Mathematics and Statistics · Ch 17 — Linear Inequations

Inequations and the Rules for Solving Them

This Maharashtra Std XI (Commerce) Mathematics and Statistics chapter studies linear inequations — statements that compare two expressions using an inequality sign rather than an equals sign. Inequations model real commercial limits (a budget of at most a certain amount, a target of at least so many units), which makes them the language of the optimisation and feasible-region work that follows in Std XII. The chapter draws on the same standard, well-established treatment of linear inequalities used in mathematics curricula nationally.

What an inequation is

Inequation

An inequation is a statement that two expressions are unequal in a specified direction. It uses one of the four order symbols:

$<\ (\text{less than}),\quad >\ (\text{greater than}),\quad \le\ (\text{less than or equal to}),\quad \ge\ (\text{greater than or equal to}).$

An inequation is linear when each variable appears only to the first power (no $x^{2}$ , no $xy$ , no $1/x$ ). Examples: $3x-5<7$ (one variable), $2x+3y\le 6$ (two variables).

Strict signs $<,>$ exclude the boundary value; the weak signs $\le,\ge$ include it. That single distinction decides an open vs. closed dot on a number line, and a dashed vs. solid line in a graph.

The rules for solving

Solving an inequation means finding every value of the variable that makes it true — its solution set. The rules mirror those for equations, with one crucial exception.

Note

The three rules

Add or subtract the same number on both sides — the sign is unchanged: if $a<b$ then $a+c<b+c$ .
Multiply or divide by a positive number — the sign is unchanged: if $a<b$ and $c>0$ then $ac<bc$ .
Multiply or divide by a negative number — the inequality sign REVERSES: if $a<b$ and $c<0$ then $ac>bc$ .

Watch out

The sign flip is the number-one error in this chapter

Dividing $-2x>6$ by $-2$ gives $x<-3$ , not $x>-3$ — the $>$ must become $<$ the moment you divide by the negative $-2$ . A quick way to avoid the flip entirely is to move the variable term to whichever side keeps its coefficient positive.

Why the flip happens: on the number line, multiplying by a negative reflects every point about $0$ , so the left-to-right order of any two numbers is reversed (e.g. $2<3$ , but $-2>-3$ ).

Definition 1Inequation

A statement comparing two expressions with $<,\ >,\ \le$ or $\ge$ . It is linear when every variable appears only to the first power. Strict signs $<,>$ exclude the boundary; weak signs $\le,\ge$ include it.

Definition 2Solution set

The set of all values of the variable(s) that make the inequation true. For a one-variable linear inequation this is a ray (interval) on the number line; for a two-variable one it is a half-plane.

Definition 3Sign-reversal rule

Multiplying or dividing both sides of an inequation by a negative number reverses the inequality sign ( $<\leftrightarrow>$ , $\le\leftrightarrow\ge$ ). Adding, subtracting, or scaling by a positive number leaves the sign unchanged.

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