Basic Concepts Connected with Solving Inequalities

Suppose we are given inequality `f(x)>g(x)`. We call it inequality with one variable. Every value of variable that converts given inequality into correct numerical equality is called a solution of inequality. To solve inequality with one variable means to find all solutions of inequality or prove that there are no solutions.

Two inequalities with one variable are called equivalent if they have same solutions; in particular two inequalities are equivalent if they have no solutions.

Fact 1. If we move some summand from one side of inequality to another and change sign of summand, then we will obtain equivalent inequality.

For example, `x^2+x-2>5` is equivalent to `x-2>5-x^2` (we moved `x^2` from left side to right and changed sign). `x^5-2x>34-x^3` is equivalent to `x^5-2x+x^3>34` (we moved `-x^3` from right side to left and changed sign).

Fact 2. If multiply or divide both sides of inequality by some positive number then we will obtain equivalent inequality.

For example, `x>5` is equivalent to `2x>10` (we multiplied both sides of inequality by 2). `6x> -36` is equivalent to `x> -6` (we divided both sides of inequality by 6).

Fact 3. If multiply or divide both sides of inequality by some negative number and change sign of inequality then we will obtain equivalent inequality.

For example, `x>5` is equivalent to `-2x<-10` (we multiplied both sides of inequality by -2 and changed sign). `6x> -36` is equivalent to `-x< 6` (we divided both sides of inequality by -6 and changed sign).

Fact 4. If multiply or divide both sides of inequality by some expression that takes for all values of x only positive values then we will obtain equivalent inequality.

For example, `x>5` is equivalent to `x(x^2+4)>5(x^2+4)` (we multiplied both sides of inequality by `x^2+4` that is positive for all x). `6x> -36` is equivalent to `x/(x^2+1)> -6/(x^2+1)` (we divided both sides of inequality by `x^2+1` that is positive for all x).

Fact 5. If multiply or divide both sides of inequality by some expression that takes for all values of x only negative values and change sign of inequality then we will obtain equivalent inequality.

For example, `x>5` is equivalent to `-x(x^2+4)< -5(x^2+4)` (we multiplied both sides of inequality by `-(x^2+4)` that is negative for all x). `6x> -36` is equivalent to `-(6x)/(x^2+1)<(36)/(x^2+1)` (we divided both sides of inequality by `-(x^2+1)` that is negative for all x).