Linear Inequation: Linear inequality is an inequality which involves a linear function and it contains one of the symbols of inequality.

  • \displaystyle < is less than
  • \displaystyle > is greater than
  • \displaystyle \leq is less than or equal to
  • \displaystyle \geq is greater than or equal to
  • \displaystyle \neq is not equal to

A linear inequality actually looks exactly like a linear equation \displaystyle (y = mx + c) , with the inequality sign replacing the equality sign \displaystyle (e.g. y > mx + c) .

Two-dimensional linear inequalities are expressions in 2 variables of the form: \displaystyle ax + by < c\text{ or  } ax + by \le c where the inequalities may either be strict or not.

A statement of any of the following forms: \displaystyle i) \hspace{0.5cm} ax + b > 0 \ \ ii) \hspace{0.5cm} ax + b < 0 \ \ iii) \hspace{0.5cm} ax + b \ge 0 \text{ or } \ \ iv) \hspace{0.5cm} ax + b \le 0 where \displaystyle a \text{ and  }  b are real numbers  and  is called a Linear Inequation in \displaystyle x .

Replacement Set or Universal Set: This is a set which contains all the values of the variable \displaystyle x , which would satisfy the inequation.

Solution Set: This is a subset of the replacement set which satisfy the given inequation.

Properties of Inequalities

Addition and Subtraction Property: A common constant c may be added to or subtracted from both sides of an inequality. For any real numbers

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

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

Multiplication and Division Property: The property states that for any real numbers, \displaystyle a, b\text{ and  non-zero }  c:

If \displaystyle c \text{ is positive } , then  multiplying or dividing by c does not change the inequality:

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

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

If \displaystyle c   is negative , then multiplying or dividing by \displaystyle c inverts the inequality:

\displaystyle \displaystyle \text{If } a \ge b\text{ and  } c < 0\text{ , then  } ac \le bc\text{ and  } \frac{a}{c} \le \frac{b}{c}

\displaystyle \displaystyle \text{If } a \le b\text{ and  } c < 0\text{ , then  } ac \ge bc\text{ and  } \frac{a}{c} \ge \frac{b}{c}

Transitive property of inequality states that for any real numbers \displaystyle a, b, c :

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

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

Converse Property: The relations ≤\text{ and  } ≥ are each other’s converse. For any real numbers \displaystyle a\text{ and  } b:

\displaystyle \text{If } a \le b\text{ , then  } b \ge a

\displaystyle \text{If } a \ge b\text{ , then  } b \le a

Additive inverse: The properties for the additive inverse state: For any real numbers \displaystyle a\text{ and  } b , negation inverts the inequality:

\displaystyle \text{If } a \le b\text{ , then  } -a \ge -b

\displaystyle \text{If } a \ge b\text{ , then  } -a \le -b

Multiplicative Inverse: The properties for the multiplicative inverse state:

For any non-zero real numbers \displaystyle a\text{ and  } b that are both positive or both negative:

\displaystyle \text{If } a\le b\text{ , then  } \frac{1}{a} \ge \frac{1}{b}

\displaystyle \text{If } a\ge b\text{ , then  } \frac{1}{a} \le \frac{1}{b}

If anyone of \displaystyle a\text{ and  } b is positive  and  the other is negative, then:

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

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

For any non-zero real numbers \displaystyle a\text{ and  } b :

\displaystyle \text{If } 0 < a\le b\text{ , then  } \frac{1}{a} \ge \frac{1}{b} > 0

\displaystyle \text{If } a \le b < 0\text{ , then  } 0 > \frac{1}{a} \ge \frac{1}{b}

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

\displaystyle \text{If }  0 > a \ge b\text{ , then  } \frac{1}{a} \le \frac{1}{b} < 0

\displaystyle \text{If } a \ge b > 0\text{ , then  } 0 < \frac{1}{a} \le \frac{1}{b}

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