Question

Can someone help me to finish this
Sign Rule
Problem : solve the inequality
3/x+2 + 5/x-2 ≤ 8
3/x+2 + 5/x-2 - 8 ≤ 0
3/(x+2)(x-2) + 5/(x+2)(x-2) - 8/(x+2)(x-2)
3(x-2) + 5(x+2) - 8(x+2)(x-2)
3x-6 + 5x+10 - 8x^2+32
= -8x^2 + 8x + 36/(x+2)(x-2)

x+2 = 0 , x - 2 = 0
x = -2 x = 2
i cant finish it i dont know what sign I will use
help me finish this pls

Answer 1

does it look like this from the beginning ?

\[\frac{ 3 }{ x+2 } + \frac{ 5 }{ x - 2 } \le 8\]

Answer 2

yes sir

Answer 3

factor the numerator to see what the roots are. we know that either the function is increasing between roots, or decreasing between roots. and since these are the only two roots, then whatever value the function is before or after, it will remain that way.

for example, one root is 1/2(1 - sqrt 19)

this is approximately -1.68

if we pick a value left of -1.68 and plug it into the function, we will find that the function returns a positive value.

to switch this to a negative, notice the denominator where x = -2 is not in the domain.. if x < -2, however, then the denominator switches signs, and hence the function itself switches signs. we arrive at the first solution:

x < -2

next check what happens between -1.68 and 2 (the other point where the function is not defined)

we find that the values here are also negative so:


1/2(1 - sqrt 19) <= x < 2

then check what happens between x = 2 and x = 1/2(1 + sqrt 19) [the other root of the numerator]

1/2(1 + sqrt 19) is approx. 2.68 so testing an x value between x = 2 and x = 2.68 tells us that the function is positive on this interval.

so when x > 2, the sign of the denominator switches, and then the function becomes negative so

x > 1/2(1 + sqrt 19)

in summary


x < -2
1/2(1 - sqrt 19) <= x < 2
x >= 1/2(1 + sqrt 19)

Answer 4

should say "x >= 1/2(1 + sqrt 19)" above "in summary"