Linear inequalities

Expressions containing variables connected by signs are called inequalities:
«greater than» (>);
«greater than or equal to» (≥);
«less than» (<);
less than or equal to (≤).

Linear inequalities with one variable x are described by expressions of the type:
xy + z > 0
xy + z < 0
xy + z ≥ 0
xy + z ≤ 0
in this case y is not equal to zero.

Characteristics of linear inequalities: contain the variable only in the first degree; division by the variable is not performed; multiplication of the variable by 0 is not performed.

To solve an inequality means to find all possible values of the variable it contains, or to prove that they do not exist.

Three rules for solving linear inequalities

When moving terms from one part to another, negative values become positive, and vice versa. The sign of the inequality itself remains.
x – y > z => x – z > y => x > z + y
for example:
x – 9 > 3 => x > 3 + 9 => x > 12

When multiplying or dividing both parts by the same positive number, the inequality remains valid and its sign does not change.
x < z => yx < yz => x/y < z/y
for example:
10x > 20 => x > 2
0,5x < 3 => x < 6

If the multiplier (divisor) is negative, the inequality sign must be replaced with the opposite.
x < z => -yx > -yz => -x/y > -z/y
For example:
9 > 3 => -9 < -3 => -3 < -1

The ability to solve linear inequalities will be useful to you in further study and research of functions. They are needed for:
• finding the maximum and minimum value of a function in a certain interval;
• determining intervals of increase and decrease of a function;
• determining the boundedness of functions.