1.Algebra Booster

Quadratic Equations and Expressions 2.7
Inequalities with the Absolute Value
Basic rules for the absolute value of the inequality.
1. |x| = 2 fi x = ±2
2. |x| = –3 fi x = f
3. |x| < 4 fi –4 < x < 4
4. |x| £ 5 fi –5 £ x £ 5
5. | x| 2 x 2, x –2 x (– , –2) (2, )
6. | x| 3 x 3, x –3 x ( , 3] [3, )
2.3 IRRATIONAL EQUATIONS
Introduction
In an equation, if the unknown quantities are under radical
sign, it is known as an irrational equation.
For example, x + x - 2 = 1, x - 3 = -2,
x - x + 1=
2,
etc., are irrational equations.
The concept of irrational equation comes from irrational
expression. This expression contains at least one fractional
power of the unknown quantity.
For example,
1. f( x)
= x is an irrational expression.
3
2. f( x)
= x is an irrational expression.
= x is an irrational expression.
Thus, we get the even root as well as odd root of the unknown
quantity of an irrational expression.
If roots of an irrational expression are even, i.e.
4 6 10 100
x, x, x, x, x , ..., it is defined for non-negative real
values of the radicand.
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If the radicand is negative ( – x, - x, ...) , the roots are
imaginary.
If all the roots are odd, that is, 3 x, 5 x, 9 x, 2013 x...,
it is
defined for all real values of the radicand.
If the odd radicand is negative, its roots are also negative.
2013
3. f( x)
2.4 IRRATIONAL IN-EQUATIONS
Type I: An in-equation is of the form
2n+ 1 2n+
1
f() x < g(),
x nŒN
fi f(x) < g(x)
Type II: An in-equation is of the form
fi
f() x < g(),
x nŒN
2n
2n
Ï f() x ≥ 0
Ô
Ì gx () > 0
Ô
Ógx
() > f()
x
Type III: An in-equation is of the form
fi
2n+
1
f() x < g(), x nŒN.
f(x) < g 2n+1 (x)
Type IV: An in-equation is of the form
2 n f() x < g(),
x nŒN
Type V: An in-equation is of the form
fi
fi
2n+
1
f() x > g(), x nŒN.
f(x) > g 2n+1 (x)
Ï f() x ≥ 0
Ô
Ì gx () > 0
Ô
2n
Ó f() x < g () x
Type VI: An in-equation is of the form 2 n
f() x > g(),
x n ΠN.
ÏÔ gx () ≥ 0 Ïgx
() < 0
fi Ì
and
2n
Ì
ÔÓ f() x > g () x Ó f() x ≥ 0
2.5 EXPONENTIAL EQUATIONS
Type I: An equation is of the form
a f(x) = 1, a > 0, a π 1
fi f(x) = 0
Type II: An equation is of the form f(ax) = 0
fi f(t) = 0, where t = a x .
If t 1
, t 2
, t 3
, … t n
are the roots of f(t) = 0, then ax = t 1
, …,
a x
= t n
.
Type III: An equation is of the form
f () x f () x f () x
a◊ a + b◊ b + g ◊ c = 0 ,
where a, b, g Œ R and a, b, g π 0 and the bases satisfy the
condition b 2 = ac.
2
Êaˆ
fi a◊ t + b◊ t + g = 0 , where t = Á
Ë
˜
b¯
Type IV: An equation is of the form
f () x f () x
f () x
a◊ a + b◊ b + c = 0,
where a, b, c Œ R, a, b, c π 0
and ab = 1 (where a and b are inverse positive numbers).
2
fi a◊ t + c◊ t + b = 0 , where t = a f(x)
Type V: An equation is of the form
a f(x) + b f(x) = c,
where a, b, c ΠR
and a 2 + b 2 = c.
The solution of the given equation is f(x) = 2.
Type VI: An equation is of the form
a f(x) + b f(x) + c f(x) = d,
where a, b, c, d ΠR and a 3 + b 3 + c 3 = d. The required solution
is f(x) = 3.
Type VII: An equation is of the form
a f(x) + a g(x) = c,
where f(x) + g(x) = 1, and
a, c (π 0)
We shall put a f(x) = t
Type VIII: An equation is of the form
a f(x) + a f(x)–1 + a f(x)–2 = b f(x) + b f(x)–1 + b f(x)–2
Then the required solution is
.