Question

Solve the absolute value of the following -

7 | 1/2x + 3 1/2 | - 2 = 5

Answer 1

So firstly, we need to isolate the absolute value onto one side. To do this, we must first add both sides by 2:

`7|(1)/(2)x+3 (1)/(2)|=7`

Next, divide both sides by 7:

`|(1)/(2)x+3 (1)/(2)|=1`

Now, we can split this equation to 2 equations:
`(1)/(2)x+3 (1)/(2)=1\ \textsf{and}\ (1)/(2)x+3 (1)/(2)=-1`

Let's start with the first equation. Firstly, subtract both sides by 3 1/2:

`(1)/(2)x=-2(1)/(2)\\\\(1)/(2)x=-(5)/(2)`

Lastly, multiply both sides by 2 and your first answer will be
`x=-5`

Now, to the second equation. Firstly, subtract both sides by 3 1/2:

`(1)/(2)x=-4(1)/(2)\\\\(1)/(2)x=-(9)/(2)`

Lastly, multiply both sides by 2 and your second answer will be
`x=-9`

Answer 2

`7 |(1)/(2)x + (7)/(2)| - 2 = 5`

+2 +2

`7 |(1)/(2)x + (7)/(2)|      = 7`

÷7 ÷7

`|(1)/(2)x + (7)/(2)|      = 1`

`(1)/(2)x + (7)/(2)= 1`
`(1)/(2)x + (7)/(2)= -1`

`-(7)/(2)`
`-(7)/(2)`
`-(7)/(2)`
`-(7)/(2)`

`(1)/(2)x = -(5)/(2)`
`(1)/(2)x = -(9)/(2)`

x = -5 x = -9

Answer: x = {-5, -9}