Question

What is the solution set for the equation `|4a + 6| − 4a = 10` ?

Answer 1

`a = -2`

Explanation:

The first thing to do here is isolate the modulus on onse side of the equation by adding `4a` to both sides

`|4a + 6| - color(red)(cancel(color(black)(4a))) + color(red)(cancel(color(black)(4a))) = 10 + 4a`

`|4a + 6| = 10 + 4a`

Now, by definition, the absolute value of a real number will only return positive values, regardless of the sign of said number.

This means that the first condition that any value of `a` must satisfy in order to be a valid solution will be

`10 + 4a >= 0`

`4a >= -10 implies a >= -5/2`

Keep this in mind. Now, since the absolute value of a number returns a positive value, you can have two possibilities

• `4a + 6 < 0 implies |4a + 6| = -(4a+6)`

In this case, the equation becomes

`-(4a + 6) = 10 + 4a`

`-4a - 6 = 10 + 4a`

`8a = - 16 implies a = ((-16))/8 = -2`

• `(4a + 6) >=0 implies |4a + 6| = 4a + 6`

This time, the equation becomes

`color(red)(cancel(color(black)(4a))) + 6 = 10 + color(red)(cancel(color(black)(4a)))`

`6 != 10 implies a in O/`

Therefore, the only valid solution will be `a = -2`. Notice that it satisfies the initial condition `a >= -5/2`.

Do a quick check to make sure that the calculations are correct

`|4 * (-2) + 6| - 4 * (-2) = 10`

`|-2| +8 = 10`

`2 + 8 = 10 color(white)(x)color(green)(sqrt())`