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Chapter 9: Problem 28

Solve each absolute value equation. \(|x|=1\)

Short Answer

Expert verified
The solutions are \(x = 1\) and \(x = -1\).

Step by step solution

01

Understanding the Absolute Value Equation

The given equation is \(|x| = 1\). Absolute value represents the distance of a number from zero, hence \(|x| = 1\) means the values of \(x\) are at a distance of 1 unit from zero on the number line.
02

Split into Two Possible Equations

An absolute value equation \(|x| = a\) can be split into two separate equations: 1. \(x = a\)2. \(x = -a\).Given \(|x| = 1\), we create the equations:1. \(x = 1\)2. \(x = -1\).
03

Solutions for x

The solutions from the equations in the previous step are:- From \(x = 1\), we have \(x = 1\).- From \(x = -1\), we have \(x = -1\).Thus, the solutions to \(|x| = 1\) are \(x = 1\) and \(x = -1\).

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Key Concepts

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

Algebra
Algebra can be seen as a bridge between arithmetic and abstract mathematics, dealing with symbols and the rules for manipulating these symbols. These symbols represent numbers and are used to express mathematical ideas and relationships.
  • Algebra helps us solve equations and find unknown values by using known quantities.
  • It involves techniques for expressing mathematical relationships using variables, constants, and operational symbols like "+" and "-".
In the given problem, the equation \(|x|=1\) is expressed using a variable, \(x\), and an absolute value symbol. Understanding algebra helps us manipulate and solve such equations effectively, enabling us to split the absolute value equation into more manageable parts. This gives us insights into the possible values that \(x\) can take.
Absolute Value
The concept of absolute value might seem a bit abstract at first, but it's quite simple once you break it down. The absolute value of a number is essentially how far that number is from zero, without considering its direction on the number line.
  • Given any number \(a\), the absolute value \(|a|\) is always positive or zero.
  • The absolute value ignores whether \(a\) is negative or positive.
For example, both 3 and -3 are 3 units away from zero on the number line, so \(|3| = 3\) and \(|-3| = 3\). In our exercise, \(|x|=1\) means \(x\) must be either 1 or -1 because both these points are 1 unit away from zero. Absolute value plays a critical role when solving equations, especially when we want to find all possible solutions that satisfy the condition set by the absolute value equation.
Solving Equations
Solving equations is a core aspect of algebra, where we aim to find the value of the unknown variable that satisfies the equation. When dealing with absolute value equations like \(|x|=1\), there are specific steps to follow:
  • Recognize the absolute value equation and understand what it represents, such as the distance from zero.
  • Split the absolute value equation into two separate linear equations based on the property that \(|x| = a\) gives \(x = a\) and \(x = -a\).
  • Solve each of these linear equations individually to find all potential values for the variable.
In our example, those steps lead directly to \(x = 1\) or \(x = -1\) as solutions. This process ensures all possible solutions are captured, making it an efficient and systematic approach to handle absolute value equations.

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Most popular questions from this chapter

Describe the location of the solution region of the system $$ \left\\{\begin{array}{l} {x>0} \\ {y>0} \end{array}\right. $$

Graph each inequality. See Examples 2 through 5. $$ 2 x+5 y>-10 $$

Scott Sambracci and Sara Thygeson are planning their wedding. They have calculated that they want the cost of their wedding ceremony \(x\) plus the cost of their reception \(y\) to be no more than 5000 dollars. a. Write an inequality describing this relationship. b. Graph this inequality. c. Why should we be interested in only quadrant I of this graph?

Graph the solution of \(\left\\{\begin{array}{l}{2 x-y \leq 6} \\ {x \geq 3} \\\ {y>2}\end{array}\right.\)

Solve each compound inequality. Graph the solution set and write it in interval notation. See Examples 7 and \(8 .\) $$ x \leq-4 \text { or } x \geq 1 $$
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