Question

How do you solve `2/3|x-4| + 2=5` for `x`?

Answer 1

`x=17/2` or `x=-1/2`

Explanation:

Given:
`color(white)("XXX")2/3abs(x-4)+2=5`

Step 1: Isolate the absolute value:
`color(white)("XXX")2/3abs(x-4)=5-2=3`

`color(white)("XXX")abs(x-4) = 3xx3/2 = 9/2`

Step 2: Evaluate the two possibilities for the absolute value argument

#{: ("if "(x-4) >= 0,color(white)("XXXXXX"),"if "(x-4) < 0), (rarr abs(x-4)=x-4,,rarr abs(x-4)=4-x), (rarr x-4=9/2,,rarr 4-x=9/2), (rarr x=17/2,,rarr -x=1/2), (,,x=-1/2) :}#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Step 3: (Optional but Desirable) Verify for the two solutions:

`2/3abs(x-4)+2= ?`

#color(white)("XXX"){: ("if " x=17/2,color(white)("XXXXXX"),"if " x=-1/2), (2/3abs(x-4)+2,,2/3abs(x-4)+2), (color(white)("XX")=2/3abs(17/2-4)+2,,color(white)("XX")=2/3abs(-1/2-4)+2), (color(white)("XX")=2/3(9/2)+2,,color(white)("XX")=2/3abs(-9/2)+2), (color(white)("XX")=3+2,,color(white)("XX")=3+2), (color(white)("XX")=5,,color(white)("XX")=5) :}#

Both answers are valid solutions: