The notion of linear inequality in one unknown

Solving equations and inequalities: Linear inequalities in one unknown

The notion of linear inequality in one unknown

A mathematical expression with an inequality sign is called an inequality. The inequality signs are: \[\begin{array}{|c|c|c|c|} \hline \lt & \text{less than} & \le & \text{less than or equal to} \\ \hline \gt & \text{greater than} & \ge & \text{greater than or equal to}\\ \hline\end{array}\]

A linear inequality of unknown \(\boldsymbol{x}\) is an inequality that can be reduced, by elementary operations, to a basic form \(ax+b<0\), where \(a\) and \(b\) numbers, and instead of \(\lt\) there may also be one of the other inequality symbols.

An elementary operation is understood to be expansion of brackets, regrouping of subexpressions, addition or subtraction of equivalent expressions on both sides of the equation, or multiplication or division on both sides of the equation by a number distinct from zero.

The solution of a linear inequality with unknown \(x\) is in the form \(x<c\), \(x\le c\), \(x\ge c\) or \(x\gt c\), where \(c\) is a number.

Examples
\[\begin{aligned}x^2&<2x+3\\ &\phantom{wxyz} \blue{\text{quadratic inequality}} \\ 2x+1&\le 4x-3\\ &\phantom{wxyz} \blue{\text{linear inequality}}\\ x>-\tfrac{1}{2} &\Leftrightarrow -\tfrac{1}{2}<x\\ &\phantom{wxyz} \blue{\text{equivalent inequalities}}\end{aligned} \]

\(\quad\)The inequality \(2x+1\le 4x-3\)
\(\quad\)has solution \(x\ge 2\).
\(\quad\)This follows from the following
\(\quad\)reduction: \[\begin{aligned} 2x+1\le 4x-3 &\Leftrightarrow\\ 2x+1-4x\le 4x-3 -4x&\Leftrightarrow\\ -2x+1\le -3&\Leftrightarrow\\ -2x+1-1\le -3-1
&\Leftrightarrow\\ -2x\le -4
&\Leftrightarrow\\ 2x\ge 4
&\Leftrightarrow\\ x\ge 2\end{aligned}\]

Form of an inequality Schematically, an inequality look like \(L<R\), where \(L\) and \(R\) are mathematical expressions and instead of \(\lt\) there may also be one of the other inequality symbols.

Combining inequalities Sometimes two inequalities are combined into a single expression. For example, \(-1<x<1\) is a more compact notation for \(-1<x\) and \(x<1\), also denoted by the logical operator \(\wedge\) for 'and' as \(-1<x\;\wedge\;x<1\). Because \(-1<x\) is equivalent to \(x>-1\), the expression \(-1<x<1\) means ' \(x\) greater than \(-1\) and smaller than \(1\) '.

Wording To avoid confusion, we give the wording of mathematical symbols that occur in equations and inequalities. \[\begin{array}{|c|c|} \hline \textit{Symbols} & \textit{Wording} \\ \hline \lt, \le, \gt, \ge & \text{inequality symbol} \\ \hline = & \text{equals sign, equality sign} \\ \hline \neq & \text{unequal sign, not equals sign}\\ \hline \wedge, \vee & \text{logical operator ('and', 'or')} \\ \hline\end{array}\]

Absolute value The absolute value is sometimes used instead of a combination of inequalities. For example, the expression \(|x|<1\) means \(-1<x<1\). The inequality \(|x|>2\) means \(x>2\) or \(x<-2\), in mathematical notation also written with the logical operator \(\vee\) for 'or' as \(x>2\;\vee\;x<-2\).

Equivalence of linear inequalities

Two linear inequalities are called equivalent when they have the same solutions, because they can be converted into each other through elementary operations.
To indicate that two inequalities are equivalent, the symbol \(\Leftrightarrow\) can be used; for example \(4x<2\Leftrightarrow 2x<1\) and \(2x<1\Leftrightarrow x<\tfrac{1}{2}\).

If two inequalities can be reduced to the same basic form, then the two inequalities are equivalent.