Modulus Equations
Overview
The modulus, or absolute value, of a number is its distance from zero, so it is never negative. That single idea, distance, is the key to solving every modulus equation. Strip away the bars and you are left with two ordinary equations to consider, one for each sign.
In this lesson you will solve equations of the form \(|ax+b|=c\), \(|ax+b|=cx+d\) and \(|ax+b|=|cx+d|\). The guiding rule is simple: if \(|\text{something}|=c\) then that something is \(+c\) or \(-c\). You will also learn the vital habit of checking your solutions, because the modulus can quietly create answers that do not work.
Objectives
- Solve equations involving the modulus, such as |ax + b| = c, |ax + b| = cx + d and |ax + b| = |cx + d|, using algebraic or graphical methods.
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Lesson Note
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Lesson Evaluation
Congratulations on completing the lesson on Modulus Equations. Now that you've explored the key concepts and ideas, it's time to put your knowledge to the test. This section offers a variety of practice questions designed to reinforce your understanding and help you gauge your grasp of the material.
You will encounter a mix of question types, including multiple-choice questions, short answer questions, and essay questions. Each question is thoughtfully crafted to assess different aspects of your knowledge and critical thinking skills.
Use this evaluation section as an opportunity to reinforce your understanding of the topic and to identify any areas where you may need additional study. Don't be discouraged by any challenges you encounter; instead, view them as opportunities for growth and improvement.
- Solve |2x - 1| = 5. A. x = 3 or x = -2 B. x = 2 or x = -3 C. x = 3 only D. x = -2 only Answer: A
- The equation |A| = c (with c >= 0) is equivalent to: A. A = c only B. A = c or A = -c C. A = -c only D. A = 0 Answer: B
- Which value is never possible for |x|? A. 0 B. 5 C. -3 D. 2.5 Answer: C
- Solve |x - 1| = |2x + 1|. A. x = -2 or x = 0 B. x = 2 or x = 0 C. x = 1 or x = -1 D. x = -2 only Answer: A
- When solving |x - 2| = 2x - 7, x = 3 is rejected because: A. it does not solve case 2 B. it makes the right-hand side negative C. it makes |x - 2| negative D. it is not an integer Answer: B
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Recommended Books
Further learning from open sources
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Textbook section
Other Types of Equations
Solve equations involving rational exponents.
"Algebra and Trigonometry: Other Types of Equations" by OpenStax, OpenStax, Rice University, licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Source: https://openstax.org/books/algebra-and-trigonometry/pages/2-6-other-types-of-equations. Access for free at https://openstax.org/books/algebra-and-trigonometry/pages/1-introduction
Published by other organisations under open licences, credited as they ask. Links open their websites.
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