# -3 ≤ 1/2 - 3x/4 ≤ 13/8; x∈NSolve the inequation and graph the solution set on the number line.

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First you can tell that there are 2 inequalities which need to be solved. The N is basically the natural numbers which are positive numbers/integers.

The first thing you should do is multiply each side by 4 to start eliminating everything. After that once it is simplified you would then of course have to add 3x+12 to both sides to move the x's and have the numbers seperate. Divide by 3 and that is the answer to the first inequality.

For the second one you have to do the following. . .

1/2 - 3x/4 ≤ 13/8.. You need to multiply each side by 8.

4-6x <= 13 Add 6x-13 to both sides:

4-13 <= 6x or -9<=6x or -9/6=-3/2 = - (1 and1/2)<= x...........(2)

The answers are (1,2,3,4) The above answer tells you why.

Hope this helped.

this inequality has three numbers in the denominators 2,4,8 their lowest common multiple ( L.C.M) is 8 , so we multiply its three sides by 8 to get rid of the denominators .

-3 ≤ 1/2 - 3x/4 ≤ 13/8 ( multiply by 8 )

-3*8 ≤ 1/2*8 - 3x/4*8 ≤ 13/8*8

-24 ≤ 4 - 6x ≤ 13 ( subtracting 4 from all sides )

-24-4 ≤ 4 - 6x - 4 ≤ 13-4

-28 ≤ - 6x ≤ 9 ( multiplying by -1 to change the sign of x

28 > 6x > -9 ( dividing by 6 )

28/6 > 6x/6 > -9/6

4.67 > x > -1.5

since x belongs to N = (0,1,2,3,.......................)

therefore x belongs to ( 0,1,2,3,4 )

To solve:

-3 ≤ 1/2 - 3x/4 ≤ 13/8: x∈N

Solution:

There are two inequalities:

-3 ≤ 1/2 - 3x/4 and 1/2 - 3x/4 ≤ 13/8 x belongs to N.

x belongs to N means solution set should be in natural numbers or positive integers. This we look at the end of the procedure.

Multiply by 4 both sides of the inequality,-3<=1/2-3x/4 to get rid of the fractional form:

-12<=2-3x . Add 3x+12 to both sides to shift x's to the left and numbers to the right:

3x<=2+12 = 14 or x <= 14/3 = 4 and 1/3..........(1)

Take the other inequality:

1/2 - 3x/4 ≤ 13/8. Multiply both sides by 8.

4-6x <= 13 Add 6x-13 to both sides:

4-13 <= 6x or -9<=6x or -9/6=-3/2 = - (1 and1/2)<= x...........(2)

From (1) and ( 2):

-(1 and 1/2) <= x<= (4 and 1/3) is the solution set. But there is one more condtion that x blongs N. So, the solution set is

-(1 and 1/2) <= x<= (4 and 1/3) AND x belongs to N

Therefore, x=1 or x = 2 or x=3,x=4 Or the solution set is {1,2,3,4}