Question

Solve the following inequation.

`(11 + 3x)/5 ≥ 3 - x > -3/2, x ∈ R`

1. Write the solution set. 
2. Represent the solution on the number line.

Answer 1

Given: `(11 + 3x)/5 ≥ 3 - x > -3/2, x ∈ R`

Step-wise calculation:

1. Split the compound inequality into two parts and solve each separately, use inequality rules for adding/multiplying, etc.

2. Right part: `3 - x > -3/2`

Subtract 3: `-x > -3/2 - 3 = -9/2`

Multiply by –1 (reverse inequality): `x < 9/2`

3. Left part: `(11 + 3x)/5 ≥ 3 - x`

Multiply both sides by 5 positive, so inequality keeps direction: 11 + 3x ≥ 15 – 5x

Bring x-terms together:

3x + 5x ≥ 15 – 11

⇒ 8x ≥ 4

Divide by 8: `x ≥ 1/2`

4. Combine the two results both conditions must hold:

`x ≥ 1/2` and `x < 9/2`

So, `1/2 ≤ x < 9/2`.

Solution set = `{x ∈ R : 1/2 ≤ x < 9/2} = (1/2, 9/2)`.

On the number line: a solid (closed) dot at `x = 1/2`, an open dot at `x = 9/2` and the segment between them shaded.