A straight line ax + by = c divides the Cartesian plane into two parts. Each part is an half plane.

**Rational Functions**

A rational expression of *x* is defined as the ratio of two
polynomials in *x*, say *P* (*x*) and *Q*(*x*) where

A straight line ax + by = c
divides the Cartesian plane into two parts. Each part is an half plane. A
vertical line x = c will divide the plane in left and right half planes and a
horizontal line y = k will divide the plane into upper and lower half planes.

A point in the cartesian plane
which is not on the line ax + by = c will lie in exactly one of the two half
planes determined by the line and satisfies one of the inequalities ax + by
< c or ax + by > c.

To identify the half plane
represented by ax + by < c, choose a point P in any one of the half planes
and substitute the coordinates of P in the inequality.

If the inequality is satisfied,
then the required half plane is the one that contains P; otherwise the required
half plane is the one that does not contain P. When c ¹ 0, it is most convenient to take P to be the
origin.

** Exercise - 2.10**

Determine the region in the plane
determined by the inequalities:

(1) x ≤ 3y, x ≥ y.

(2) y ≥ 2x, −2x + 3y ≤ 6.

(3) 3x + 5y ≥ 45, x ≥ 0, y ≥ 0.

(4) 2x + 3y ≤ 35, y ≥ 2, x ≥ 5.

(5) 2x + 3y ≤ 6, x + 4y ≤ 4, x ≥
0, y ≥ 0.

(6) x − 2y ≥ 0, 2x − y ≤ −2, x ≥
0, y ≥ 0.

(7) 2x + y ≥ 8, x + 2y ≥ 8, x + y
≤ 6.

Tags : Rational Inequalities, Partial Fractions, Graphical Representation | Definition, Formula, Solved Example Problems, Exercise | Algebra | Mathematics , 11th Mathematics : UNIT 2 : Basic Algebra

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11th Mathematics : UNIT 2 : Basic Algebra : Rational Functions | Rational Inequalities, Partial Fractions, Graphical Representation | Definition, Formula, Solved Example Problems, Exercise | Algebra | Mathematics

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