When does the absolute value inequality become a compound inequality versus a union?

Less-than form, such as the absolute value of A less than c, becomes a compound inequality where A is between negative c and positive c. Greater-than form, such as the absolute value of A greater than c, becomes a union where A is less than negative c or greater than positive c.

What happens when the right side is negative?

Since an absolute value is always non-negative, comparison to a negative number is automatic. Less-than form with a negative right side has no solution; greater-than form with a negative right side holds for all real numbers. The solver detects both edge cases without applying the case split.

How do you write the answer in interval notation?

Compound form (between the two boundary values) uses a single interval, open or closed depending on whether the inequality is strict or non-strict. Union form uses two intervals joined by the union symbol, both extending to infinity. Endpoints use parentheses for strict, brackets for non-strict.

How do you handle an absolute value inequality with a coefficient or added constant outside?

Isolate the absolute value first. Subtract any added constants. Divide by any coefficient multiplying the absolute value. If the coefficient is negative, the inequality direction flips once during the division. Once isolated to the absolute value compared to a constant, apply the case split.

Free Absolute Value Inequality Solver - Step by Step

Getting Started

Demo|x − 3| < 5, Solve, Clear, no solution

Step 0 of 4

The solver shows an inequality display at the top with a blinking yellow caret marking the cursor position. Click anywhere in the display to place the cursor, or use arrow keys, Home, and End for keyboard navigation. Type directly or use the button panel below.

Three quick experiments:

• Type

|x - 3| < 5

and press Enter — the right panel rewrites as

-5 < x - 3 < 5

, solves both sides, and reports the compound inequality

-2 < x < 8

with the interval

(-2, 8)

.
• Click the "Simple Greater Than" example template — an inequality like

|x + 2| > 4

loads. The solver rewrites as a union,

x + 2 < -4

or

x + 2 > 4

, and reports the union answer

x < -6

or

x > 2

.
• Try the "No Solution" template — an inequality with a negative right side loads. The solver detects the impossibility immediately and reports no solution.

The Solve button is disabled until you enter something. The special row includes the absolute value bars button (rendered as

|

|

). An expression without an inequality symbol is refused with "Input must be an inequality (use <, >, ≤, or ≥)", and one without bars with "No absolute value found".

|x| + 3 < 1

isolates to

|x| < -2

and ends in No solution. What the bars mean for an inequality is explained in what is an absolute value inequality.

Building Inequalities with Buttons and Keyboard

DemoA coefficient and a constant, undo, delete, a union

Step 0 of 4

The button panel groups inputs by type. All inputs can be made either by clicking or typing on the keyboard.

• Variable row —

x

,

y

,

n

.
• Number row — digits 0 through 9 and the decimal point.
• Operator row — multiplication, division, plus, minus.
• Ineq row — the four inequality symbols.
• Special row — absolute value bars and parentheses.

The absolute value bars button inserts a single

|

character; click it twice to surround an expression. Keyboard users can type the vertical-bar character directly. For multi-term expressions inside the bars, type the terms between the two bars:

|2x + 1|

.

Keyboard shortcuts: type letters and digits directly, use star or slash for multiplication and division, type

<

and

>

directly, type

<=

or

>=

to insert

\leq

and

\geq

, type the vertical bar for absolute value, and press Enter to solve. Ctrl+Z undoes up to fifty edits back. The Clear button empties the display entirely; the curved-arrow button steps through the undo stack one edit at a time; the ⌫ button deletes the character before the caret.

2 \cdot |x + 1| - 4 > 6

is isolated to

|x + 1| > 5

and gives the union

x < -6

or

x > 4

. Why a greater-than form becomes a union is explained in the two cases.

Try an Example — Eight Form Templates

DemoOpen the templates, load two forms

Step 0 of 4

Click the "Try an Example" header to expand the template panel. Each card generates a random inequality of that form. Clicking again produces a new random version.

• Simple Less Than —

|x + a| < b

. The compound case with coefficient 1 on the variable.
• Simple Greater Than —

|x + a| > b

. The union case with coefficient 1.
• Less or Equal —

|ax + b| \leq c

. Non-strict compound case with bracketed endpoints.
• Greater or Equal —

|ax + b| \geq c

. Non-strict union case with bracketed endpoints.
• Isolate First —

|x + a| + b < c

. Requires subtracting

b

before applying the case split.
• Coefficient Outside —

a \cdot |x + b| > c

. Requires dividing by

a

before applying the case split.
• No Solution —

|expression| <

negative. Tests the impossible-case branch.
• All Reals —

|expression| \geq 0

. Tests the always-true branch.

Roughly 80 percent of generated inequalities produce clean integer boundary values. Loading a template clears the previous solution; click Solve to work the new inequality. The four stages applied to every template are listed in the solving process explained.

Reading the Step-by-Step Solution

DemoA single point, then a non-strict union

Step 0 of 3

The solution panel shows each algebraic move as a labeled step. The exact sequence depends on the structural case.

• Rearrange (when applicable) — if the absolute value is on the right, the solver swaps sides and flips the direction.
• Isolate Absolute Value — subtracts an additive constant or divides by a coefficient outside the bars to leave

|expression| \square c

alone on one side. Dividing by a negative coefficient flips the direction.
• No Solution or All Real Numbers (for special cases) — handles negative right sides and zero edge cases without applying the case split.
• Compound Inequality (for less-than form) — rewrites

|A| < c

as

-c < A < c

.
• Union of Two Inequalities (for greater-than form) — rewrites

|A| > c

as

A < -c

or

A > c

.
• Solve Both Sides (compound) or Solve Both Branches (union) — finishes each linear sub-inequality.
• Solution Interval or Solution — states the final compound or union answer.

The final-answer card shows the answer in three forms: the inequality form (

a < x < b

or

x < a

or

x > b

), a plain-language note about the bounds, and the interval notation.

|x - 5| \leq 0

is the Zero Case and holds only at

x = 5

;

|2x + 1| \geq 3

splits into

2x + 1 \leq -3

or

2x + 1 \geq 3

and gives

(-\infty, -2] \cup [1, \infty)

. The box under the solver, How the solver works, summarises the method in a few lines and links to the theory below. The split behind every answer is explained in the two cases.

The Two Cases — Why They Work

Every absolute value inequality reduces to one of two structural cases. Knowing which one is half the work.

Case 1: $|A| < c$ becomes a compound inequality (AND). The absolute value of

A

is less than

c

exactly when

A

is closer to zero than

c

units. That is,

A

lies in the open interval from

-c

to

c

:

|A| < c \;\iff\; -c < A < c

Geometrically: on the number line,

|A|

is the distance from zero, so

|A| < c

means "within

c

of zero".

Case 2: $|A| > c$ becomes a union (OR). The absolute value of

A

is greater than

c

exactly when

A

is farther from zero than

c

units. That is,

A

lies outside the interval from

-c

to

c

:

|A| > c \;\iff\; A < -c \text{ or } A > c

Geometrically: on the number line, "more than

c

away from zero" means either far to the left or far to the right.

Non-strict variants. The strict

<

and

>

produce open endpoints; the non-strict

\leq

and

\geq

produce closed endpoints. So

|A| \leq c

becomes

-c \leq A \leq c

(closed interval), and

|A| \geq c

becomes

A \leq -c

or

A \geq c

(union with closed endpoints).

Edge cases when $c \leq 0$ .

•

|A| < c

with

c \leq 0

: impossible because

|A| \geq 0

never lies below a non-positive number.
•

|A| > c

with

c < 0

: always true because

|A| \geq 0 > c

.
•

|A| \leq 0

: only when

A = 0

, giving a single point.
•

|A| \geq 0

: always true.
•

|A| > 0

: all reals except where

A = 0

.
•

|A| < 0

: no solution.

For deeper coverage see absolute value inequality and compound inequality.

The two forms are set out in two fundamental forms, solving the less-than form and solving the greater-than form.

What is an Absolute Value Inequality?

An absolute value inequality is an inequality containing an expression inside absolute value bars compared to another quantity. The standard form is

|f(x)| \;\square\; c

where

f(x)

is a linear expression (usually

ax + b

),

c

is a real number, and

\square

is one of

<

,

>

,

\leq

,

\geq

.

Solution shapes. Depending on the inequality direction and the sign of

c

, the solution takes one of several forms:

• A bounded interval (compound) — less-than form with

c > 0

. Example:

|x| < 3

gives

-3 < x < 3

.
• A union of two rays — greater-than form with

c > 0

. Example:

|x| > 3

gives

x < -3

or

x > 3

.
• A single point — non-strict less-than with

c = 0

. Example:

|x - 5| \leq 0

gives

x = 5

.
• All reals except one point — strict greater-than with

c = 0

. Example:

|x| > 0

gives all

x \neq 0

.
• All real numbers — greater-than form with

c \leq 0

(and non-strict

c = 0

).
• No solution — less-than form with

c \leq 0

(and strict

c = 0

).

Geometric meaning.

|x - a|

is the distance from

x

to

a

on the number line. So

|x - a| < c

means "

x

is within

c

of

a

" (an interval centered at

a

), and

|x - a| > c

means "

x

is more than

c

from

a

" (the complement, two rays extending outward).

Standard form for non-trivial cases. Once isolated to

|ax + b| \square c

with

c > 0

, the linear-isolation case split applies cleanly: compound for less-than, union for greater-than.

For deeper coverage see absolute value inequality, absolute value, and interval notation.

The place of these inequalities among the types is absolute value inequalities, the distance reading is geometric interpretation, and the bars themselves are defined in definition of absolute value.

The Solving Process Explained

Four stages handle any standard absolute value inequality.

• Stage 1: Isolate the absolute value. Subtract any constants added to the absolute value expression. Divide by any coefficient multiplying it (flipping the direction once if the coefficient is negative). The goal is to reach the form

|f(x)| \square c

where

c

is a constant.

• Stage 2: Examine the right side.

• $c < 0$ with less-than form: no solution.
• $c < 0$ with greater-than form: all real numbers.
• $c = 0$ , special cases: single point (non-strict less-than), all reals except a point (strict greater-than), no solution (strict less-than), all reals (non-strict greater-than).
• $c > 0$ : proceed to stage 3.

• Stage 3: Apply the case split.

• Less-than form ( $|A| < c$ or $|A| \leq c$ ): rewrite as the compound inequality

-c \square A \square c

with the same strictness on both sides.
• Greater-than form ( $|A| > c$ or $|A| \geq c$ ): rewrite as the union

A \square -c

or

A \square c

with the appropriate strictness.

• Stage 4: Solve the resulting linear sub-inequalities. For compound form, isolate the variable in the middle by performing the same operation on all three parts. For union form, solve each branch independently. Direction flips once in each branch if the linear coefficient is negative.

The implementation. Internally, the solver collects linear terms inside the absolute value to extract coefficient

a

and constant

b

. It then performs the case split symbolically, solves each sub-inequality, and assembles the final answer in the appropriate compound, union, point, or special-case form. The final-answer card adapts to the matched branch.

For comprehensive treatment see solving absolute value inequalities and case analysis.

The stages are worked in worked examples with linear expressions and summarised in summary of absolute value inequality forms; the chain form of the answer is compound inequalities.

Related Concepts

Absolute Value — the distance from zero on the number line. Defined piecewise as

x

if

x \geq 0

and

-x

if

x < 0

. The foundational concept for any absolute value inequality.

Absolute Value Equation — the equality counterpart

|f(x)| = c

. Solves by splitting into two cases (

f(x) = c

and

f(x) = -c

) rather than the compound or union form of the inequality.

Compound Inequality — a chain like

-c < A < c

that captures a bounded region. Arises naturally from less-than absolute value inequalities.

Union of Inequalities — two inequalities joined by "or", producing a disconnected solution set. Arises naturally from greater-than absolute value inequalities.

Interval Notation — the standard way to write the solution. Parentheses for strict bounds, brackets for non-strict, the union symbol

\cup

for disconnected pieces.

Linear Inequality — the inequality type that the post-case-split sub-inequalities are. The solver routes each branch through standard linear-inequality logic.

Distance on the Number Line — the geometric interpretation.

|x - a| < c

asks for points within distance

c

of

a

;

|x - a| > c

asks for points farther than

c

from

a

. This perspective makes absolute value inequalities intuitive.

Triangle Inequality — the fundamental relation

|x + y| \leq |x| + |y|

. The cornerstone result connecting absolute value to addition; appears throughout analysis.

Piecewise Function — the formal way to define the absolute value function. The absolute value is the canonical example, splitting at

x = 0

.

Quadratic Inequality — squaring both sides of

|f(x)| > c

gives

f(x)^2 > c^2

, a quadratic inequality with the same solution set when

c \geq 0

. An alternative solving path for some problems, though usually less direct than the case split.

Modulus — the generalization to complex numbers.

|a + bi| = \sqrt{a^2 + b^2}

. Real absolute value is the special case where the imaginary part is zero.

Number Line Graphing — the geometric representation of the solution. A compound solution is a single shaded segment; a union solution is two shaded rays extending outward.

The related tools are the absolute value equation solver, the linear inequality solver and the quadratic inequality solver; the lessons are absolute value inequalities with quadratic or other expressions, inequalities with absolute value on both sides and interval notation.