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

sciencesolve | Certified Educator

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/` .

kfcarter | Student

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

-2<2x                                            2x<6

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

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

{x|-1<x<3}

OR

Option 2:

Leave as one equation and perform identical operations:

-3  <  2x-1  <  5

+1       +1      +1

The key is to still isolate your variable, so you would begin by adding one in each area of the problem (3 times in this instance, as illustrated)

-2   <  2x  <  6

Divide by two on all sides

-1  <  x  <  3

{x|-1<x<3}