Absolute Value Equations
∣x∣ = a splits into two cases: x = a or x = −a.
About this topic
Because absolute value measures distance from zero, an equation saying the absolute value of x − 3 equals 5 asks for numbers 5 units from 3. That gives two solutions, x = 8 and x = −2. Isolate the absolute value first, then split into two cases: the inside equals the positive value, or the negative value.
For example, to solve 2 times the absolute value of x + 1, minus 4, equals 6, first add 4 and divide by 2 to isolate the absolute value, getting the absolute value of x + 1 equals 5. Then x + 1 = 5 gives x = 4, and x + 1 = −5 gives x = −6. Always check both answers in the original equation.
If the isolated absolute value equals a negative number, there is no solution, since distance cannot be negative. If it equals zero, there is exactly one solution. Think of a thermostat that must stay within 2 degrees of 21: the absolute value of T − 21 equals 2 gives exactly the two edge temperatures, 19 and 23. Tolerances in manufacturing, such as a bolt that must be 10 mm plus or minus 0.1, use the same idea.
A frequent error is solving only the positive case and missing the second answer, or splitting before isolating the absolute value, which gives wrong equations. Another is dropping the negative sign when forming the second case: the correct second equation is x + 1 = −5, not −x + 1 = 5 unless you carefully distribute.
Key ideas
- Isolate the absolute value first
- Solve both cases
- A negative right side means no solution
Quick check
Use cases: where Absolute Value Equations helps
- Homework: review the key idea behind Absolute Value Equations before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Absolute Value Equations without notes.
- Visual learning: play with the interactive lab to see how Absolute Value Equations changes when you adjust its inputs.
- Teaching a friend: explain Absolute Value Equations in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Absolute Value Equations?
∣x∣ = a splits into two cases: x = a or x = −a.
What will I learn in Absolute Value Equations?
Isolate the absolute value first. Solve both cases. A negative right side means no solution.
Is Absolute Value Equations part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Absolute Value Equations lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Absolute Value Equations?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.