Question

PLZ HELP IM 100% LOST ON THIS

Answer 1

The equation
`f(x)=g(x)` is

`|x-4|-11=-√(5x)`

Some observations:

• 
  `√(5x)` is defined only as long as
  `5x\ge0`, or
  `x\ge0`
• wherever
  `√(5x)` is defined, its value must be non-negative, so that
  `-√(5x)` is never positive
• by the definition of absolute value, we have
  `|x-4|=x-4` if
  `x\ge4`, and
  `|x-4|=-(x-4)=4-x` if
  `x<4`. Then

`|x-4|-11=\begin{cases}x-15&\text{for }x\ge4\\-x-7&\text{for }x<4\end{cases}`

If
`x<4`, the equation becomes

`-x-7=-√(5x)\implies x+7=√(5x)`

Taking the square of both sides gives

`(x+7)^2=\left(√(5x)\right)^2\implies x^2+14x+49=5x\implies x^2+9x+49=0`

but since the discriminant is
`9^2-4\cdot1\cdot49<0`, there are no real solutions.

If
`x\ge4`, then

`x-15=-√(5x)`

Taking squares gives

`(x-15)^2=\left(-√(5x)\right)^2\implies x^2-30x+225=5x`

and solving by the quadratic formula gives two potential solutions,

`x=\frac{35\pm5√(13)}2`

which have approximate values of 8.49 and 26.51.

We know for any value of
`x` that
`g(x)\le0`. We have
`f(8.49)\approx-6.51` and
`f(26.51)\approx11.51`, so only the first solution 8.49 is valid.