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Chapter 6: Problem 21

Solve the equation. $$|x|=25$$

Short Answer

Expert verified
The solutions to the equation are \(x\) = 25 and \(x\) = -25.

Step by step solution

01

Identify the absolute value

The absolute value equation is |x| = 25
02

Find the positive solution

First consider the case when the value inside the absolute value bars is positive. If \(x\) is positive, then \(x\) = 25 is a solution.
03

Find the negative solution

Next, consider the case when the value inside the absolute value bars is negative. This occurs when -\(x\) is positive, so -\(x\) = 25, and \(x\) = -25 is a solution.

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Key Concepts

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

Solving Equations
When solving equations, one of the essential skills is understanding the meaning and behavior of absolute values. Absolute value represents the distance a number is from zero on the number line. It always yields a non-negative number regardless of whether the input is positive or negative. For example, the absolute value of both 25 and -25 is 25.

To solve the equation \(|x| = 25\), you must consider both possible scenarios for \(x\) based on the definition of absolute values. Equations with absolute values typically involve two steps:

  • Find the solution assuming the expression inside the absolute value is non-negative.
  • Find the solution assuming the expression inside the absolute value is negative.
By understanding these fundamental principles, you can approach absolute value equations systematically and confidently.
Positive and Negative Solutions
In absolute value equations, positive and negative solutions are derived due to the nature of absolute values themselves. Taking the equation \(|x| = 25\) as an example, solving it involves evaluating two scenarios:

  • First, assume \(x\) is positive, meaning the equation simplifies to \(x = 25\). This gives us our first solution: \(x = 25\).
  • Next, consider the possibility that \(x\) is negative. Here, you solve \(-x = 25\), which results in \(x = -25\). This is the second solution.

These steps highlight that an absolute value equation can have more than one solution. Understanding why both positive and negative results are feasible helps build a strong foundation in solving more complex equations.
Mathematical Concepts
Absolute value equations like \(|x| = 25\) offer insight into broader mathematical concepts. They reveal how mathematical expressions can have multiple solutions based on their structure. The use of absolute value also reinforces the principle that mathematical definitions often serve as tools to simplify complex ideas for practical use.

These equations also illustrate:
  • The importance of symmetry in mathematics 鈥� both \(x = 25\) and \(x = -25\) are equidistant from zero on the number line.
  • How numbers can change based on the context given by mathematical operations like absolute value.
By understanding these concepts, you enrich your ability to approach problems creatively and logically in mathematics.

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