Question

Solve the inequation given below. Write the solution set and represent it on the number line:

`-3 + x ≤ (8x)/3 + 2 ≤ 14/3 + 2x, x ∈ I`

Answer 1

Given,

`-3 + x ≤ (8x)/3 + 2 ≤ 14/3 + 2x`

Solving L.H.S. of the inequation,

⇒ `-3 + x ≤ (8x)/3 + 2`

⇒ `x - (8x)/3 ≤ 2 + 3`

⇒ `x - (8x)/3 ≤ 5`

⇒ `(3x - 8x)/3 ≤ 5`

⇒ 3x – 8x ≤ 5 × 3

⇒ –5x ≤ 15

Dividing by –5 on both sides we get,

⇒ x ≥ –3 (As on dividing by negative number the sign reverses)   ...(1)

Solving R.H.S. of the inequation,

⇒ `(8x)/3 + 2 ≤ 14/3 + 2x`

⇒ `(8x + 6)/3 ≤ (14 + 6x)/3`

⇒ 8x + 6 ≤ 14 + 6x

⇒ 8x – 6x ≤ 14 – 6

⇒ 2x ≤ 8

⇒ `x ≤ 8/2`

⇒ x ≤ 4   ...(2)

From (1) and (2) we get,

–3 ≤ x ≤ 4

Since, x ∈ I

Hence, solution set = {–3, –2, –1, 0, 1, 2, 3, 4}.

Solution on the number line is: