Question

How do you solve `-3| a - 1| = - 15`?

Answer 1

-4 and 6.

Explanation:

Start by isolating the absolute value (it contains the variable you're solving for, a.)

Dividing both sides by -3 we get |a-1| = 5

Then, split this into two parts:

`a-1 = 5`
and
`a-1 = -5`

Solve for a in both of those, and you should have your answers.

The reason we split it into two equations to remove the absolute value symbols is because those symbols make a negative number positive. If the variable inside is negative, it will be made positive. This must be reflected when we're solving an equation like this.

For example, `|-30| = 30`.
But `|30| = 30` too.
Now imagine if a variable a was in those bars.
`|a| = ?` The answer could be positive OR negative!

Hope this makes sense.

Answer 2

`a=-4 " " " " \text{or} " " " " a=6`

are the required solutions.

Explanation:

`-3|a-1|=-15`

Divide both sides by `-3` to isolate the absolute value on the left hand side:

`\frac{-3|a-1|}{-3}=\frac{-15}{-3}`

Simplify:

`|a-1|=5`

For an absolute value equation, there are two solutions.

`|f(a)|=a " " rightarrow " " f(a)=-a " " " " \text{or} " " " "f(a)=a`

`a-1=-5 " " " " \text{or} " " " " a-1=5`

`a=-4 " " " " \text{or} " " " " a=6`

That's it!