1. Square inequalities
The type inequality:
аx² + b + c> 0 or ах² + bx + c <0 resp
ectively
аx² + bx + c ≥ 0 or аx² + bx + c ≤ 0
where a, b, c are real numbers and a ≠ 0 is called a quadratic inequality with one variable.
Solve one of the quadratic inequalities аx² + b + c> 0 or аx² + b + c <0
comes down to determining those values of x for which the corresponding quadratic function
f (x) = аx² + bx + c
gets a positive or negative value.
The set of solutions to the inequality
аx² + bx + c ≥ 0 or аx² + bx + c ≤ 0
contains the zeros of the corresponding quadratic function
Example. Let us solve the inequality х² – 2x -3 <0
The corresponding quadratic function is f (x) = х² – 2х – 3. We sketch its graph with the help of its zeros, which are solutions of the equation х² – 2х – 3. These are the numbers x₁ = -1 and x₂ = 3
(Fig.1).
From the graph we see that it is negative, ie. the graph is below the x-axis, for values of x from the interval (-1, 3).
2. SYSTEM SQUARE INEQUALITIES
The general type of a system of quadratic inequalities with one variable is:
Instead of the ">" sign in the inequalities it can be either '' <", or" ≤ ", or" ≥ ".
If MZ and MZ are the sets of solutions of the first, ie the second inequality, then the set of solutions M of the system is the intersection of those sets, ie.
M = M₁ ∩ M₂
