Solve the following inequation, write the solution set and represent it on the number line. -3(x - 7) >= 15 - 7x > (x + 1) / 3, x ∈ R

Given:

−3(x − 7) ≥ 15 − 7x > (x + 1)/3

First: −3(x − 7) ≥ 15 − 7x

−3x + 21 ≥ 15 − 7x

4x ≥ −6

x ≥ −3/2

Now: 15 − 7x > (x + 1)/3

45 − 21x > x + 1

44 > 22x

x < 2

Combining:

−3/2 ≤ x < 2

Solution set: {x ∈ R : −3/2 ≤ x < 2}

On the number line: closed circle at −3/2, open circle at 2, and shade the interval between them.

Solve the two inequalities separately and take their common interval. The left endpoint is included because of ≥, while 2 is excluded because the second inequality is strict.