How do you solve (x(x-4))/(2-3x)<= 3x(x4)23x3?

2 Answers
Jun 17, 2015

-6<=X<=16X1

Explanation:

Multiplying the numerator by x gives (x^2-4x)/(2-3x)<=3x24x23x3
Multiplying by the denominator gives x^2-4x<=6-9xx24x69x
Rearranging gives x^2 +5x-6<=0x2+5x60
Factoring gives (x+6)(x-1)<=0(x+6)(x1)0
For the inequality to be true one term must be positive and the other negative
This means (x+6)<=0 and (x-1)>=0(x+6)0and(x1)0 or (x+6)>=0 and (x-1)<=0(x+6)0and(x1)0
This means x<=-6 and x>=1x6andx1 which is impossible
Or x>=-6 and x<=1x6andx1 which gives -6<=x<=16x1

Jun 17, 2015

Compare to 00 and do a sign analysis (sign chart, sign table, sign diagram, whatever you were taught to call it).

Explanation:

(x^2-4x)/(2-3x) <= 3x24x23x3 if and only if:

(x^2-4x)/(2-3x) -3 <= 0x24x23x30

Rewrite to get a single ratio on the left.

(x^2-4x)/(2-3x) -3/1 ((2-3x)/(2-3x)) <= 0x24x23x31(23x23x)0

((x^2-4x) -3 (2-3x))/(2-3x) <= 0(x24x)3(23x)23x0

((x^2-4x -6+9x))/(2-3x) <= 0(x24x6+9x)23x0

(x^2+5x -6)/(2-3x) <= 0x2+5x623x0

Find the key numbers (partition numbers, unnamed special numbers) for the expression on the left. These are the places where the expression might change sign. We find them by finding the zeros and the places where the expression is undefined.

In the end we solve "TOP" = 0TOP=0 and "BOTTOM" = 0BOTTOM=0

x^2+5x-6 = (x+6)(x-1) =0x2+5x6=(x+6)(x1)=0 at x=-6, 1x=6,1

2-3x = 023x=0 at x=3/2x=32

The key numbers are:-66, 11, and 3/232.

They cut the real number line into intervals:

(-oo, -6)(,6), (-6, 1)(6,1), (1, 3/2)(1,32), and (3/2, oo)(32,)

The expression: ((x+6)(x-1))/(2-3x)(x+6)(x1)23x is:

positive on (-oo, -6)(,6), (test x=-10x=10)
negative on (-6, 1)(6,1) (test x=0x=0)
positive on (1, 3/2)(1,32), (test x=5/4x=54)
negative on (3/2, oo)(32,) (test x=5x=5)

We want the value of xx that give negative values for the expression, so the solution is:

(-6, 1) uu (3/2, oo)(6,1)(32,).