive and negative of the given number:
\(|x+2| = 6\) splits into two cases:
  • \tCase 1: \(x + 2 = 6\)
  • \tCase 2: \(x + 2 = -6\)
This step essentially removes the absolute value signs, resulting in two separate linear equations that can be solved independently.
solving linear equations
Once you have split the absolute value equation into two linear equations, your task becomes solving each of them. Solving linear equations involves isolating the variable on one side of the equation.
Here鈥檚 how you solve the two equations from our example \(|x+2| = 6\):
  • \tFor \(x + 2 = 6\): subtract 2 from both sides to get \(x = 4\)
  • \tFor \(x + 2 = -6\): subtract 2 from both sides to get \(x = -8\)

So the solutions to the original absolute value equation \(|x+2| = 6\) are \(x = 4\) and \(x = -8\). These solutions mean that plugging 4 or -8 back into the original absolute value equation will satisfy \(|x+2| = 6\).

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