How do you solve `abs(7-2x)=5`?

The use of the special brackets of | | means that whatever is inside them is considered as positive.

On the right side of the equals we have +5.

So the left side must end up as `|+-5|` giving:

`|+-5|=+5`

Ok!

Lets consider what will turn `7-2x` into -5

Set `color(white)("d")7-2x=-5`

`2x=7+5`

`x=12/2=+6`

Check `(color(white)(2/2)7-[2xx6]color(white)("d") ) -> 7-12 = -5` as required

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Lets consider what will turn `7-2x` into +5

Set `color(white)("d")7-2x=+5`

`2x=7-5`

`x=2/2=1`

Check `(color(white)(2/2)7-[2xx1]color(white)("d")) -> 5xx1=+5` as required

The absolute value function takes any term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent.

Solution 1

`7 - 2x = -5`

`-color(red)(7) + 7 - 2x = -color(red)(7) - 5`

`0 - 2x = -12`

`-2x = -12`

`(-2x)/color(red)(-2) = (-12)/color(red)(-2)`

`(color(red)(cancel(color(black)(-2)))x)/cancel(color(red)(-2)) = 6`

`x = 6`

Solution 2

`7 - 2x = 5`

`-color(red)(7) + 7 - 2x = -color(red)(7) + 5`

`0 - 2x = -2`

`-2x = -2`

`(-2x)/color(red)(-2) = (-2)/color(red)(-2)`

`(color(red)(cancel(color(black)(-2)))x)/cancel(color(red)(-2)) = 1`

`x = 1`

The Solutions Are: `x = 6` and `x = 1`