Question

Solve the following inequation, write the solution set and represent it on the real number line.

`5x - 21 < (5x)/7 - 6 ≤ -3 3/7 + x, x ∈ R`

Answer 1

Given the inequation: `5x - 21 < (5x)/7 - 6 ≤ -3 3/7 + x`

Solving the L.H.S. of the inequation:

⇒ `5x - 21 < (5x)/7 - 6`

⇒ `5x - (5x)/7 < 21 - 6`

⇒ `(35x - 5x)/7 < 15`

⇒ `(30x)/7 < 15`

⇒ `x < (7 xx 15)/30`

⇒ `x < 7/2` or 3.5  ....(1)

Solving the R.H.S. of the inequation:

⇒ `(5x)/7 - 6 ≤ -3 3/7 + x`

⇒ `(5x)/7 - 6 ≤ - 24/7 + x`

⇒ `x - (5x)/7 ≥ -6 + (24)/7`

⇒ `(7x - 5x)/7 ≥ (-42 + 24)/7`

⇒ `(2x)/7 ≥ (-18)/7`

⇒ 2x ≥ −18

⇒ `x ≥ (-18)/2`

⇒ x ≥ –9   ......(2)

From equation (1) and (2),

Solution set = `{x: -9 ≤ x < 7/2, x ∈ R}`