Linear Inequalities

Key Terms

Definition and Standard Form

Inequality Notation

Solving with the Properties of Inequality

Worked Examples

Graphing the Solution

Compound Linear Inequalities

Inequalities with Fractions and Decimals

Special Cases

Literal Inequalities

Summary of Linear Inequality Forms

Linear Inequalities FAQ

One Direction, One Boundary, One Ray

A linear inequality in one variable is solved by the same operations that solve a linear equation — with one additional rule. Dividing or multiplying by a negative number flips the direction of the comparison. That single rule is the only point of divergence, and once it is internalized, every first-degree inequality reduces to a statement of the form

x < c

x > c

, whose solution is a ray on the number line. The boundary point is always the solution of the corresponding equation, and the inequality determines which side of it to shade.

Key Terms

Core Concepts

Linear Inequality—

ax + b < 0

with

a \neq 0

; solution is always a ray

Inequality— direction reverses when multiplying or dividing by a negative

Interval Notation— open parenthesis for strict, closed bracket for non-strict

Extensions

Compound Inequality— AND produces a bounded interval, OR produces two rays

See All Algebra Definitions →

Definition and Standard Form

A linear inequality in one variable is any inequality that can be written as

ax + b < 0

(or with

>

\leq

, or

\geq

in place of

<

), where

a

and

b

are real constants and

a \neq 0

. The variable

x

appears only to the first power — no squares, no roots, no

x

in a denominator.

The solution is always a ray: a half-line beginning at one point and extending to infinity in one direction. The boundary point is

x = -\frac{b}{a}

, the solution of the corresponding linear equation

ax + b = 0

. The inequality symbol determines which side of this boundary the solution occupies and whether the boundary itself is included.

Not every linear inequality arrives in standard form. The inequality

3x - 7 > 2x + 5

is linear because, after collecting terms, it reduces to

x - 12 > 0

. Recognition is the same as for equations: if the variable appears only to the first power with no compositions, the inequality is linear regardless of its initial presentation.

Inequality Notation

Notation

Inequality Notation

What the little bar under the sign promises, which chains are legal, and the bracket rules every interval answer obeys.

Interval glyphs and their set-builder translations — expressing domains;

\cup

for joined rays — set operations.

<

\leq

>

\geq

Less than; less than or equal to; greater than; greater than or equal to

Two strict marks and their two closed cousins — the bar underneath is a one-stroke “or equal”. The open end always faces the larger side, so

a < b

and

b > a

are the same statement written from opposite ends.

CasesStrict versus inclusive decides everything downstream: whether the boundary point belongs, whether the dot is open or solid, whether the interval bracket is round or square — the full correspondence runs through Graphing the Solution below.

Also written

\leqslant

and

\geqslant

— the slanted-bar forms, standard in British, French, and Russian typesetting; identical in meaning.

Do not confuse

\neq

. “Not equal” refuses equality but says nothing about order — it is not a fifth member of this family, and it cannot be graphed as a ray.

-3 < 2x + 1 \leq 7

Two x plus one is greater than minus three and at most seven

A three-part chain is a compressed AND: both outer conditions hold at once, and every operation must be applied to all three parts, as in Compound Linear Inequalities below.

CasesLegal chains point one way only — both signs

<

\leq

, or both

>

\geq

. A mixed chain like

5 > x < 8

is malformed: it compresses no single statement, and every style guide bans it.

Also writtenThe uncompressed pair —

-3 < 2x+1

and

2x+1 \leq 7

— which is also the only legal way to write an OR: disjunctions never chain.

Do not confuseA chain of separate comparisons read pairwise. The chain asserts simultaneity — it is one condition on one number, not a running commentary.

(-\infty, 9]

From minus infinity to nine, inclusive

Solution rays and intervals obey two bracket rules: square means the endpoint belongs, round means it does not — and

\infty

always takes a round bracket, because an unreachable end cannot belong to anything.

CasesStrict inequality → round, inclusive → square, matching the marks above. Two rays joined by OR write their union:

(-\infty, -4) \cup (2, \infty)

Also writtenReversed brackets —

\left]-2, 3\right]

— the Bourbaki convention, standard in French and some European texts, where an outward-turned bracket marks the excluded end.

Do not confuseThe coordinate pair

(4, \infty)

versus the point

(4, 7)

— identical glyphs. Context decides, which is exactly why the Bourbaki school switched to reversed brackets.

All algebra symbols →↑ Inequalities

Solving with the Properties of Inequality

Two properties govern the manipulation of linear inequalities, directly paralleling the properties used for equations.

The addition property states that adding or subtracting the same quantity on both sides preserves the direction of the inequality. If

x - 5 > 3

, then adding

5

to both sides gives

x > 8

. The direction

>

is unchanged.

The multiplication property has two cases. Multiplying or dividing by a positive number preserves the direction: from

3x \leq 21

, dividing by

3

gives

x \leq 7

. Multiplying or dividing by a negative number reverses the direction: from

-2x \leq 8

, dividing by

-2

gives

x \geq -4

. The inequality flips from

\leq

\geq

.

The reversal is the entire difference between solving equations and solving inequalities. Every other step — collecting terms, distributing, clearing fractions — is mechanically identical. Missing the flip when dividing by a negative is the single most common error, and it produces the exact complement of the correct solution: the wrong half of the number line.

Operation on both sides	Direction	Note
add or subtract any real number	preserved	always safe; behaves exactly like equations
multiply or divide by a positive constant	preserved	behaves like equations
multiply or divide by a negative constant	reversed	the defining rule: 3 < 5 becomes −3 > −5

Worked Examples

A single-step inequality requires one operation. The inequality

3x > 12

is solved by dividing both sides by

3

(positive, no flip):

x > 4

. The solution in interval notation is

(4, \infty)

.

A multi-step inequality demands the same sequence as a multi-step equation. Consider

4x - 7 \leq 2x + 11

. Subtract

2x

from both sides:

2x - 7 \leq 11

. Add

7

2x \leq 18

. Divide by

2

x \leq 9

. The solution is

(-\infty, 9]

.

An inequality requiring a direction flip:

-2x + 5 > 13

. Subtract

5

-2x > 8

. Divide by

-2

and reverse the inequality:

x < -4

. The solution is

(-\infty, -4)

.

A more involved example:

5 - 3(2x + 1) \geq 4x - 8

. Distribute:

5 - 6x - 3 \geq 4x - 8

. Simplify the left:

2 - 6x \geq 4x - 8

. Add

6x

2 \geq 10x - 8

. Add

8

10 \geq 10x

. Divide by

10

1 \geq x

, or equivalently

x \leq 1

. The solution is

(-\infty, 1]

Step	Action	Example
1	clear parentheses (distribute)	3(x − 2) < 9 → 3x − 6 < 9
2	clear fractions or decimals (multiply by LCD or 10ⁿ — must be positive)	x/3 + 1 < 5 → ×3 → x + 3 < 15
3	gather variable terms on one side	4x − 7 ≤ 2x + 11 → 2x − 7 ≤ 11
4	gather constants on the other side	2x − 7 ≤ 11 → 2x ≤ 18
5	isolate the variable — flip direction if dividing by a negative	2x ≤ 18 → x ≤ 9 \| −2x > 8 → x < −4

Graphing the Solution

The solution to a linear inequality is represented on the number line by marking the boundary point and shading the appropriate direction.

For

x > 4

: place an open dot at

4

(excluded because the inequality is strict) and shade to the right. The interval is

(4, \infty)

.

For

x \leq 9

: place a solid dot at

9

(included because the inequality is non-strict) and shade to the left. The interval is

(-\infty, 9]

.

The boundary point is always the solution of the corresponding equation. The inequality

4x - 7 \leq 2x + 11

has the boundary at

x = 9

, the solution of

4x - 7 = 2x + 11

. The inequality symbol tells which side to shade, and whether the dot is solid or open.

This connection is not a coincidence. The equation

ax + b = 0

identifies the single value where the two sides of the inequality are equal. On one side of that value,

ax + b

is positive; on the other, it is negative. The inequality selects the appropriate side.

Compound Linear Inequalities

A compound inequality joins two linear inequalities into a single condition. The conjunction (AND) and disjunction (OR) produce fundamentally different solution sets.

A three-part chain like

-3 < 2x + 1 \leq 7

is a conjunction: both

-3 < 2x + 1

and

2x + 1 \leq 7

must hold. Operations are applied to all three parts simultaneously. Subtract

1

-4 < 2x \leq 6

. Divide by

2

-2 < x \leq 3

. The solution is the half-open interval

(-2, 3]

.

A disjunction like

x + 3 < -1

x + 3 > 5

requires solving each part independently. From the first:

x < -4

. From the second:

x > 2

. The solution is the union

(-\infty, -4) \cup (2, \infty)

— two separate rays with a gap between them.

The choice between AND and OR depends on the problem's structure. Conjunctions narrow the solution set by requiring both conditions; the result is the intersection of two rays, which is either a bounded interval or empty. Disjunctions broaden it by accepting either condition; the result is the union of two rays, which is either two separate rays or the entire line.

Inequalities with Fractions and Decimals

Fractional coefficients are handled exactly as in linear equations: multiply every term on both sides by the least common denominator to clear all fractions. The critical requirement is that the LCD must be positive, which it always is when the denominators are numerical constants.

Consider

\frac{x}{3} - \frac{x + 2}{4} > 1

. The LCD of

3

and

4

12

. Multiplying every term by

12

4x - 3(x + 2) > 12

Distributing and collecting:

4x - 3x - 6 > 12

x - 6 > 12

x > 18

The solution is

(18, \infty)

. The multiplication by

12

is safe because

12 > 0

, so the direction is preserved.

Decimal coefficients are cleared in the same way, by multiplying by the appropriate power of

10

. The inequality

0.4x - 1.5 \leq 0.1x + 3

becomes

4x - 15 \leq x + 30

after multiplying by

10

, which simplifies to

3x \leq 45

and then

x \leq 15

.

If the variable appeared in a denominator, the inequality would no longer be linear — it would be a rational inequality, requiring sign analysis rather than simple clearing.

Special Cases

When the variable terms cancel during simplification, the inequality reduces to a comparison between two constants. The result is either universally true or universally false.

An always-true inequality arises when the remaining statement is valid. The inequality

2(x + 4) > 2x + 3

distributes to

2x + 8 > 2x + 3

. Subtracting

2x

gives

8 > 3

, which holds regardless of

x

. Every real number satisfies the original inequality, and the solution set is

(-\infty, \infty)

.

A never-true inequality arises when the remaining statement is false. The inequality

5(x - 1) \leq 5x - 9

distributes to

5x - 5 \leq 5x - 9

. Subtracting

5x

gives

-5 \leq -9

, which fails for every

x

. The solution set is empty — no real number satisfies the inequality.

These cases parallel the identity and contradiction outcomes in linear equations. The mechanism is the same: the variable disappears, and the truth or falsity of the resulting constant statement determines whether the solution set is everything or nothing.

Outcome	When it occurs	Example	Solution set
Always true	variable terms cancel; remaining constant statement is true	2(x + 4) > 2x + 3 → 8 > 3	(−∞, ∞) — all real numbers
Never true	variable terms cancel; remaining constant statement is false	5(x − 1) ≤ 5x − 9 → −5 ≤ −9	∅ — empty

Literal Inequalities

A literal inequality contains multiple variables, and the task is to isolate one variable in terms of the others. The procedure follows the same rules as numerical inequalities, but with an added complication: when dividing by a variable, its sign may not be known.

Consider solving

ax + b > c

for

x

. Subtracting

b

gives

ax > c - b

. To divide by

a

, the sign of

a

matters. If

a > 0

, the direction is preserved:

x > \frac{c - b}{a}

. If

a < 0

, the direction reverses:

x < \frac{c - b}{a}

. Both cases must be stated, because without knowing

a

, neither can be discarded.

This case-splitting does not arise in numerical inequalities, where the coefficient's sign is always visible. In literal inequalities, the answer is often conditional: the solution takes one form when a parameter is positive and another when it is negative. Stating both cases explicitly is not optional — omitting one produces an answer that is wrong half the time.

The same caution applies to multiplying by a variable expression whose sign is unknown. This is why, in more complex inequalities, sign analysis replaces direct manipulation: it avoids the need to know the sign of any expression in advance.

Summary of Linear Inequality Forms

All linear inequalities reduce to the same standard multi-step procedure once a single preparation step is applied — and a flip when dividing by a negative. The table below collects each form, an example, the preparation that converts it to a clean multi-step inequality, and the technique that finishes it.

Form	Example	Preparation step	Then solve as
Single-step	3x > 12	none	apply one property of inequality
Multi-step	4x − 7 ≤ 2x + 11	none (already simplified)	gather, then isolate
With parentheses	5 − 3(2x + 1) ≥ 4x − 8	distribute	multi-step
With fractions	x/3 − (x + 2)/4 > 1	multiply every term by the LCD (positive — no flip)	multi-step
With decimals	0.4x − 1.5 ≤ 0.1x + 3	multiply by 10ⁿ to clear decimals	multi-step
Literal	ax + b > c, solve for x	divide by a; case-split on sign of a (preserve if a > 0, reverse if a < 0)	multi-step + cases

Linear Inequalities FAQ

Is $5 > x < 8$ a valid way to write an inequality?

No, it is malformed. A three-part chain compresses a single AND, so both signs must point the same way:

-3 < 2x + 1 \leq 7

is legal, a mixed chain is not. An OR cannot be chained at all, whatever the direction, and has to be written as two separate statements joined by the word or.Read more →

Does $(4, 7)$ mean an interval or a point?

Context alone decides, since the glyphs are identical. On a number line

(4, 7)

is the open interval between

4

and

7

; in the plane it is a single coordinate pair. Note that

\infty

always takes a round bracket, as in

(4, \infty)

, because an endpoint you can never reach cannot belong to the set.Read more →

Is $\neq$ an inequality?

Not in this family. The four order marks

<

\leq

>

\geq

each rank two quantities, and the bar underneath is a one-stroke "or equal". By contrast

\neq

only refuses equality and says nothing about which side is larger, so it has no direction to solve for and cannot be graphed as a ray.Read more →

Linear Inequalities

One Direction, One Boundary, One Ray

Key Terms

Core Concepts

Extensions

Definition and Standard Form

Inequality Notation

Inequality Notation

Solving with the Properties of Inequality

Worked Examples

Graphing the Solution

Compound Linear Inequalities

Inequalities with Fractions and Decimals

Special Cases

Literal Inequalities

Summary of Linear Inequality Forms

Linear Inequalities FAQ

Is 5>x<85 > x < 85>x<8 a valid way to write an inequality?

Does (4,7)(4, 7)(4,7) mean an interval or a point?

Is ≠\neq= an inequality?

Is $5 > x < 8$ a valid way to write an inequality?

Does $(4, 7)$ mean an interval or a point?

Is $\neq$ an inequality?