Question

How do you solve `abs(7 + 2x) = 9`?

Answer 1

`x = 1, -8`.

Explanation:

As The Absolute Value of the Expression is 9, we will have to solve the equation twice, once for positive and once for negative.

As. `|a| = a = |-a|`.

So, Case 1 (Taking Positive):

`color(white)(xxx)(7 + 2x) = 9`

`rArr cancel7 + 2x cancel(- 7) = 9 - 7` [Subtract `7` from both sides]

`rArr 2x = 2`

`rArr (2x)/2 = 2/2` [Dividing both sides by `2`]

`rArr x = 1`

Case 2 (Taking Negative) :

`color(white)(xxx)-(7 + 2x) = 9`

`rArr - 7 - 2x = 9` [Distributive Property]

`rArr cancel(-7) - 2x + cancel7 = 9 + 7` [Add `7` to both sides]

`rArr -2x = 16`

`rArr (-2x)/-2 = 16/-2` [Dividing both sides by `-2`]

`rArr x = -8`

So, `x` has two values, `1, -8`.

Hope this helps.

Answer 2

`x=-8" or "x=1`

Explanation:

`"the expression inside the absolute value bars can be"`
`"positive or negative so there are 2 possible solutions"`

`color(magenta)"Positive expression"`

`7+2x=9`

`"subtract 7 from both sides and divide by 2"`

`rArr2x=9-7=2rArrx=2/2=1`

`color(magenta)"Negative expression"`

`-(7+2x)=9`

`rArr-7-2x=9`

`"add 7 to both sides and divide by "-2`

`rArr-2x=9+7=16rArrx=16/(-2)=-8`

`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.

`x=1to|7+2|=|9|=9`

`x=-8to|7-16|=|-9|=9`

`rArrx=-8" or "x=1" are the solutions"`