Question

Solve the following inequation and represent the solution set on the number line:

`4x - 19 < (3x)/5 - 2 <= (-2)/5 + x, x ∈ R`

Answer 1

Consider the given inequation

`4x - 19 < (3x)/5 - 2 <= (-2)/5 + x, x ∈ R`

`=> 4x - 19 + 2 < (3x)/5 - 2 + 2 <= (-2)/5 + x + 2, x ∈ R`

`=> 4x - 17 < (3x)/5 <= x + 8/5, x ∈ R`

`=> 4x - (3x)/5 < 17 and (-8)/5 <= x - (3x)/5, x ∈ R `

`=> (20x  - 3x)/5 < 17 and (-8)/5 <= (5x - 3x)/5, x ∈ R`

`=> (17x)/5 < 17 and (-8)/5 <= (2x)/5, x ∈ R`

`=> x/5 < 1` and `-4 <= x, x ∈ R`

`=> x < 5` and `-4 <= x, x ∈ R`

`=> -4 ≤ x < 5; "where"  x ∈ R`

The solution set can be represented on a number line as follows: