English:Absolute Value Equations
Absolute Value Equations
Introduction
Absolute value is a way to describe distance without direction. On a number line, both 5 and -5 are five units from 0, so |5| = 5 and |-5| = 5. This distance idea is the key to understanding absolute value equations.
In this aiMOOC for Grades 9–10, you will learn how to interpret, solve, graph, check, and apply absolute value equations. The main focus is on equations built from linear expressions, such as |2x - 3| = 7, as well as equations in which you must first isolate the absolute value expression.
By the end of the course, you should be able to explain why an absolute value equation may have two solutions, one solution, or no real solution; solve equations accurately; verify candidate solutions; and connect equations to distance, graphs, and real-world tolerances.
Learning Goals
You will learn to:
- Absolute value: Interpret |x| as the nonnegative distance from x to 0.
- Linear equations: Solve the two branch equations created by an absolute value equation.
- Number line: Explain solutions as points at a fixed distance from a center.
- Graphs: Identify solutions as intersection points.
- Modeling: Translate tolerance and distance situations into absolute value equations.
- Verification: Substitute candidate solutions into the original equation and reject invalid values.
Prerequisite Knowledge
You should already be comfortable with signed numbers, the order of operations, solving one-step and multi-step linear equations, distributing a negative sign, and checking a solution by substitution. If any of these ideas are unfamiliar, review Integer, Linear equation, and Distributive property first.
Understanding Absolute Value
Distance from Zero
For a real number x, |x| is its distance from 0. Because distance cannot be negative, |x| is always greater than or equal to 0. The two numbers x and -x are opposites, but they have the same absolute value.
For example, |-7| = 7 because -7 is seven units from 0. Similarly, |7| = 7. This symmetry is why many absolute value equations split into two cases.
The same idea can be written as a piecewise rule: when x is nonnegative, |x| = x; when x is negative, |x| = -x. You do not need to memorize the piecewise notation to solve every equation, but it explains why the graph forms a V.
The Absolute Value Function
The graph of y = |x| has its vertex at the origin. The left and right branches rise at the same rate because opposite inputs have equal absolute values.
When the expression inside the bars is shifted or stretched, the vertex and steepness can change. The V-shape is especially useful when you solve an equation graphically: solutions appear where the absolute value graph meets another graph.
The Core Equation Pattern
The basic algebraic pattern is |A| = c, where A is an algebraic expression and c is a real number.
If c is positive, then A can equal c or -c. If c is zero, then A must equal zero. If c is negative, there is no real solution because an absolute value cannot be negative.
For a linear equation |ax + b| = c with a not equal to 0:
- If c is positive, there are two real solutions.
- If c is zero, there is one real solution.
- If c is negative, there is no real solution.
This three-case statement applies to the linear form above. More complicated absolute value equations can have different solution patterns, so always analyze the actual equation.
The visual above illustrates the central idea: an absolute value equation can be understood by comparing distances or by looking for intersections.
Why Two Branches Appear
Consider |x - 4| = 3. The expression x - 4 must be three units from zero. Therefore it can be 3 or -3.
First branch: x - 4 = 3, so x = 7.
Second branch: x - 4 = -3, so x = 1.
Both values work. On the number line, 1 and 7 are each three units from 4. The equation can therefore also be read as: Which numbers are three units from 4?
A Reliable Solving Method
Use this process for standard absolute value equations.
- Isolate the absolute value expression. Move constants and divide by coefficients outside the bars until the absolute value stands alone.
- Check the target value. If the isolated absolute value equals a negative number, stop: there is no real solution.
- Create the branches. For a positive target c, write the inside expression equal to c and equal to -c.
- Solve each branch. Use ordinary linear-equation methods.
- Check the candidates. Substitute each value into the original equation, especially when variables also appear outside the absolute value.
A common mistake is to split the equation too early. For example, in 3|x - 2| + 1 = 13, you should first subtract 1 and divide by 3. Only after you obtain |x - 2| = 4 should you create the two branches.
Worked Example: A Direct Equation
Solve |2x - 3| = 7.
First branch: 2x - 3 = 7, so 2x = 10 and x = 5.
Second branch: 2x - 3 = -7, so 2x = -4 and x = -2.
Check x = 5: |2 times 5 - 3| = |7| = 7.
Check x = -2: |2 times -2 - 3| = |-7| = 7.
Therefore the solution set is {-2, 5}.
Worked Example: Isolate First
Solve 2|3x + 1| - 4 = 10.
Add 4 to both sides: 2|3x + 1| = 14.
Divide by 2: |3x + 1| = 7.
First branch: 3x + 1 = 7, so x = 2.
Second branch: 3x + 1 = -7, so 3x = -8 and x = -8/3.
Both values satisfy the original equation, so the solution set is {2, -8/3}.
Worked Example: Zero Target
Solve |4x + 8| = 0.
An absolute value equals zero only when the expression inside it equals zero. Therefore 4x + 8 = 0, so x = -2.
There is only one solution because zero has no distinct positive and negative versions.
Worked Example: Negative Target
Solve |2x - 5| = -3.
The left side is always nonnegative, but the right side is negative. Therefore the equation has no real solution. Do not create two branches.
Equations with Variables Outside the Bars
Some equations have a variable on the other side, such as |2x - 1| = x + 5. Here the right side is not automatically nonnegative, so checking becomes especially important.
Because an absolute value is nonnegative, any genuine solution must make x + 5 greater than or equal to 0.
For |2x - 1| = x + 5:
First branch: 2x - 1 = x + 5, giving x = 6.
Second branch: 2x - 1 = -(x + 5), giving 2x - 1 = -x - 5, so 3x = -4 and x = -4/3.
Both candidates make x + 5 nonnegative and both satisfy the original equation, so both are solutions.
Now consider |x + 2| = x - 4. Any solution must make x - 4 nonnegative, so x must be at least 4. Solving the branch equations produces no valid value. The equation therefore has no real solution. This example shows why domain reasoning and substitution checks matter.
Graphical Interpretation
An equation says that two expressions have equal values. You can therefore solve an absolute value equation by graphing both sides and finding their intersection points.
For example, to solve |x - 1| = 3, graph y = |x - 1| and y = 3. The horizontal line meets the V-shaped graph at x = -2 and x = 4. Those x-coordinates are the solutions.
A graph also makes the three basic cases visible. A positive horizontal target can cross both arms of a V, a target at the vertex may touch once, and a horizontal line below the minimum of the V does not intersect it.
Graphing is useful for checking algebra, but an exact algebraic solution is usually preferred when the intersection coordinates are simple.
Distance Between Two Numbers
Absolute value can measure distance between any two numbers a and b. The distance is |a - b|, which is the same as |b - a|.
This idea allows you to model statements such as "x is 6 units from 10" with |x - 10| = 6. The two solutions are x = 4 and x = 16.
Modeling with Absolute Value Equations
Absolute value equations are useful when direction is less important than the size of a difference. Common contexts include measurement error, manufacturing tolerances, temperature differences, timing deviations, and distances along a straight route.
Suppose a machine part is designed to be 50 millimeters long, and you want to find measurements exactly 0.2 millimeters away from the target. The equation is |L - 50| = 0.2. Its solutions are L = 49.8 and L = 50.2.
Suppose a bus is scheduled to arrive at 8:00, and a record shows an arrival exactly 6 minutes from the scheduled time. If t is the signed number of minutes relative to 8:00, then |t| = 6, so t = -6 or t = 6. The bus arrived either six minutes early or six minutes late.
A model should always include a clear definition of the variable and a check that the mathematical solutions make sense in the context.
Common Errors and How to Avoid Them
Error 1: Forgetting the negative branch. If |A| = 5, both A = 5 and A = -5 must be considered.
Error 2: Splitting before isolating. In 2|x| + 3 = 11, first obtain |x| = 4.
Error 3: Treating |A| = -5 as solvable. Absolute value is never negative, so the equation has no real solution.
Error 4: Dropping parentheses in the negative branch. If |2x - 3| = x + 4, the second branch is 2x - 3 = -(x + 4), which becomes 2x - 3 = -x - 4.
Error 5: Skipping verification. Candidate values can fail when another variable expression appears outside the bars. Substitute into the original equation.
Error 6: Confusing equations and inequalities. The equation |x| = 4 asks for exact distances and has x = -4 or x = 4. The inequality |x| < 4 asks for all numbers closer than 4 units to zero, which is a different problem.
Strategy Summary
When you see an absolute value equation, first interpret the bars as distance. Then isolate the absolute value expression, inspect the value on the other side, create branches only when appropriate, solve carefully, and check each candidate in the original equation.
Interactive Tasks
Quiz: Test Your Knowledge
What does absolute value represent on a number line? (Distance from zero) (!Direction from zero) (!A negative coordinate) (!The slope of a line)
What are the solutions of |x| = 6? (x equals 6 or negative 6) (!x equals 6 only) (!x equals negative 6 only) (!There is no solution)
What should you do first when solving 3|x - 2| + 1 = 13? (Isolate the absolute value expression) (!Create two branches immediately) (!Square both sides immediately) (!Change every sign)
What is true when an isolated absolute value equals a negative number? (There is no real solution) (!There are always two solutions) (!There is always one solution) (!Every real number is a solution)
What are the solutions of |x - 4| = 3? (x equals 1 or 7) (!x equals 3 or 4) (!x equals 1 only) (!x equals 7 only)
What is the solution of |4x + 8| = 0? (x equals negative 2) (!x equals 2) (!x equals negative 8) (!There are two solutions)
Why should candidate solutions be checked in the original equation? (A candidate may fail the original equation) (!Checking changes the equation) (!Checking removes the absolute value) (!Checking always creates another solution)
Which statement describes the graph of y = |x|? (It is V shaped with a vertex at zero) (!It is a horizontal line) (!It is a circle centered at zero) (!It is a downward opening parabola)
What does |x - 10| = 6 mean? (x is six units from ten) (!x is ten units from six) (!x must equal sixteen only) (!x must be greater than ten)
When |ax + b| = c has a nonzero linear expression and c is positive, how many real solutions does it have? (Two real solutions) (!One real solution) (!No real solutions) (!Infinitely many real solutions)
Memory Game
| Absolute value | Nonnegative distance from zero |
| Opposite | Number located the same distance on the other side of zero |
| Isolation | Moving other terms so the bars stand alone |
| Branch equation | One of the ordinary equations created from a positive target |
| Verification | Substituting a candidate into the original equation |
| Vertex | Turning point of a V shaped graph |
Drag and Drop
| Match the correct terms. | Topic |
|---|---|
| Two solutions | Positive target in a nonconstant linear absolute value equation |
| One solution | Zero target in a nonconstant linear absolute value equation |
| No real solution | Negative target after isolation |
| Isolate first | Terms remain outside the absolute value bars |
| Check candidates | A variable expression also appears outside the bars |
Match each strategy phrase with the situation in which it is most useful.
Crossword Puzzle
| Distance | What does absolute value measure from zero on a number line? |
| Modulus | What is another mathematical name for absolute value? |
| Opposite | What do you call a number with the same magnitude but the other sign? |
| Isolate | What should you do to the absolute value expression before creating branches? |
| Branch | What do we call one of the ordinary equations created from an absolute value equation? |
| Verify | What should you do by substitution after finding candidate solutions? |
LearningApps
Cloze Text
Open-Ended Tasks
Easy
- Number line poster: Create a one-page visual showing why 5 and negative 5 have the same absolute value, and include two additional examples of your own.
- Two-branch explanation: Write a short explanation for a classmate showing why |x - 3| = 4 creates two ordinary linear equations.
- Solution card set: Make six study cards, each with an absolute value equation on the front and a complete checked solution on the back.
- Mini tutorial video: Record a two-minute video in clear English demonstrating how to solve one equation of the form |x - a| = b.
Standard
- Tolerance investigation: Measure the same classroom object several times, choose a target length, and write absolute value equations for measurements at selected fixed deviations from the target.
- Error analysis poster: Create an illustrated poster showing three common mistakes in absolute value equations and a corrected example for each mistake.
- Graph and algebra comparison: Solve one absolute value equation algebraically, graph both sides, and explain how the intersection points confirm your solutions.
- Mathematics interview: Interview a teacher, technician, craftsperson, or engineer about situations where tolerance or deviation matters, then translate one example into an absolute value model.
Advanced
- Piecewise derivation: Derive the two branches of |ax + b| = c from the piecewise definition of absolute value and state clearly when the argument applies.
- Variable target investigation: Explore three equations in which a variable expression appears outside the absolute value bars, solve them, and explain why checking candidates is essential.
- Local measurement field study: Visit an appropriate school workshop, lab, makerspace, sports facility, or other supervised setting and document a measurement standard that can be modeled by distance from a target.
- Instructional media project: Produce a polished slide deck, infographic, or five-minute teaching video that connects number-line distance, algebraic branches, graph intersections, and a real-world tolerance example.
Learning Assessment
- Reasoning from distance: Explain why |x - 8| = 5 has two solutions without first using algebra, then solve it and connect each solution to a position on a number line.
- Method selection: Compare solving 2|x + 1| - 3 = 9 by algebra and by graphing, and justify which method gives the clearest exact answer.
- Error diagnosis: A learner solves |3x - 2| = 7 using only 3x - 2 = 7. Identify the missing reasoning, correct the work, and verify both solutions.
- Constraint analysis: Solve |2x + 1| = x - 5 and explain how the nonnegative nature of absolute value restricts possible solutions before or during your work.
- Modeling transfer: Create a realistic tolerance or timing situation that leads to an equation of the form |x - a| = b, solve it, and interpret both solutions in context.
- Graphical justification: Sketch two functions whose intersections represent the solutions of |x + 2| = 4 and explain why the number of intersections matches the algebraic solution count.
Evidence of Learning
Knowledge: You can define absolute value as nonnegative distance, distinguish positive, zero, and negative target cases, and explain why a positive target creates two branches in a nonconstant linear absolute value equation.
Skills: You can isolate absolute value expressions, solve both linear branches, distribute a negative sign correctly, check candidate solutions, and interpret graph intersections.
Products: Strong evidence may include a worked problem set, a number-line explanation, a graph comparison, a tolerance model, an interview summary, an infographic, or a short instructional video.
Reasoning: You can justify each step rather than applying a memorized rule without explanation, and you can diagnose common errors in another solution.
Transfer: You can recognize new situations involving fixed distance or deviation from a target and represent them with an appropriate absolute value equation.
OERs on the Topic
The English Wikipedia article on Absolute value provides a broad reference on definitions, properties, distance, and the absolute value function.
For additional openly accessible practice and explanations, you can also consult OpenStax College Algebra: Absolute Value Functions.
Linked Learning Areas
Absolute value equations connect algebraic procedures with geometric distance, function graphs, modeling, and logical case analysis. These links help you see the topic as more than a rule for manipulating symbols.