How do you solve these kinds of inequalities?

Question

I'm having trouble solving these types of inequalities:

2-1/3x>3

4<2(1-3x)<10

|x-3|<4

Answer 1

2-1/3x>3
adding -2
-1/3x>1
-x>3
x<-3

2.4<2-6x<10
adding -2
2<-6x<8
dividing by 2
1<-3x<4
multiplying by -1
-1>3x>-3

3.x-3<4
x<7
x-3>-4
x>-1
-1

Answer 2

2-1/(3x)>3 (assume this is the right form.)

-1>1/(3x) (collect constant terms in one side, variable terms in the other side.)
-1<3x (flip it and change the direction of the inequality sign.)
x>-1/3 (divide by 3)


4<2(1-3x)<10

2<(1-3x)<5 (divide by 2)
1<-3x<4 (subtract 1)
-1>3x>-4 (change all the signs and the direction of the inequality at the same time.)
-1/3>x>-4/3 (divide by 3)


|x-3|<4

-4 -1

Answer 3

2 - 1/3x > 3

-1/3x > 3-2

-1/3x > 1

-x > 1/(1/3) This is a reciprocal technique

-x > 3

x > -3

x is any number higher than negative 3.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - -
4 < 2(1 - 3x) < 10

2 < 1 - 3x < 5

1 < -3x < 5

-1/3 < x < -1 2/3

x is any number between but not including -1/3 and -1 2/3 (or -1.66666666...)
- - - - - - - - - - - - - - - - - - - - - - - - - - - - -
|x-3|<4

|x|<4+3

|x|<7

x = any number greater than -7 and less than 7, because the |x| means the absolute of x, where -x is converted to positive.

Answer 4

2 - (1/3)x > 3
2-3 > (1/3)x (subtract/add terms to other side)
-1 > (1/3)x (combine like terms)
-3 > x (multiply both sides by 3)

You can write this as x < -3 or (-infinity , -3)

4 < 2 - 6x < 10
2 < -6x < 8 (subtract 2 from each)
2/(-6) > x > 8/(-6) (divide by -6 but switch all the signs because it's NEGATIVE)
-1/3 > x > -4/3 which is the same as -4/3 < x < -1/3
in interval notation it's (-4/3 , -1/3)

The third one needs to be rewritten as -4 < x-3 < 4
-1 < x < 7 (adding 3 to each)
interval notation is (-1 , 7)

Answer 5

1. first: get rid of the fraction > take the denominator and multiply it by everything:

3x(2) - 3x(1/3x) > 3x(3)
6x - 1 > 9x

Second: add 1 to both sides:

6x -1+1 > 9x + 1
6x > 9x + 1

Third: subtract 9x from both sides and divide everything by (-3) > change the sign when multiplying by a negative number:

6x - 9x > 9x - 9x + 1
-3x > 1
-3x/-3 > 1/-3
x < -1/3

2. follow the same steps from number 1

Answer 6

these type of inequalities can be solved in the following steps.
1) consider the denominator portion of any inequality on one side is positive and den solve for the equation n include only dos values for which the denominator is positive
2)repeat the above step considering for the denominator to be negative but remember to change the sign of the inequality while cross multiplying
3)if you contain any other operations like modulus you should again consider for both positive and negative values of modulus and hence union of all des cases gives you the solution