Solving Modulus Equations: Theoretical Definition and Step-by-Step Solutions

Definition and Fundamental Principles of Modulus Equations

• The absolute value or modulus definition is divided into two distinct domain conditions:

  • Case 1: When $x$ is greater than or equal to zero ($x \ge 0$).
  • Case 2: When $x$ is less than zero ($x < 0$).

• Solving a modulus equation requires analyzing the problem from two separate mathematical angles to yield two distinct solutions.

• Principles for Case 1 ($x \ge 0$):

  • The expression is written without the modulus sign.
  • No terms or signs within or outside the modulus are altered except for the removal of the modulus sign itself.
  • For the equation $|3x| = 18$, dropping the modulus sign yields:     $3x = 18$

• Principles for Case 2 ($x < 0$):

  • The term $3x$ is retained as it is written, while the opposite side of the equation is multiplied by $-1$.
  • Multiplying the right-hand side constant $18$ by $-1$ gives $-18$.
  • This produces the second algebraic equation:     $3x = -18$

Step-by-Step Solution of the Modulus Equation

• Solving the First Linear Equation ($x \ge 0$):

  • Equation:     $3x = 18$
  • Divide both sides of the equation by $3$ to isolate $x$:     $x = \frac{18}{3}$
  • Simplify the quotient:     $x = 6$

• Solving the Second Linear Equation ($x < 0$):

  • Equation:     $3x = -18$
  • Divide both sides of the equation by $3$ to isolate $x$:     $x = \frac{-18}{3}$
  • Simplify the quotient:     $x = -6$

• Final Result:

  • The two solutions derived for $x$ in the equation $|3x| = 18$ are $x = 6$ and $x = -6$

Questions and Discussion

• Initial Explanation Clarification:

  • Question / Feedback: A participant expressed confusion during the initial explanation, stating, "Yes, sir. No, sir. You can start over. Start over."
  • Request for Review: A request was made to repeat the material from the beginning: "We can go back a little bit. Didn't catch, like, half of it. Let's go again."
  • Re-explanation: The formal definition of the modulus sign was repeated, clearly stating the two domain cases ($x < 0$ and $x \ge 0$) and highlighting that for $x \ge 0$, nothing changes in $3x = 18$ except removing the modulus sign.

• Intermediate Comprehension Check:

  • Question / Check: After demonstrating how to multiply the right side of the equation by $-1$ for the $x < 0$ case to get $3x = -18$, understanding was confirmed.
  • Response: "Okay? Yes, sir. I got it now. I got I got got somewhere?"

• Final Solution and Writing Directive:

  • Question / Check: After completing both calculations to obtain $x = 6$ and $x = -6$, the final inquiry was made: "Make any sense to you?"
  • Response: "Yes, sir."
  • Final Instructions: "Alright. Clear to everybody? Could you please write it down for me, please?"