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Chapter 1: Problem 71

Solve each equation. Check your solutions. \(|x+5|=18\)

Short Answer

Expert verified
The solutions are \(x = 13\) and \(x = -23\).

Step by step solution

01

Understand Absolute Value Equations

The absolute value of a number is its distance from zero on the number line, regardless of direction. So, \(|x+5| = 18\) means there are two possibilities: \(x+5 = 18\) and \(x+5 = -18\). We need to solve each equation separately.
02

Solve the First Equation

The first equation is \(x+5 = 18\). To find \(x\), we subtract 5 from both sides, which gives:\[x = 18 - 5\]\[x = 13\]
03

Solve the Second Equation

The second equation is \(x+5 = -18\). To find \(x\), we subtract 5 from both sides, which gives:\[x = -18 - 5\]\[x = -23\]
04

Check the Solutions

Now we'll check if both solutions satisfy the original equation \(|x+5|=18\).For \(x = 13\):\(|13 + 5| = |18| = 18\) - This is correct.For \(x = -23\):\(|-23 + 5| = |-18| = 18\) - This is also correct.

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Key Concepts

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

Solving Equations
When solving absolute value equations like \(|x+5| = 18\), we treat the absolute value as representing two separate cases. An absolute value denotes the distance from zero, thus both positive and negative outcomes are viable. For instance, solving \(|x+5| = 18\) leads to two scenarios:
  • \(x+5 = 18\)
  • \(x+5 = -18\)
These two cases need solving individually.
For \(x+5 = 18\), subtract 5 to isolate \(x\), resulting in \(x = 13\). For \(x+5 = -18\), again subtract 5, yielding \(x = -23\).
Therefore, the solutions are \(x = 13\) and \(x = -23\). Understanding these steps allows for precise solving of both equations, ensuring the right solutions are obtained.
Checking Solutions
Checking solutions is crucial in solving equations to verify accuracy. Once potential solutions \(x = 13\) and \(x = -23\) are found, substitute them back into the original equation \(|x+5| = 18\).
Start with \(x = 13\). Plugging into the equation gives:
  • \(|13 + 5| = 18\)
  • Which simplifies to \(|18| = 18\)
  • This confirms 18 equals 18, verifying \(x = 13\) as a valid solution.
Next, try \(x = -23\):
  • \(|-23 + 5| = 18\)
  • Which simplifies to \(|-18| = 18\)
  • Again confirming 18 equals 18, thus \(x = -23\) is also correct.
Checking solutions reinforces understanding and ensures no errors were made.
Algebraic Equations
Algebraic equations often involve expressions with variables that need isolating to discover their values. In absolute value equations, algebraic manipulation helps unravel potential solutions.
For \(|x+5|=18\), algebraic skills transform \(x+5=18\) and \(x+5=-18\). The process includes:
  • Identifying expressions inside the absolute value.
  • Setting up corresponding algebraic equations for both positive and negative cases.
  • Simplifying through basic operations like addition or subtraction.
These algebraic techniques are indispensable for tackling complex equations. They provide a systematic path to finding solutions, equipping learners to handle diverse problems efficiently. Comprehending these fundamentals aids in understanding care beyond just absolute value, paving the way through various algebra principles.

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