Linear Inequalities: Solving & Graphing Step by Step

Definition and Standard Form

You can already solve a linear equation.

The equation

3x - 2 = 4

has exactly one answer. Add

2

to both sides, divide by

3

, and

x = 2

. The graph shows why there is only one: the line

y = 3x - 2

reaches the level

4

at a single point and never returns to it.

The line

y = 3x - 2

meets the level

n = 4

once, at

x = 2

: the one answer of

3x - 2 = 4

. The same state is on the equation visualizer.

The equation cannot say where the left side is bigger.

Change the question: for which

x

is

3x - 2

greater than

4

? The equation's method finds

x = 2

and stops there. Trying values shows that

2

is not the answer but the border between two kinds of value.

x	3x − 2	greater than 4?
0	−2	no
1	1	no
2	4	no — equal: the border
3	7	yes
4	10	yes

Every value above

2

works, every value below fails, and

2

itself fails because

4 > 4

is false. The answer is not one number. It is everything to the right of

2

, and no equation can write that down.

The solid part of the line is where

3x - 2

is above the level

4

— the "yes" rows of the table. The open dot at

x = 2

is the "equal" row: on the level, not above it.

The definition

A linear inequality in one variable is an inequality that can be written in the form below, with

<

or any of

>

,

\leq

,

\geq

, where

a

and

b

are real constants and

a \neq 0

.

Definition and Standard Form

ax + b < 0

The equation it widens: linear equations.

A linear inequality is a statement about x, and its answer is a set.

It is not a number and not a function. Like an equation, it is a sentence that is true for some values of

x

and false for others, and solving it means collecting the values that make it true. For

3x - 2 > 4

that collection is the ray

(2, \infty)

: a half-line that starts at the border and runs off to infinity. Every linear inequality has a ray for its answer.

The open dot at

2

is the border, left out because

4 > 4

is false. The arrow is "every

x

greater than

2

" from the table.

Every move that solves the equation still works.

Solve the equation and the inequality side by side and the steps are the same.

Step	Equation	Inequality
start	3x − 2 = 4	3x − 2 > 4
add 2	3x = 6	3x > 6
divide by 3	x = 2	x > 2
answer	one number, 2	a ray, (2, ∞)

The border is always the answer of the matching equation

ax + b = 0

, which is

x = -\frac{b}{a}

. In standard form our inequality is

3x - 6 > 0

, and

-\frac{-6}{3} = 2

. Telling a linear inequality apart also works as for equations:

3x - 7 > 2x + 5

is linear, because collecting terms turns it into

x - 12 > 0

.

One move changes: multiplying or dividing by a negative number reverses the sign.

Multiply both sides of

3x - 2 > 4

by

-1

. The values of

x

that work stay the same, but the statement now reads

-3x + 2 < -4

. Solving it, subtract

2

to get

-3x < -6

, then divide by

-3

: the sign flips back, and

x > 2

. Keep the sign instead and the result is

x < 2

, exactly the wrong half of the line. The other loss is the check: the answer is no longer one number to substitute. Test one value from each side of the border instead —

x = 3

gives

7 > 4

, true;

x = 0

gives

-2 > 4

, false.

The line

y = -3x + 2

falls, so it is below the level

-4

to the right of

2

: the same ray as before, written with

<

.

The rest of the page works with this one idea: the two properties of inequality and the flip, worked examples, graphing the ray, compound inequalities joined by AND and OR, fractions and decimals, the special cases where

a = 0

and there is no border at all, and literal inequalities.

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 open interval

(4, 7)

versus the point

(4, 7)

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

All algebra symbols →↑ Inequalities

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Notation · Cheat sheet

Polynomials, Equations and Inequalities: Notation at a Glance

The marks from the notation cards across the polynomial, equation and inequality pages: how each is read and what it means. Every group links to the full card on its page.

Polynomial roots

a_n x^n + \cdots + a_0

a-n x to the n, down to a-zero

Standard form; the subscript of each coefficient is its power of

x

.

\deg P = n

the degree of P is n

The highest power with a nonzero coefficient.

(x - r)^m

x minus r, to the multiplicity m

The root

r

counted

m

times.

Polynomial rules

P = (x - c)\,Q + R

quotient and remainder

Division as a product: quotient

Q(x)

, remainder

R = P(c)

.

P(c), \; P(-x)

P of c; P of minus x

Put in, not multiply: a value, and a new polynomial.

\pm\frac{p}{q}

plus or minus p over q

Rational root candidates:

p \mid a_0

,

q \mid a_n

.

r_1, \ldots, r_n

r sub one to r sub n

The roots, listed once per multiplicity.

Quadratic equations

ax^2 + bx + c = 0

standard form

Descending powers, everything on the left,

a \neq 0

.

\Delta = b^2 - 4ac

delta, the discriminant

Its sign tells how many real roots there are.

x_1, \; x_2

x one and x two

The two roots; the subscripts only count.

Absolute value equations

|x|

the absolute value of x

Distance of

x

from

0

; never negative.

|a + b|

absolute value of a plus b

The bars cover the whole quantity: add first, then take the size.

|x| = \sqrt{x^2}

the root of the square

A second definition, handy with radicals.

Inequalities

(a, b)

open interval from a to b

Endpoints left out.

[a, b]

closed interval from a to b

Endpoints included.

(a, \infty)

·

(-\infty, a]

rays

Infinity always takes a round bracket.

\cup

union

Joins two intervals into one solution set.

Linear inequalitiesThis page

<

·

\leq

·

>

·

\geq

less than … greater than or equal to

The bar under the sign adds equality.

-3 < 2x + 1 \leq 7

a compound inequality

Both conditions at once: an "and".

(-\infty, 9]

from minus infinity to nine, inclusive

The solution written as an interval.

Same marks, different job

|x|, \ |z|, \ |A|

Absolute value of a number, modulus of a complex number, determinant of a matrix: the same two bars.

(a, b)

An open interval here; a point in the plane in coordinate geometry.

p

in

\frac{p}{q}

Any divisor of

a_0

, not necessarily a prime.

Polynomial, equation and inequality symbols Algebra

Polynomials, Equations and Inequalities: Notation at a Glance·/algebra/polynomials/rootsLearn Math Class

Solving with the Properties of Inequality

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

RULEAddition property

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.

RULEMultiplication property

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

to

\geq

.

Pitfall: Flipping the sign for a negative number

Solving

3x > -12

as

x < -4

flips the sign because

-12

is negative. Only the sign of the divisor matters, and

3

is positive, so

x > -4

.

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 mirror image of the correct solution across the boundary point: 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

An inequality is solved like an equation, except that multiplying or dividing by a negative number reverses its direction.

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)

.

Pitfall: Keeping the sign when dividing by a negative

Dividing

-2x > 8

by

-2

without reversing gives

x > -4

, which includes

x = 0

, where

-2(0) + 5 = 5 > 13

is false. Reversing gives

x < -4

.

In that example the direction changes at exactly one step, the division by

-2

. The subtraction of

5

before it leaves the sign as it was, and so would any addition, subtraction, or multiplication by a positive number elsewhere in the working.

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]

.

To solve your own linear inequality with each step shown, use the linear inequality solver.

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.

Fact: No first number after 4

The solutions of

x > 4

have no smallest member: for any solution, the midpoint between it and

4

is a smaller one. The shading starts at a point that no solution reaches.

The boundary point is where the shading starts, and the one question there is whether the point itself belongs to the solution. A strict sign (

<

or

>

) leaves it out, and a non-strict sign (

\leq

or

\geq

) takes it in. The graph shows the difference with the dot at the boundary, and interval notation shows it with the bracket: round when the point is left out, square when it is taken in.

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]

.

RULE

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

Pitfall: Using an open dot for a non-strict sign

Drawing

x \leq 9

with an open dot, or writing

(-\infty, 9)

, leaves out

x = 9

, where both sides equal

29

and

29 \leq 29

holds. The answer is

(-\infty, 9]

.

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.

Where the boundary comes from

The section's 4x − 7 ≤ 2x + 11, with each side drawn as a line. They cross at x = 9, the solution of the equation 4x − 7 = 2x + 11: that is the boundary point. To the left of 9 the steeper line 4x − 7 is below the other, so the inequality holds there; the crossing itself counts because the inequality is non-strict. The solution is (−∞, 9].

The equation finds the boundary; one test value or the inequality symbol finds the 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.

Example

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]

.

Pitfall: Changing only the middle of a chain

Subtracting

1

from the middle alone turns

-3 < 2x + 1 \leq 7

into

-3 < 2x \leq 7

, so

-\frac{3}{2} < x \leq \frac{7}{2}

. All three parts change, which gives

-2 < x \leq 3

.

The chain is solved as one statement, with

x

in the middle part. Each step is chosen to undo something around

x

, and after the last step

x

stands alone between two numbers that are read off directly as the endpoints of the interval.

Example

A disjunction like

x + 3 < -1

or

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 a bounded interval or empty when the rays point in opposite directions, and a single ray when they point the same way. Disjunctions broaden it by accepting either condition; the result is the union of two rays, which is two separate rays or the entire line when the rays point in opposite directions, and a single ray when they point the same way.

AND versus OR on the number line

The section's two compound inequalities. The chain −3 < 2x + 1 ≤ 7 needs both parts at once, which leaves the overlap −2 < x ≤ 3: open at −2, closed at 3. The disjunction x + 3 < −1 or x + 3 > 5 needs only one part, which leaves two separate rays, x < −4 and x > 2, with the gap between them excluded.

An AND of two rays pointing the wrong way is empty; an OR of overlapping rays is the whole 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

is

12

. Multiplying every term by

12

:

Inequalities with Fractions and Decimals

4x - 3(x + 2) > 12

Distributing and collecting:

Inequalities with Fractions and Decimals

4x - 3x - 6 > 12

Inequalities with Fractions and Decimals

x - 6 > 12

Inequalities with Fractions and Decimals

x > 18

Pitfall: Distributing a minus to the first term only

Opening

4x - 3(x + 2)

as

4x - 3x + 6

gives

x + 6 > 12

, so

x > 6

. The

-3

multiplies both terms,

-3x - 6

, so

x - 6 > 12

and

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

.

Example

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.

Clearing numerical fractions or decimals multiplies by a positive number, so the direction of the inequality never changes.

Special Cases

COROLLARY

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.

Example

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.

Example

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

Once the variable cancels, the inequality is a plain true-or-false statement: its solution set is every real number or none.

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.

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Why the answer splits

Solving

ax + b > c

for

x

: subtracting

b

gives

ax > c - b

. To divide by

a

, its sign matters, and in a literal inequality that sign may not be known.

Literal inequality

ax > c - b

split by the sign of $a$

+

Case 1

a > 0

Direction preserved

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

−

Case 2

a < 0

Direction reverses

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

Takeaway

Both cases must be stated, because without knowing

a

neither can be discarded. Omitting one produces an answer that is wrong half the time.

Literal inequalities: the sign of the coefficient·/algebra/inequalities/linearLearn Math Class

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

Common Pitfalls

Five mistakes account for most wrong answers to linear inequalities. Each one is marked where it happens in the sections above; here each is worked through in full, with the wrong and the right working side by side.

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1Flipping the sign for a negative number

The division is by a positive coefficient, but a negative number elsewhere in the inequality, here on the right side, is taken as the signal to reverse the direction.

Start

3x > -12

Wrong turn

Error here

\implies x < -4

Right path

The divisor

3

is positive, so the direction is kept.

\implies x > \frac{-12}{3} = -4

The solution is

(-4, \infty)

.

Why it happensThe flip rule is remembered as "negative means flip", without the part that says which number has to be negative: the one both sides are multiplied or divided by.

How to checkTest one value from each answer.

x = 0

lies in the right answer and gives

0 > -12

, true.

x = -5

lies in the wrong answer and gives

-15 > -12

, false.

2Keeping the sign when dividing by a negative

Both sides are divided by a negative coefficient, and the inequality sign is copied unchanged, as it would be in an equation.

Start

-2x + 5 > 13

\implies -2x > 8

Wrong turn

Error here

\implies x > -4

Right path

Dividing by

-2

reverses the sign.

\implies x < -4

The solution is

(-\infty, -4)

.

Why it happensEvery other step of the working is the same as for an equation, so the one step that differs is done out of habit.

How to checkTest one value on each side of the boundary

-4

.

x = -5

gives

-2(-5) + 5 = 15 > 13

, true.

x = 0

gives

5 > 13

, false. The solutions lie to the left of

-4

.

3Using an open dot for a non-strict sign

The boundary of a non-strict inequality (

\leq

or

\geq

) is drawn as an open dot or written with a round bracket, so the point where both sides are equal drops out of the answer.

Wrong turn

Error here

x \leq 9

is graphed with an open dot at

9

and written as

(-\infty, 9)

.

Right path

The bar under

\leq

allows equality, so

9

belongs to the answer. The graph has a solid dot at

9

with shading to the left, and the interval is

(-\infty, 9]

.

Why it happensThe boundary is found by solving an equation, and the separate question of whether it belongs is skipped when the dot is drawn.

How to checkSubstitute the boundary into the original inequality.

x = 9

gives

4(9) - 7 = 29

and

2(9) + 11 = 29

, and

29 \leq 29

is true, so

9

is in the answer. With a strict sign,

29 < 29

would be false and the dot would be open.

4Changing only the middle of a chain

In a three-part chain, an operation is applied to the middle part, where

x

is, and the two outer bounds are left as they were.

Start

-3 < 2x + 1 \leq 7

Wrong turn

Error here

\implies -3 < 2x \leq 7

\implies -\frac{3}{2} < x \leq \frac{7}{2}

Right path

Subtract

1

from all three parts, then divide all three by

2

.

-4 < 2x \leq 6

\implies -2 < x \leq 3

The solution is

(-2, 3]

.

Why it happensThe attention is on isolating

x

, and the outer numbers look like fixed limits rather than sides of the inequality.

How to checkTest an endpoint of the answer.

x = \frac{7}{2}

gives

2\left(\frac{7}{2}\right) + 1 = 8

, and

8 \leq 7

is false. The right endpoint

x = 3

gives

7 \leq 7

, true.

5Distributing a minus to the first term only

After the fractions are cleared, the bracket in

-3(x + 2)

is opened and the second term keeps its plus sign, as if the

-3

reached only the

x

.

Start

4x - 3(x + 2) > 12

Wrong turn

Error here

\implies 4x - 3x + 6 > 12

\implies x > 6

Right path

The

-3

multiplies both terms, and

-3 \cdot 2 = -6

.

4x - 3x - 6 > 12

\implies x - 6 > 12

\implies x > 18

The solution is

(18, \infty)

.

Why it happensThe minus sign in front of

\frac{x + 2}{4}

covers the whole numerator, and once the fraction bar is gone that is no longer visible.

How to checkTest a value from the wrong answer that lies outside the right one.

x = 10

gives

\frac{10}{3} - \frac{12}{4} = \frac{1}{3}

, which is not greater than

1

. A value from the right answer,

x = 24

, gives

8 - \frac{26}{4} = \frac{3}{2} > 1

.

Common pitfalls in linear inequalities·/algebra/inequalities/linearLearn Math Class

Two of the five slips are about the direction of the sign: it reverses when both sides are multiplied or divided by a negative number, and only then. The other three each lose a piece: the sign of the second term in a bracket, the outer parts of a chain, or the endpoint of a non-strict answer. A test value catches every one of them: take a number from the wrong answer that is not in the right one, or substitute the boundary itself, and the original inequality gives the verdict.

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 →

How do you check the answer to an inequality?

+

Test one value from each side of the boundary in the original inequality, then test the boundary itself. For

3x - 2 > 4

,

x = 3

gives

7 > 4

, true, and

x = 0

gives

-2 > 4

, false, so the answer

(2, \infty)

lies on the correct side. At

x = 2

both sides equal

4

, and

4 > 4

is false, so the endpoint stays out.Read more →

Is there a smallest number greater than 4?

+

No. Whatever solution of

x > 4

you name, the midpoint between it and

4

is also greater than

4

and smaller than the one you named, so the solutions never reach a first member. That is why the answer is written with a round bracket,

(4, \infty)

, and drawn with an open dot: the shading starts at a point that no solution actually reaches.Read more →

Can a compound inequality have no solution?

+

Yes, an AND can. Its solution is the overlap of two rays, and two rays that point away from each other without meeting, such as

x < 1

and

x > 3

, have no overlap, so the set is empty. An OR goes the other way: two rays pointing in opposite directions that overlap, such as

x < 3

or

x > 1

, cover the whole number line.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

Common Pitfalls

Linear Inequalities FAQ

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

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

Is $\neq$ an inequality?

How do you check the answer to an inequality?

Is there a smallest number greater than 4?

Can a compound inequality have no solution?

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

Common Pitfalls

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?

How do you check the answer to an inequality?

Is there a smallest number greater than 4?

Can a compound inequality have no solution?

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

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

Is $\neq$ an inequality?