Square inequalities

1. Square inequalities

The type inequality:

аx² + b + c> 0 or ах² + bx + c <0 resp
e
ctively

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

193-c1

(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:

194-f1

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 ∩ M

– square neravenki –
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