Question

what can x equal if

4 < 2x+1/x-3 < 7

Answer 1

multiply by x-3

4(x-3) < 2x + 1 < 7(x-3) , if x-3 > 0 or x > 3

4x-12 < 2x+1 or 2x+1 < 7x - 21
2x < 13 ----- or -5x < -22
x < 6.5 ------ or x > 22/5

So we have 3 critical values, x = 6.5 , x = 4.4 and x = 3
dividing our number line into 4 segments.
I then take any value in each of the segments and test it in the original

left segment , x < 3, e.g let x = 0
4 < 1/-3 < 7 false
segment between 3 and 4.4, x = 4
4 < 9/1 < 7 false
segment between 4.4 and 6.5, x = 5
4 < 11/2<7
4<5.5<7 true
segment > 6.5, x = 10
4 < 21/8<7
4 < 2.625<7 false

so 4.4 < x < 6.5

btw, if you have a graphing calculator
graph
y = (2x+1)(x-3) , y = 4 and y = 7 and you will see that they intersect at (4.4 , 7) and (6.5 , 4)
so the graph is between 4 and 7 for my values