Question

Solve the rational inequality (1/x+2)+(1/2-x)<= (3x+8/x^2-4) by providing full algebraic solutions using an interval chart. Express your answer in proper interval notation.

Answer 1

guessing that you forgot the brackets and you meant:

1/(x+2) + 1/(2-x) ≤ (3x+8)/(x^2-4)
1/(x+2) - 1/(x-2) ≤ (3x+8)/(x^2-4) , clearly x ≠ ± 2

multiply by x^2 - 4
x-2 - (x+2) ≤ 3x+8
-4 ≤ 3x + 8
-12 ≤ 3x
x ≥ -4

can't make a chart here, but the "critical" values are ± 2, and -4

on a number line we have

<----- -4----- -2 ------------------------ +2 ------------->

testing for a value of x in each section
x ≤ -4, I will use x = -6
-1/4 + 1/9 ≤ -10/32
-.139 ≤ -.3125 , false

between -4 and -2, I will use x = -2.5
1/-.5 + 1/4.5 < .5/2.25
-1.777.. ≤ .2222 , true

between -2 and 2, I pick x = 0
1/2 +1/2 ≤ -2
1 ≤ -2 , false

x > 2, I pick x = 3
1/5 -1 ≤ 17/5 , true

so -4 ≤ x < -2 OR x > 2

notice that x = -8/3 makes the right side equal to zero and the left
side negative, but since -8/3 lies between -4 and -2, and we found
those values to work, we don't have to worry about the -8/3

enter : 1/(x+2) + 1/(2-x) ≤ (3x+8)/(x^2-4)
into Desmos at
www.desmos.com/calculator

to see my answer is correct