# how can i solve an inequality with 2 or more greter than or less than signs?

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Considering that you need to solve the system of simultaneous linear inequalities, yields:

`c < ax + b < d => {(ax + b>c),(ax + b<d):} => {(ax + b - c>0),(ax + b - d<0):} `

You need to attach and solve the following equations such that:

`{(ax + b - c=0),(ax + b - d=0):} => {(ax= c - b),(ax = d - b):} => {(x = (c - b)/a),(x = (d - b)/a):}`

Notice that the expression `ax + b - c = 0` is positive for `x in ((c - b)/a, oo)` and the expression `ax + b - d=0` is negative for `x in (-oo,(d - b)/a)` .

You need to intrsect the resulted intervals to come up with a solution, such that:

`x in ((c - b)/a, oo) nn (-oo,(d - b)/a)`

**If `(c - b)/a < (d - b)/a => x in ((c - b)/a, oo) nn (-oo,(d - b)/a) = ((c - b)/a,(d - b)/a)` **

**If `(c - b)/a> (d - b)/a` , the system has no solution, hence `x in` `O/` .**

Here is an example:

-3<2x-1<5

You can do this in a few ways. I will show you the most common two and you can decide which one works best for you.

Option 1:

1. Split into two equations

-3<2x-1 2x-1<5

2. Solve for x on both equations

-3<2x-1 2x-1<5

(add one to both sides) (add 1 to both sides)

-2<2x 2x<6

(divide by 2 to isolate x) (divide by 2 to isolate x)

-1<x (or x>-1) x<3

Answer:

{x|-1<x<3}

**OR**

Option 2:

Leave as one equation and perform identical operations:

-3 < 2x-1 < 5

*+1 +1 +1*

Answer:

{x|-1<x<3}

Maybe you have to solve what in between the signs first