Question

How do you solve the equation `abs(4/3-2/3x)=3/4`?

Answer 1

`x=25/8` and `7/8`

Explanation:

Absolute value equations are a little tough. Some teachers say that they "make numbers positive", but that doesn't mean `x` is always a positive number. Absolute value bars concern distance. That's why they "make the number positive"; because there's no such thing as a negative distance (we can't have `-262` feet).

When we have `sqrt(x)=y`, we take the inverse of the square root to "undo" it. But what's the inverse of absolute value bars? Nothing. But, we do know that whatever `x` equals, it can be either positive or negative.

So, the way we solve for absolute value equations is to let one equation be positive and another be negative:

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

Positive situation
`(4/3-2/3x)=3/4`

subtract `4/3` on both sides

`-2/3x=-7/12`

divide by `-2/3`

`x=7/8`

Negative solution
`-(4/3-2/3x)=3/4`

divide by `-1` on both sides

`4/3-2/3x=-3/4`

subtract `4/3` on both sides

`-2/3x=-25/12`

divide by `-2/3` on both sides

`x=25/8`

So, our solutions are `x=25/8` and `7/8`. Just to double check, let's grahp our equation:

graph{y=abs(4/3-2/3x)-3/4}

Yep, intercepts at `0.875` and `3.125`, or `7/8` and `25/8`.

Answer 2

`x=7/8" or " x=25/8`

Explanation:

`"the value of the "color(blue)"absolute value function"` is always positive.

`"However, the value of the expression inside the bars"`
`"can be positive or negative"`

`"This means there are 2 possible solutions to the equation"`

`color(red)(+-)(4/3-2/3x)=3/4`

`color(blue)"First possible solution"`

`4/3-(2x)/3=3/4larrcolor(red)" positive value"`

`rArr(2x)/3=4/3-3/4=7/12`

`rArr24x=21larrcolor(red)" cross-multiplying"`

`rArrx=21/24=7/8larrcolor(magenta)" first possible solution"`

`color(blue)"Second possible solution"`

`(2x)/3-4/3=3/4larrcolor(red)" negative value"`

`rArr(2x)/3=3/4+4/3=25/12`

`rArr24x=75larrcolor(red)" cross-multiplying"`

`rArrx=75/24=25/8larrcolor(magenta)" second possible solution"`

`color(blue)"As a check"`

Substitute these values into the left side of the equation and if equal to the right side then they are the solutions.

`|4/3-(2/3xx7/8)|=|4/3-7/12|=|3/4|=3/4`

`|4/3-(2/3xx25/8)|=|4/3-25/12|=|-3/4|=3/4`

`rArrx=7/8" or " x=25/8" are the solutions"`