INEQUATION
Inequation : Inequation or inequality is a statement in which one thing is not equal to the other e.g. x ¹ 8, 3y < 19, 2x > 15 or If A(x) and B(x) are two polynomials in x then the expression A(x) < B(x) or A(x) > B(x) are called inequalities.
- Replacement or universal set : Consider 5x < This inequation will remain satisfied if the variable x is replaced by the number -3, -2, -1, 0, 1.
The various numbers from which the value of the variable x is chosen, is called the replacement set and it is indicatied by Î.
- Solution set : The subset of the replacement set which satisfies the above given statement is called solution set or truth
The solution of an inequation depends upon the replacement set used. An inequation may have one, many or no solution depending upon the replacement set.
Inequality Principles : 1. Addition or subtraction from both sides of an inequality of same thing gives an equivalent inequality. e.g. A(x) > B(x) then A(x) + C(x) > B(x) + C(x) or A(x) – C(x) > B(x) – C(x).
- Multiplication or division of both sides of an inequality by the same positive number gives an equivalent e.g. A(x) > B(x)
l l
K.A(x) > K.B(x) or — A(x) > — B(x).
k k
- Multiplication on both sides of an inequality by the same negative number gives an inequality with its direction
(1) 3 < 5 is equivalent to (- 1) (3) > (-1) (5) i.e., -3 > – 5.
- The statement ax + c > b can also written as b < ax + c 1 1 1 1
- If a > b, then — < — and if a < b, then — > —.
a b a b
Graphical representation on the number line : A number line has real num- bers marked on it.
Graph of (x : – 2 < x < 5, x Î z)
| | | | | | | |
-2 -1 0 1 2 3 4 5
Graph of (x : -3 £ x < 4)
| | | | | | | |
-3 -2 -1 0 1 2 3 4
Graph of (x : x ³ 3, x Î N)
| | | | |
-1 0 1 2 3 Solid line Graph of (x : x £ 2, x Î Z)
| | | | Solid line -1 0 1 2
First degree whole rational Inequalities : Such inequalities can be reduced in the following forms
ax + b > 0
when a > 0
or ax + b < 0
these can be solved to get b b
x >—— or x < – —.
a a
Second Degree Whole Rational Inequalities :
ax2 + bx + c > 0
when a > 0
or ax2 + bx + c < 0
Such inequalities can be solved as under.
|
The discriminant D = b2 – 4ac (for the equation ax2 + bx + c = 0) and x the solution for the equation
ax2 + bx + c = 0.
Then solutions of inequalities given above are –
- Solution to inequality ax2 – bx + c > 0 (1) a > 0, x < x1 x > x2
b (2) a = 0, x ¹ – —-
2a
(3) a < 0, No solution are possible
- Solution to inequality ax2 + bx + < 0 (1) a > 0, x1 < x < x2
- a = 0, all real values of x
- a < 0, No solutions are
INEQUALITY SYSTEMS
and x2
are
Such systems can be expressed as
(x) > 0
Q (x) > 0
R (x) > 0
The solution to systems are any values of x that are capable of satisfying simul- taneously all the inequalities of a system.
BROKEN RATIONAL INEQUALITIES
Such inequalities can always be expressed as A(x)
—– > 0
B(x)
A(x)
or < 0
B(x)
Resolution of Inequalities
A(x)
—— > 0 can be resolved in two systems as B(x)
A(x) > 0 A(x) < 0 B(x) > 0 or B(x) < 0
A(x)
Similarly the inequality——- < 0 can be resolved in the following two systems.
B(x)
A(x) > 0 A(x) < 0 B(x) < 0 or B(x) > 0
IRRATIONAL INEQUALITIES
Any inequality containing only a single radical can be expressed as
A(x) > nÖB(x) or A(x) < nÖB(x) Such inequalities can be resolved as follows A(x) > nÖB(x)
Can be resolved to yield [A(x)n > B(x), when n is odd
or A(x) > 0
B(x) ³ 0 , when n is even [A(x)]n > B(x)
Similarly the inequality A(x) < nÖB(x) can be resolved to give [A(x)]n < B(x), n is odd
or A(x) > 0 and A(x) ³ 0 [A(x)]n < B(x), when n is even.
B(x) ³ 0
