Question

How do you solve the equation `abs(3x-1)=10`?

Answer 1

Please see the explanation.

Explanation:

Use `|a| ={ (a; a>=0), (-a; a <0) :}`

Substitute `3x-1` for a:

`3x -1 = 10; 3x-1>=0 and -(3x-1) = 10; 3x-1<0`

Simplify the inequalities:

`3x -1 = 10; x>=1/3 and -(3x-1) = 10; x<1/3`

Solve the equations:

`3x -1 = 10; x>=1/3 and 3x-1 = -10; x<1/3`

`3x = 11; x>=1/3 and 3x = -9; x<1/3`

`x = 11/3; x>=1/3 and x = -3; x<1/3`

The equations do not violate the inequalities, therefore, both are equations are true:

`x = 11/3 and x = -3`

Check:

`|3(11/3)-1| = 10`
`|3(-3)-1| = 10`

`|11-1| = 10`
`|-9-1| = 10`

`|10| = 10`
`|-10| = 10`

`10 = 10`
`10 = 10`

This checks.