SOLUTION: Solve the inequalities for the range of x (if any). (6x-5)/(4x+1) ≤ (2x)/(1-x)

Algebra ->  Inequalities -> SOLUTION: Solve the inequalities for the range of x (if any). (6x-5)/(4x+1) ≤ (2x)/(1-x)      Log On


   

Question 1192698: Solve the inequalities for the range of x (if any).
(6x-5)/(4x+1) ≤ (2x)/(1-x)
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

%286x-5%29%2F%284x%2B1%29+%3C=%282x%29%2F%281-x%29+

%286x-5%29%2F%284x%2B1%29+-%282x%29%2F%281-x%29+%3C=0

%286x-5%29%281-x%29+-%282x%29%284x%2B1%29+%29%2F%28%284x%2B1%29+%281-x%29+%29%3C=0

%28-8+x%5E2+%2B+9+x+-+5%29%2F%28%284x%2B1%29+%281-x%29+%29%3C=0........will have solution only if denominator is not equal to zero

%284x%2B1%29+%281-x%29=0

=>4x%2B1=0 =>4x=-1=>x=-1%2F4
=>1-x=0 =>x=1

solution;
x%3E-1%2F4 or x%3C1
so, -1%2F4%3Cx%3C1
Interval notation: (-1%2F4,1)


Answer by ikleyn(52815) About Me  (Show Source):
You can put this solution on YOUR website!
.

    %286x-5%29%2F%284x%2B1%29 <= %282x%29%2F%281-x%29+


    %286x-5%29%2F%284x%2B1%29+-%282x%29%2F%281-x%29 <= 0


    %286x-5%29%281-x%29+-%282x%29%284x%2B1%29+%29%2F%28%284x%2B1%29+%281-x%29+%29 <= 0


    %28-8x%5E2+%2B+9x+-+5%29%2F%28%284x%2B1%29%2A%281-x%29+%29 <= 0     (*)


The nuerator of the fraction in the left side is a quadratic polynomial

    -8x^2 + 9x - 5.


It has the discriminant  d = 9%5E2+-+4%2A%28-8%29%2A%28-5%29 = 81 - 160 = - 79.

Since the discriminant is negative, the polynomial has no real roots; hence, it does not change sign;
and its value at x= 0 is -5, so the polynomial in the numerator is ALWAYS NEGATIVE for all x.


Hence, inequality (*) is true if anf only if the denominator in (*) is positive


    %284x%2B1%29%2A%281-x%29 > 0


It implies that the given / (original) inequality has the solution set

    -1%2F4 < x < 1.      ANSWER

Solved.