Prove that the following equations has no solutions.
(i) `sqrt((2x+7))+sqrt((x+4))=0`

Explanation

(i) We have `sqrt((2x+7))+sqrt((x+4))=0`
This equation is defined for `2x+7ge0`
and `x+4ge0implies{(xge-7/2),(xge-4):"`
`:.xge-7/2`
For `xge-7/2` , the left hand side of the original equation is positive, but right hand side is zero. Therefore, the equation has no roots.
(ii) We have `sqrt(x-4))=-5`
The equation is defined for `x-4ge0`
`:.xge4`
For `xge4`, the left hand side of the original equation is positive but right hand side is negative.
Therefore, the equation has no roots.
(iii) We have `sqrt((5-x))-sqrt(x-8)=2`
The equation is defined for
`6-xge0` and `x-8ge0`
`:.{(xle6),(xge8):}`
Consequently, there is no `x` for which both expressions would have sense. Therefore, the equation has no roots.
(iv) We have `sqrt((-2-x))=root(5)((x-7))`
This equation is defined for
`-2-xge0impliesxle-2`
For `xle-2` the left hand side is positive, but right hand side is negative.
Therefore, the equation has no roots.
(v) We have `sqrt(x)+sqrt((x+16))=3`
The equation is defined for
`xge0` and `x+16ge0implies{(xge0),(xge-16):}`
Hence `xge0`
For `xge0` the left hand side `ge4`, but right hand side is 3. Therefore, the equation has no roots.
(vi) We have `7sqrt(x)+8sqrt(-x)+15/(x^(3))=98`
For `xlt0` the expression `7sqrt(x)` is meaningless,
For `xgt0`, the expressionn `8sqrt(-x)` is meaningles
and for `x=0` the expression `15/(x^(3))` is meaningless.
Consequently the left hand side of the original equation ils meaningless for any `x epsilonR`. Therefore the equation has no roots.
(vii) We have `sqrt((x-3))-sqrt((x+9))=sqrt(x-1)`
This equation is defined for
`{(x-3ge0),(x+9ge0),(x-1ge0):}implies{(xge3),(xge-9),(xge1):}`
Hence `xge3`
For `xge3, sqrt(x-3)ltsqrt(x+9)` i.e. `sqrt((x-3))-sqrt((x+9))lt0`
Hence for `xge3`,the left hand side of the original equation is negative and right hand side is positive.
Therefore, the equation has no roots.