wavy curve method
If p, q and r are real numbers, then
and equality holds for a = 1
and equality holds for a = –1
Inequation Involving Exponential Expression:
If k > 0, then kx > 0 for all real x.
If k > 1, then kx > 1, when x > 0
If 0 < k < 1, then kx < 1, when x > 0 and kx > 1, when x < 0.
How to solve inequalities by wavy curve method
In order to solve the inequalities of the form
where n1, n2, ……. , n k , m1, m2, ……. , mp are real numbers and a1, a2, ……. , ak, b1, b2, ……., bp are any real number such that ai ≠ bj where i = 1, 2, 3, ….k and j = 1, 2, 3, ….p.
Step - 1→ First arrange all values of x at which either numerator or denominator is becomes zero, that means a1, a2,….., ak, b1, b2, ….bp in increasing order say c1, c2, c3,……. cp + k. Plot them on real line
Step -2 → Value of x at which numerator becomes zero should be marked with dark circles.
Step - 3 → All pints of discontinuities (x at which denominator becomes zero) should be marked on number line with empty circles. Check the value of ƒ(x) for any real number greater than the right most marked number on the number line.
Step - 4 → From right to left draw a wavy curve (beginnings above the number line in case of value of ƒ(x) is positive in step–3 otherwise from below the number line), passing thoroughly all the marked points. So that when passes through a point (exponent whose corresponds factor is odd) intersects the number line, and when passing thoroughly a point (exponent whose corresponds factor is even) the curve doesn’t intersect the real line and remain on the same side of real line.
Step - 5 → The appropriate intervals are chosen in accordance with the sign of inequality (the function ƒ(x) is positive wherever the curve is above the number line, it is negative if the curve is found below the number line). Their union represents the Detail Explanation of inequality
Example : Let
Detail Explanation :
Step - 1 → make on real line all x at which numerator becomes zero with dark circles.
Step - 2 → mark point of discontinuity (value of x at which denominator becomes zero) with empty circles
Step - 3 → Check ƒ(x) for x > 7, ƒ(8) > 0
Exponents of factors of –1, 3, 4 is even, hence wave will not change the direction at these points.