Solve the following inequation, write the solution set and represent it on the real number line. 2x - 5/3 < 3x/5 + 10 ≤ 4x/5 + 11; x ∈ R

Given:

2x − 5/3 < 3x/5 + 10 ≤ 4x/5 + 11, x ∈ R

Solve the left inequality:

2x − 5/3 < 3x/5 + 10

(6x − 5)/3 < (3x + 50)/5

Multiply both sides by 15:

5(6x − 5) < 3(3x + 50)

30x − 25 < 9x + 150

30x − 9x < 150 + 25

21x < 175

x < 175/21

x < 25/3 ...(1)

Now solve the right inequality:

3x/5 + 10 ≤ 4x/5 + 11

3x + 50 ≤ 4x + 55

50 − 55 ≤ 4x − 3x

−5 ≤ x

x ≥ −5 ...(2)

Combining (1) and (2):

−5 ≤ x < 25/3

Solution set: {x ∈ R : −5 ≤ x < 25/3}

Number line: closed circle at −5, open circle at 25/3, and shade the interval between them.

Both parts of the compound inequation are solved separately. The final interval is their intersection.