Solve, `sqrt(x-6)-sqrt(10-x) ge 1`.

Answer

`x in [(16+sqrt(7))//2,10]`

Explanation

We have` sqrt(x-6)-sqrt(10-x) ge 1`
We must have `x ge 6` and `x le 10`
Then `sqrt(x-6) ge 1+sqrt(10-x)`
Squaring given inequality,
`x-6 ge 1+10-x+2sqrt(10-x)`
`implies 2x-17 ge 2sqrt(10-x)`
Clearly LHS must be positive.
`:. x ge (17)/(2)`
So we have
`4x^(2)+289-68x ge 4(10-x)`
`implies 4x^(2)-64x+249 ge 0`
`implies x^(2)-16x+(249)/(4) ge 0`
`implies (x-8)^(2)-((sqrt(7))/(2))^(2) ge 0`
`implies(x-8-(sqrt(7))/(2))(x-8+(sqrt(7))/(2)) ge 0`

From the sign scheme, `x in [(16+sqrt(7))//2,10]`