Inequality Mathematical Definition: Symbols, Strict and Non-Strict, Examples

The runners give the picture: an order, known before any measuring. On a number line, $b$ sits to the right of $a$ - greater, though neither time was recorded.

Wentworth's 1898 New School Algebra gives inequality mathematical definition in one line: a statement in symbols that one of two quantities is greater than the other. Durell's 1911 School Algebra echoes it, naming the sides the first and second members. Olney's 1873 University Algebra frames it as an expression, in symbols, of the inequality of two numbers.

So the definition has three parts: two members, one symbol, a claim of order. Read the table as inequality mathematical definition, part by part.

Sameness belongs to equations instead - see the sister page what-is-an-equation-in-math.

| Symbol | Read aloud | Example | |---|---| | $>$ | is greater than | $7>3$ | | $<$ | is less than | $2<9$ | | $\geq$ | is greater than or equal to | $x\geq 1$ | | $\leq$ | is less than or equal to | $y\leq 3$ |

Durell lists the four signs of inequality mathematical definition with exactly these readings. Say $7>3$ the long way - seven is greater than three - and the symbol has done its job.

Wentworth adds the memory trick: the sign always points toward the lesser quantity. One glance at the small end therefore checks any symbol in inequality mathematical definition, whichever side it sits on.

The shapes in inequality mathematical definition are old. Durell & Robbins's 1897 A School Algebra Complete credits Harriot with inventing the signs in 1631 - the pointing rule is their oldest part.

Inequality mathematical definition splits by one question: is equality allowed?

A strict form uses only $>$ or $<$. On a number line, $x>2$ gets an open dot at 2 - the boundary itself is not a solution. An "or equal to" form is non-strict: $x\geq 2$ takes a closed dot, because 2 belongs to the answer.

The two halves, side by side:

| Statement | Boundary $2$ included? | Read as | |---|---| | $x>2$ | no, open dot | strictly greater than 2 | | $x\geq 2$ | yes, closed dot | 2 or anything greater |

Which half of inequality mathematical definition a statement occupies is decided by its symbol alone.

With a letter inside, an inequality becomes an open sentence, true or false only once a number is chosen. The values that make inequality mathematical definition true form its solution set, worked through on what-is-the-solution-to-the-inequality.

Search boxes see "inequality equations" all the time - usually just a name for inequalities. The mixing is understandable: both statements share a shape - two members, one sign, the outline of inequality mathematical definition.

But Chrystal's 1904 Algebra draws the line. Turn $7=3+4$ around and it survives; turn $7>3$ into $3>7$ and the claim dies. Inequality mathematical definition is directional by nature, while an equation reports sameness either way.

| Move | Equation $7=3+4$ | Inequality $7>3$ | |---|---| | swap the two members | still true, $3+4=7$ | false, $3>7$ fails | | what holds | balance, either way | order, one way |

So treat "inequality equations" as a doorway, not a verdict. The no-interchange rule of inequality mathematical definition is what it meets. Solving is another craft, kept on how-to-solve-inequalities.

Adding or subtracting the same number on both members keeps the sense - Wentworth turns $7>3$ into $4>0$ by subtracting 4.

Multiplying is where inequality mathematical definition demands care: a positive multiplier keeps the sense, a negative one reverses it. In letters: if $a<b$ and $c<0$, then $ac>bc$.

$2<5\quad\xrightarrow{\times(-1)}\quad-2>-5$

Why flip? Multiplying by $-1$ reflects both numbers through zero, swapping their sides. Inequality mathematical definition is a claim about position on the number line, so any move that repositions the members is checked against that claim.

Read the inequality mathematical definition under two multipliers:

| Multiplier | Effect on the sense | Example from $2<5$ | |---|---| | $+3$ | unchanged | $6<15$ | | $-1$ | reversed | $-2>-5$ |

One flip per negative multiplier - forget it, and the statement lies.

Inequality mathematical definition also extends to compound form: two inequalities joined by the word "and" or "or", such as $x>-3$ and $x\leq 2$.

The connective decides the answer set; each piece stays a full case of inequality mathematical definition. With "and", keep only numbers that make both parts true - the intersection. With "or", gather every number that makes either part true - the union.

What Does the Word Disjunction Mean?

In logic, a disjunction is an "or" statement, true when at least one piece is true; a conjunction, an "and" statement, true only when all pieces are. Both names attach to inequality mathematical definition used twice in one statement: $x<-1$ or $x>3$ solves by union, while $x>-1$ and $x\leq 2$ solves by intersection.

| Connective | Logic name | Answer set | Example | |---|---| | or | disjunction | union of the parts | $x<-1$ or $x>3$ | | and | conjunction | intersection of the parts | $x>-1$ and $x\leq 2$ |

Problem. A movie theater admits anyone aged 13 or older; a soccer club takes players under 15. Write each rule with an inequality symbol, read it aloud - two everyday jobs for inequality mathematical definition.

Step 1, the theater. "13 or older" allows exactly 13, so inequality mathematical definition puts this rule in its non-strict half: $\text{Age}\geq 13$, read "age is greater than or equal to 13".

Step 2, the club. "Under 15" bars 15 itself, so the symbol stays strict: $\text{Age}<15$, read "age is less than 15".

Step 3, check the pointing habit. In both lines the small end faces the smaller side - the signs sit where the definition puts them.

What decided everything here? Inequality mathematical definition: it chose which half allows 13, which one bars 15. The readings here are filled in Durell's way.

Problem. Solve $-2x<8$ and state the boundary's status.

Step 1, isolate $x$. Divide both members by $-2$. The multiplier is negative, so inequality mathematical definition forces a reversal: $x>-4$.

Step 2, test the flip. Try $x=0$: the original reads $-2\times0<8$, which is $0<8$ - true, and 0 sits inside $x>-4$. Try $x=-5$: the original gives $10<8$, false - and $-5$ sits outside, agreeing with what inequality mathematical definition predicted.

Step 3, the boundary. At $-4$ itself: $-2\times(-4)=8$, and $8<8$ fails, so the dot stays open - the statement was strict all along. Boundary checks are part of inequality mathematical definition's discipline.

The same flip discipline governs every one-step statement like this one.

1. Treating $\geq$ as strictly greater. The "or equal" half is real: $x\geq 2$ admits 2, and the closed dot says so. Fix: read the symbol aloud, both halves, every time.

2. Keeping the sign after a negative multiply. From $2<5$, multiplying by $-1$ must give $-2>-5$; keeping $<$ turns a true claim false. Fix: under inequality mathematical definition, flip exactly once per negative multiplier.

3. Swapping members like an equation. Chrystal's warning applies: the two sides cannot be interchanged, because inequality mathematical definition claims a direction. $7>3$ reversed is simply wrong.

4. Erasing the boundary without checking strictness. Solving $3x<15$ into $x<5$ is safe; whether 5 belongs needs the original symbol. Fix: carry the strictness to the answer.

5. Reading an unsolved inequality as true or false. With a letter inside, it is an open sentence. Nothing in inequality mathematical definition promises a verdict before a number shows up - name the status, then test.