Absolute-Value Equations

The Central Idea

Consider

Absolute value measures distance from zero, so the equation asks:

Which real numbers are 5 units from zero?

There are two:

and

So

Absolute-value equations often create multiple branches because distance does not record direction.

Absolute Value as Distance

For a real number \(x\),

is the distance from \(x\) to \(0\).

More generally,

is the distance from \(x\) to \(a\).

For example,

asks for the points 4 units from \(3\).

Those points are

and

Therefore,

or

The Basic Two-Case Rule

If

with

then

or

Symbolically,

This is the fundamental solving rule for absolute-value equations.

Why There Are Two Cases

If

then \(A\) could be \(5\) because

or \(A\) could be \(-5\) because

Absolute value removes sign information, so solving must restore both sign possibilities.

When the Right Side Is Zero

If

then the only possibility is

For example,

gives

so

An absolute-value equation with right side zero has at most one real solution.

When the Right Side Is Negative

Absolute value is never negative.

Therefore, an equation such as

has no real solution.

Its solution set is

This is often the fastest case to recognize: if an isolated absolute value equals a negative number, stop immediately.

Three Basic Outcomes

One-Step Absolute-Value Equation

Solve

Split into two cases:

or

Therefore, the solution set is

Absolute Value Around a Linear Expression

Solve

Case 1:

Then

Case 2:

Then

So

Interpretation on the Number Line

The equation

asks for points whose distance from \(4\) is \(7\).

Move 7 units right:

Move 7 units left:

The algebraic two-case rule and the geometric distance interpretation describe the same relationship.

Isolate the Absolute Value First

Consider

Before splitting into cases, isolate the absolute-value expression.

Divide by \(2\):

Now split:

or

So

or

Multistep Absolute-Value Equation

Solve

Add \(4\):

Divide by \(3\):

Now split into two cases.

Case 1:

so

Case 2:

so

Therefore,

Do Not Split Too Early

Suppose

The absolute value is not yet isolated.

First subtract \(3\):

Then divide by \(4\):

Only now split into the two cases.

This keeps the case structure simple and reduces unnecessary work.

Example with a Negative Coefficient Outside

Solve

Divide by \(-2\):

Then

or

So

or

Detect No-Solution Cases Early

Consider

Subtract \(7\):

Divide by \(2\):

This is impossible over the real numbers.

Therefore,

A Shortcut Using Minimum Values

Because

we know

So the expression can never equal \(3\).

Recognizing the range of an absolute-value expression can sometimes classify an equation before any detailed algebra is needed.

Absolute Value with Variables on Both Sides

Consider

Because the left side is nonnegative, any solution must satisfy

Now split into cases:

Case 1:

This gives

so there is no solution from this branch.

Case 2:

Then

Check:

Therefore,

Why Verification Matters

When absolute value is isolated and the two-case rule is applied correctly, the branches are logically equivalent to the original equation.

Even so, checking remains useful because it can catch:

• arithmetic mistakes;
• sign mistakes;
• incorrect case setup;
• domain restrictions from other expressions in the equation.

Verification becomes especially important when absolute value appears alongside fractions, radicals, or other restricted operations.

Piecewise Definition of Absolute Value

Absolute value can be defined piecewise:

This explains why solving absolute-value equations naturally involves cases.

For example,

means

when \(x\ge3\), and

when \(x<3\).

Solving by Piecewise Reasoning

Solve

If

then

so

This satisfies \(x\ge3\).

If

then

So

This satisfies \(x<3\).

The piecewise method produces the same two solutions as the basic two-case rule.

Absolute Value as a Function

The graph of

is a V-shape with vertex at

Solving

means finding where the graph

intersects the horizontal line

There are two intersection points:

and

The algebraic two-solution pattern has a direct graphical interpretation.

Shifted Absolute-Value Graphs

The graph of

has vertex at

So the equation

with \(k>0\) asks for the two points at horizontal distance \(k\) from \(h\):

and

This gives the compact solution rule

Equations with Absolute Value on Both Sides

Consider

This asks for points that are equally distant from \(1\) and \(-3\).

Geometrically, the answer is the midpoint:

Algebraically, square both sides:

Expand:

Then

so

Because both sides are already nonnegative, squaring preserves equivalence here.

A Faster Rule for Equal Absolute Values

For real expressions \(A\) and \(B\),

is equivalent to

or

For example,

gives two cases:

or

This preserves both possible sign relationships.

Worked Equal-Absolute-Value Example

Solve

Case 1:

Then

Case 2:

Then

so

Therefore,

Absolute Value and Squaring

For real \(A\) and \(k\ge0\),

is equivalent to

For example,

can be squared:

Then

This is valid, but the direct two-case rule is usually simpler.

Absolute Value and Distance Between Two Quantities

The expression

is the distance between \(a\) and \(b\).

Therefore,

means that the distance between \(x\) and \(10\) is \(3\).

The solutions are

and

This interpretation is useful in measurement, tolerance, error bounds, and geometry.

Application: Measurement Tolerance

Suppose a machine part is designed to have length \(50\) mm, and the actual length \(x\) differs from the target by exactly \(0.2\) mm.

This is modeled by

So

or

Therefore,

or

Application: Equal Distance

Suppose a point \(x\) on a number line is equally distant from \(-4\) and \(10\).

We can write

The point equidistant from two locations is their midpoint:

So

Worked Example: Isolate First

Solve

Add \(10\):

Divide by \(5\):

Case 1:

so

Case 2:

so

Therefore,

Worked Example: No Real Solution

Solve

Subtract \(1\):

Divide by \(3\):

An absolute value cannot equal a negative real number.

Therefore,

Worked Example: One Solution

Solve

Divide by \(4\):

So

Then

and

There is only one solution because the required distance is zero.

Worked Example: Variable on the Right

Solve

The right side must be nonnegative:

so

Case 1:

Then

Case 2:

Then

Both candidates satisfy \(x\ge-1\), and both check in the original equation.

Therefore,

A Reliable Solving Strategy

1. Isolate the absolute-value expression.
2. Inspect the value on the other side.
3. If it is negative, conclude no real solution.
4. If it is zero, set the inside equal to zero.
5. If it is positive, create the positive and negative cases.
6. Solve each branch.
7. Apply any domain or range restrictions.
8. Check candidates in the original equation.

Common Errors

Dropping the Absolute-Value Bars

From

writing only

loses the valid solution \(x=-5\).

Forgetting to Isolate First

For

do not immediately write \(x-3=\pm9\). First isolate the absolute value.

Creating Two Cases When the Right Side Is Zero

If

both “cases” are the same:

Trying to Solve an Impossible Negative Case

If

there is no real solution.

Forgetting Parentheses in the Negative Branch

From

the negative branch is

not

Ignoring Restrictions on the Other Side

If

then every solution must satisfy

Practice

1. Basic Absolute Value

Solve:

2. Shifted Absolute Value

Solve:

3. Isolate First

Solve:

4. No Solution

Solve:

5. One Solution

Solve:

6. Multistep

Solve:

7. Absolute Values on Both Sides

Solve:

Key Ideas

• Absolute value measures distance and is therefore always nonnegative.
• \(|A|=k\) with \(k>0\) produces two branches: \(A=k\) or \(A=-k\).
• \(|A|=0\) produces one branch: \(A=0\).
• \(|A|=k\) with \(k<0\) has no real solution.
• Isolate the absolute-value expression before splitting into cases.
• The two-case rule comes directly from the piecewise definition of absolute value.
• \(|x-a|\) represents the distance between \(x\) and \(a\).
• Equal absolute values represent equal distances and can often be solved by case splitting or midpoint reasoning.
• If \(|A|=B\), every solution must satisfy \(B\ge0\).
• Parentheses are essential when forming a negative branch.
• Verification is useful for catching sign errors and enforcing other restrictions.
• Absolute-value equations connect algebraic solving, piecewise functions, geometry, and distance on the number line.

Where This Leads

Absolute-value equations introduce a major new solving pattern: one relationship can naturally split into multiple valid cases.

This case-based reasoning will become even more important in absolute-value inequalities, piecewise functions, distance constraints, and equations where different domains require different algebraic rules.