\displaystyle \textbf{Linear Inequation}

\displaystyle \text{A linear inequation or linear inequality is an inequality involving a linear}
\displaystyle \text{expression. It contains one of the symbols }<,\ >,\ \leq,\text{ or }\geq.

\displaystyle <\quad\text{means ``less than''}
\displaystyle >\quad\text{means ``greater than''}
\displaystyle \leq\quad\text{means ``less than or equal to''}
\displaystyle \geq\quad\text{means ``greater than or equal to''}
\displaystyle \neq\quad\text{means ``not equal to''}

\displaystyle \text{Note: The symbol }\neq\text{ represents inequality, but an expression such as}
\displaystyle ax+b\neq0\text{ is generally not called a linear inequation in the usual school}
\displaystyle \text{classification of linear inequations.}

\displaystyle \text{For example, }2x+3<7,\quad 5x-4\geq11,\quad x+2y\leq6\text{ are linear inequations.}

\displaystyle \textbf{Linear Inequation in One Variable}

\displaystyle \text{A linear inequation in one variable }x\text{ can be written in forms such as}

\displaystyle ax+b>0,\qquad ax+b<0,\qquad ax+b\geq0,\qquad ax+b\leq0,

\displaystyle \text{where }a,b\text{ are real numbers and }a\neq0.

\displaystyle \text{For example, }3x-5>7\text{ is a linear inequation in one variable.}

\displaystyle \text{Solving an inequation means finding all values of the variable that make the}
\displaystyle \text{inequality true. Unlike a linear equation, an inequation generally has many}
\displaystyle \text{solutions rather than a single solution.}

\displaystyle \text{For example, }x+2<5\Rightarrow x<3.

\displaystyle \text{Thus every real number less than }3\text{ is a solution of the inequation.}

\displaystyle \textbf{Linear Inequation in Two Variables}

\displaystyle \text{A linear inequation in two variables }x\text{ and }y\text{ has forms such as}

\displaystyle ax+by<c,\qquad ax+by\leq c,\qquad ax+by>c,\qquad ax+by\geq c,

\displaystyle \text{where }a,b,c\text{ are real numbers and }a\text{ and }b\text{ are not both zero.}

\displaystyle \text{For example, }y>mx+c\text{ is a linear inequation in two variables.}

\displaystyle \text{The corresponding equation }y=mx+c\text{ represents the boundary line.}
\displaystyle \text{The solutions of the inequation form a region or half-plane on one side}
\displaystyle \text{of this boundary line.}

\displaystyle \text{For }<\text{ or }>,\text{ the boundary line is not included in the solution set.}
\displaystyle \text{For }\leq\text{ or }\geq,\text{ the boundary line is included in the solution set.}

\displaystyle \textbf{Replacement Set or Universal Set}

\displaystyle \text{The replacement set or universal set is the set from which the possible values}
\displaystyle \text{of the variable are to be chosen.}

\displaystyle \text{For example, consider }x+2<7\text{ with the replacement set}
\displaystyle U=\{1,2,3,4,5,6,7\}.

\displaystyle \textbf{Solution Set}

\displaystyle \text{The solution set is the set of all values from the replacement set that satisfy}
\displaystyle \text{the given inequation. Therefore, the solution set is a subset of the}
\displaystyle \text{replacement set.}

\displaystyle x+2<7\Rightarrow x<5.

\displaystyle \text{For }U=\{1,2,3,4,5,6,7\},\text{ the solution set is}
\displaystyle S=\{1,2,3,4\}.

\displaystyle \text{If the universal set is }\mathbb{R},\text{ then the solution set is }x<5.

\displaystyle \textbf{Properties of Inequalities}

\displaystyle \textbf{1. Addition and Subtraction Property}

\displaystyle \text{Adding or subtracting the same real number on both sides of an inequality}
\displaystyle \text{does not change the direction of the inequality.}

\displaystyle \text{If }a\leq b,\text{ then }a+c\leq b+c \text{ and }a-c\leq b-c.

\displaystyle \text{If }a\geq b,\text{ then }a+c\geq b+c \text{ and }a-c\geq b-c.

\displaystyle \text{Example: If }x-4<7,\text{ then }x<11.

\displaystyle \textbf{2. Multiplication and Division Property}

\displaystyle \text{Multiplying or dividing both sides of an inequality by a positive number does}
\displaystyle \text{not change the direction of the inequality.}

\displaystyle \text{If }a\geq b\text{ and }c>0,\text{ then } ac\geq bc\text{ and }\frac{a}{c}\geq\frac{b}{c}.

\displaystyle \text{If }a\leq b\text{ and }c>0,\text{ then } ac\leq bc\text{ and }\frac{a}{c}\leq\frac{b}{c}.

\displaystyle \text{Multiplying or dividing both sides by a negative number reverses the}
\displaystyle \text{direction of the inequality.}

\displaystyle \text{If }a\geq b\text{ and }c<0,\text{ then } ac\leq bc\text{ and }\frac{a}{c}\leq\frac{b}{c}.

\displaystyle \text{If }a\leq b\text{ and }c<0,\text{ then } ac\geq bc\text{ and }\frac{a}{c}\geq\frac{b}{c}.

\displaystyle \text{Example: }-2x<8.

\displaystyle \text{Dividing both sides by }-2\text{ reverses the inequality sign.}

\displaystyle x>-4.

\displaystyle \text{This reversal of the inequality sign is one of the most important rules when}
\displaystyle \text{solving inequations.}

\displaystyle \textbf{3. Transitive Property}

\displaystyle \text{For any real numbers }a,b,c:

\displaystyle \text{If }a\geq b\text{ and }b\geq c,\text{ then }a\geq c.

\displaystyle \text{If }a\leq b\text{ and }b\leq c,\text{ then }a\leq c.

\displaystyle \text{Similarly, if }a>b\text{ and }b>c,\text{ then }a>c.

\displaystyle \text{If }a<b\text{ and }b<c,\text{ then }a<c.

\displaystyle \textbf{4. Reversal or Converse Property}

\displaystyle \text{The order of an inequality may be reversed by interchanging its two sides and}
\displaystyle \text{reversing the inequality symbol.}

\displaystyle \text{If }a\leq b,\text{ then }b\geq a.

\displaystyle \text{If }a\geq b,\text{ then }b\leq a.

\displaystyle \text{Similarly, }a<b\iff b>a\text{ and }a>b\iff b<a.

\displaystyle \textbf{5. Additive Inverse Property}

\displaystyle \text{Taking the additive inverse of both sides reverses the inequality.}

\displaystyle \text{If }a\leq b,\text{ then }-a\geq-b.

\displaystyle \text{If }a\geq b,\text{ then }-a\leq-b.

\displaystyle \text{For example, if }3<7,\text{ then }-3>-7.

\displaystyle \textbf{6. Multiplicative Inverse or Reciprocal Property}

\displaystyle \text{The reciprocal property must be used carefully because neither number may}
\displaystyle \text{be zero and the signs of the numbers must be considered.}

\displaystyle \text{If }0<a\leq b,\text{ then } 0<\frac{1}{b}\leq\frac{1}{a}.

\displaystyle \text{Equivalently, if }0<a<b,\text{ then } \frac{1}{a}>\frac{1}{b}>0.

\displaystyle \text{If }a\leq b<0,\text{ then } \frac{1}{a}\geq\frac{1}{b}.

\displaystyle \text{For example, }-5<-2,\text{ but } -\frac{1}{5}>-\frac{1}{2}.

\displaystyle \text{Thus, when two non-zero numbers have the same sign, taking reciprocals}
\displaystyle \text{reverses their order.}

\displaystyle \text{If the two numbers have opposite signs, their reciprocals retain the same}
\displaystyle \text{order because taking a reciprocal does not change the sign of a number.}

\displaystyle \text{If }a<0<b,\text{ then } \frac{1}{a}<0<\frac{1}{b}.

\displaystyle \text{If }a>0>b,\text{ then } \frac{1}{a}>0>\frac{1}{b}.

\displaystyle \textbf{Representation of Solutions on a Number Line}

\displaystyle \text{The solution of a linear inequation in one variable can be represented on a}
\displaystyle \text{number line.}

\displaystyle x>a:\quad\text{all numbers greater than }a.

\displaystyle x<a:\quad\text{all numbers less than }a.

\displaystyle x\geq a:\quad\text{all numbers greater than or equal to }a.

\displaystyle x\leq a:\quad\text{all numbers less than or equal to }a.

\displaystyle \text{An open circle is used at }a\text{ for }x>a\text{ or }x<a \text{ because }a\text{ is excluded.}

\displaystyle \text{A closed or filled circle is used at }a\text{ for }x\geq a\text{ or }x\leq a \text{ because }a\text{ is included.}

\displaystyle \textbf{Important Rule to Remember}

\displaystyle \text{When solving a linear inequation, addition and subtraction do not reverse the}
\displaystyle \text{inequality sign. Multiplication or division by a positive number also does not}
\displaystyle \text{reverse it. However, multiplication or division by a negative number always}
\displaystyle \text{reverses the inequality sign.}

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