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Chapter 1: Problem 14

Find the real solutions, if any, of each equation. $$ |1-2 z|+6=9 $$

Short Answer

Expert verified
The solutions to the equation are \(z = -1\) and \(z = 2\).

Step by step solution

01

Isolate the Absolute Value Expression

Start by isolating the absolute value expression on one side of the equation. Subtract 6 from both sides: \[|1 - 2z| + 6 - 6 = 9 - 6\]This simplifies to: \[|1 - 2z| = 3\]
02

Define Two Cases for the Absolute Value

The absolute value equation \(|1 - 2z| = 3\) can be split into two separate equations: Case 1: \[1 - 2z = 3\]Case 2: \[1 - 2z = -3\]
03

Solve the First Equation

Solve Case 1: \[1 - 2z = 3\]Subtract 1 from both sides: \[-2z = 2\]Then divide both sides by -2: \[z = -1\]
04

Solve the Second Equation

Solve Case 2: \[1 - 2z = -3\]Subtract 1 from both sides: \[-2z = -4\]Then divide both sides by -2: \[z = 2\]
05

Combine the Solutions

The solutions from both cases are: \[z = -1\] and \[z = 2\]Therefore, the real solutions to the equation are \(z = -1\) and \(z = 2\)

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Key Concepts

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

absolute value
Absolute value refers to the distance of a number from zero on the number line. It is always non-negative.
For any real number x, the absolute value is denoted as |x|. It can be defined as:
  • If x is positive or zero, \(|x| = x\)
  • If x is negative, \(|x| = -x\)

This means that \(|x|\) is always positive, even if x itself is negative. Understanding this property is crucial in working with absolute value equations.
isolation of absolute value expression
Isolating the absolute value expression is often the first step in solving an absolute value equation. This means rearranging the equation so that the absolute value term is alone on one side. For instance, in our original exercise:
  • We start with |1 - 2z| + 6 = 9
  • Subtract 6 from both sides to get |1 - 2z| = 3

Isolating the absolute value term simplifies the process and prepares us to handle the equation inside the absolute value term.
splitting into cases
Once the absolute value expression is isolated, the next step is to split the equation into two cases. This is because \(|x|\) equals some value (say, a) means that x can be either a or -a. In our exercise:
  • |1 - 2z| = 3
  • This splits into two cases: 1 - 2z = 3 and 1 - 2z = -3

This method leverages the definition of absolute value, helping us solve for the variable within the absolute value expression.
solving linear equations
After splitting the absolute value equation into two cases, each case becomes a regular linear equation. Solving these linear equations gives us the solutions for the original absolute value equation. In our exercise:
  • For Case 1: \(1 - 2z = 3\) \t
      \t
    • \tSubtract 1 from both sides: \(-2z = 2\)
      • \t
    • \tDivide by -2: \(z = -1\)
      • \t
  • For Case 2: \(1 - 2z = -3\) \t
      \t
    • \tSubtract 1 from both sides: \(-2z = -4\)
      • \t
    • \tDivide by -2: \(z = 2\)
      • \t

Hence, the solutions to the equation \(|1 - 2z| + 6 = 9\) are \(z = -1\) and \(z = 2\).

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