Question

How do you solve `(4x)/(x+7)<x`?

Answer 1

`x > 0` or `x in (-3,-7)`

Explanation:

Note: for the given expression to be meaningful `x!=-7`
and `x!=0` (since this would make both sides equal).

Case 1: `color(black)(x > 0)`

`color(white)("XXX")(4x)/(x+7) < x`

`color(white)("XXX")rarr 4/(x+7) < 1`

`color(white)("XXX")rarr 4 < x+7`

`color(white)("XXX")rarr x > -3`

`color(white)("XXX")`but by the Case 1 limitation `x > 0` (which is more limiting).

Case 2: `color(black)(x < 0)`

`color(white)("XXX")(4x)/(x+7) < x`

`color(white)("XXX")rarr 4/(x+7) > 1` (dividing by a negative reverses the inequality)

`color(white)("XXX"){: (color(black)("Case 2a: "),color(white)("XX"),color(black)("Case 2b: ")), (color(white)("X")color(black)(x < -7),,color(white)("X")color(black)(x > -7)), (color(white)("XX")4/(x+7) > 1,,color(white)("XX")4/(x+7) > 1), (color(white)("XX")rarr 4 < x+7,,color(white)("XX")4 > x+7), (color(white)("XX")rarr -3 < x,,color(white)("XX")-3 > x), (color(white)("XX")"impossible; since ",,color(white)("XX")rarr x in (-7,-3)), (color(white)("XX")color(white)("XXXX")x < 7,,) :}`

The following graph image might help understand this relationship.
(Note the asymptote for `color(red)((4x)/(x+7))`)

The region for which `color(red)((4x)/(x+7)) < color(blue)(x)` is shaded in `color(green)("green")`