Solving Quadratic And Rational Inequalities(Wavy Curve Method) :
`text(Wavy Curve Method)` (Generalised Method of Intervals)
Wave Curve Method is used for solving inequalities of the form
`f(x) = ((x-a_1)^(k_1)(x-a_2)^(k_2)...........(x-a_m)^(k_m))/((x-b_1)^(p_1)(x-b_2)^(p_2)...........(x-b_n)^(p_n)) > 0 (< 0 , >= text(or) <= 0 )`
where , `k_1 , k_2 , k_3 ,..............., k_m , p_1, p_2 ,p_3,.................,p_n` are natural numbers and such that `a_i ne b_j ,`
where `i = 1,2, .............. , m` and `j = 1,2,...............,n.`
`text(We use the following methods:)`
`1.` Solve `(x-a_1)^(k_1) (x-a_2)^(k_2)............(x-a_m)^(k_m) =0` and
`(x-b_1)^(p_1)(x-b_2)^(p_2).............(x-b_n)^(p_n)=0,` then we get
`x = a_1 , a_2 , ................ ,a_m , b_1 , b_2 , b_3 ,. ..............,b_n` [critical points]
`2.` Assume `a_1 < a_2 < ...... < a_m < b_1 < b_2 < .......... < b_n`
Plot them on the real line. Arrange inked (black) circles `(•)` and un-inked
(white) circles (0), such that
`tt[(,a_1a_2 .. a_m, b_1b_2 .. b_n),(f{x}>0,0 \ \ 0 ... 0,0 \ \ 0 ... 0),(f{x}<0,0 \ \ 0 ... 0,0 \ \ 0 ... 0),(f{x}>=0, • • ... •,0 \ \ 0 ... 0),(f{x}=< 0,• • ... •,0 \ \ 0 ... 0)]`
`3.` Obviously, `b_n` is the greatest root. If in all brackets before x positive sign and
expression has also positive sign, then wave start from :right to left,
beginning above the number line, i.e. `fig 1`
`+((x-a_1)^(k_1)(x-a_2)^(k_2) ............ (x-a_m)^(k_m))/((x-b_1)^(p_1)(z-b_2)^(p_2).........(x-b_n)^(p_n)) , ` then
and if in all brackets before x positive sign and expression has negative sign,
then wave start from right to left, beginning below the number line, i.e. `fig 2`
`-((x-a_1)^(k_1)(x-a_2)^(k_2) ............ (x-a_m)^(k_m))/((x-b_1)^(p_1)(z-b_2)^(p_2).........(x-b_n)^(p_n)) , ` then
4. If roots occur even times, then sign remain same from right to left side of the
roots and if roots occur odd times, then sign will change from right to left
through the roots of `x = a_1, a_2 , ... , a_m , b_1, b_2 , ... , b_n .`
5. The solution of `f(x) > 0` or `f(x) >= 0` is the union of all intervals in which we
have put the plus sign and the solution of `f(x) < 0` or `f(x) <= 0` is the union of all
intervals in which we have put the minus sign.
`text(Important Result :)`
1. The point where denominator is zero or function approaches infinity, will never
be included in the answer.
2. For `x^2 < a^2` or `|x| < a`
`⇔ -a < x < a`
i.e., `x in (-a , a)`
3. For `0 < x^2 < a^2` or `0 < | x | < a`
`⇔ - a < x < a ~ {0}`
i.e., `x in (-a,a)~{0}`
4. For `x^2 >= a^2 ` or `| x| >= a`
`⇔ x <= -a ` or `x >= a`
i.e., `x in (- oo,- a] uu [a, oo)`
5. For `x^2 > a^2 ` or `| x| > a`
`⇔ x < -a ` or `x > a`
i.e., `x in (- oo,- a] uu [a, oo)`
6. For `a^2 <= x^2 <= b^2 ` or `a <= |x| <= b`
`⇔ a <= x <= b ` or `-b <= x <= -a`
i.e., `x in [-b,-a] uu [a,b]`
7. For `a^2 < x^2 <= b^2 ` or `a < |x| <= b`
`⇔ a < x <= b ` or `-b <= x < -a`
i.e., `x in [-b,-a] uu [a,b]`
While solving such inequarions following steps to be taken.
`(i)` Factorise given-expression into linear factors
`(ii)` Make the coefficient of `x` positive in all factors
`(iii)` Plot the points where given expression vanishes or undefined (denominator becomes zero) on number line in increasing order
`(iv)` Start the number line from right to left taking positive or negative value.
While solving rational inequalities different situtations a rive.
`Type-1:` Inequalities involving non-repeated linear factors
`Type-2 :` Quadratic inequality involving Repeated linear factos
`Type-3 :` Quadratic / algebraic inequality of the type of `(f(x))/(g(x))` .(Rational inequality) involving modulus also.
`Type-4 :` Double inequality and biquadratic inequality
Wave Curve Method is used for solving inequalities of the form
`f(x) = ((x-a_1)^(k_1)(x-a_2)^(k_2)...........(x-a_m)^(k_m))/((x-b_1)^(p_1)(x-b_2)^(p_2)...........(x-b_n)^(p_n)) > 0 (< 0 , >= text(or) <= 0 )`
where , `k_1 , k_2 , k_3 ,..............., k_m , p_1, p_2 ,p_3,.................,p_n` are natural numbers and such that `a_i ne b_j ,`
where `i = 1,2, .............. , m` and `j = 1,2,...............,n.`
`text(We use the following methods:)`
`1.` Solve `(x-a_1)^(k_1) (x-a_2)^(k_2)............(x-a_m)^(k_m) =0` and
`(x-b_1)^(p_1)(x-b_2)^(p_2).............(x-b_n)^(p_n)=0,` then we get
`x = a_1 , a_2 , ................ ,a_m , b_1 , b_2 , b_3 ,. ..............,b_n` [critical points]
`2.` Assume `a_1 < a_2 < ...... < a_m < b_1 < b_2 < .......... < b_n`
Plot them on the real line. Arrange inked (black) circles `(•)` and un-inked
(white) circles (0), such that
`tt[(,a_1a_2 .. a_m, b_1b_2 .. b_n),(f{x}>0,0 \ \ 0 ... 0,0 \ \ 0 ... 0),(f{x}<0,0 \ \ 0 ... 0,0 \ \ 0 ... 0),(f{x}>=0, • • ... •,0 \ \ 0 ... 0),(f{x}=< 0,• • ... •,0 \ \ 0 ... 0)]`
`3.` Obviously, `b_n` is the greatest root. If in all brackets before x positive sign and
expression has also positive sign, then wave start from :right to left,
beginning above the number line, i.e. `fig 1`
`+((x-a_1)^(k_1)(x-a_2)^(k_2) ............ (x-a_m)^(k_m))/((x-b_1)^(p_1)(z-b_2)^(p_2).........(x-b_n)^(p_n)) , ` then
and if in all brackets before x positive sign and expression has negative sign,
then wave start from right to left, beginning below the number line, i.e. `fig 2`
`-((x-a_1)^(k_1)(x-a_2)^(k_2) ............ (x-a_m)^(k_m))/((x-b_1)^(p_1)(z-b_2)^(p_2).........(x-b_n)^(p_n)) , ` then
4. If roots occur even times, then sign remain same from right to left side of the
roots and if roots occur odd times, then sign will change from right to left
through the roots of `x = a_1, a_2 , ... , a_m , b_1, b_2 , ... , b_n .`
5. The solution of `f(x) > 0` or `f(x) >= 0` is the union of all intervals in which we
have put the plus sign and the solution of `f(x) < 0` or `f(x) <= 0` is the union of all
intervals in which we have put the minus sign.
`text(Important Result :)`
1. The point where denominator is zero or function approaches infinity, will never
be included in the answer.
2. For `x^2 < a^2` or `|x| < a`
`⇔ -a < x < a`
i.e., `x in (-a , a)`
3. For `0 < x^2 < a^2` or `0 < | x | < a`
`⇔ - a < x < a ~ {0}`
i.e., `x in (-a,a)~{0}`
4. For `x^2 >= a^2 ` or `| x| >= a`
`⇔ x <= -a ` or `x >= a`
i.e., `x in (- oo,- a] uu [a, oo)`
5. For `x^2 > a^2 ` or `| x| > a`
`⇔ x < -a ` or `x > a`
i.e., `x in (- oo,- a] uu [a, oo)`
6. For `a^2 <= x^2 <= b^2 ` or `a <= |x| <= b`
`⇔ a <= x <= b ` or `-b <= x <= -a`
i.e., `x in [-b,-a] uu [a,b]`
7. For `a^2 < x^2 <= b^2 ` or `a < |x| <= b`
`⇔ a < x <= b ` or `-b <= x < -a`
i.e., `x in [-b,-a] uu [a,b]`
While solving such inequarions following steps to be taken.
`(i)` Factorise given-expression into linear factors
`(ii)` Make the coefficient of `x` positive in all factors
`(iii)` Plot the points where given expression vanishes or undefined (denominator becomes zero) on number line in increasing order
`(iv)` Start the number line from right to left taking positive or negative value.
While solving rational inequalities different situtations a rive.
`Type-1:` Inequalities involving non-repeated linear factors
`Type-2 :` Quadratic inequality involving Repeated linear factos
`Type-3 :` Quadratic / algebraic inequality of the type of `(f(x))/(g(x))` .(Rational inequality) involving modulus also.
`Type-4 :` Double inequality and biquadratic inequality