Question

How do you solve `| 1/2-3/4x | + 1=3`?

Answer 1

`-`2 and 10/3.

Explanation:

The given equation is equivalent to the pair 1/2`-`3 x/4 = `+-`2.

Answer 2

`x = -2" or "x=10/3" "`Note that `10/3 = 3.33bar3`

Explanation:

The very nature of this type of problem is that it usually has 2 solutions for `x`

We need to end up with the 'Absolute' part of the equation on its own and on the left of the equals sign and everything else on the other.

Given:`" "color(brown)(|1/2-3/4 x|+1=3)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Simplifying")`

Subtract `color(blue)(1)` from both sides

`" "color(brown)(|1/2-3/4 x|+1color(blue)(-1)" "=" "3color(blue)(-1))`

`" "|1/2-3/4 x|+0" "=" "2`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Solving for "x)`

Think of this as:

`" "|1/2-3/4 x|" "=" "|+-2|`

`color(brown)("Consider: "1/2-3/4 x" "=" "-2" "`

Multiply by (-1 ) giving

`" "-1/2+3/4 x=+2`

`" "3/4 x=2+1/2`

`color(blue)(" " x" "=" "4/3xx5/2" " = " "10/3)`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(brown)("Consider: "1/2-3/4 x=+2`

Multiply by (-1 ) giving

`" "-1/2+3/4 x=-2`

`" "3/4 x=-2+1/2`

`" "color(blue)(x=4/3xx(-3/2) = -2)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`" "color(magenta)(x = -2" or "x=10/3)`

Tony B