Question

How do you solve `11+ | - 2+ x | = 30`?

Answer 1

Always you have a equation with absolute values involved, use the definition of absolute value function and solve the equation using all the possibilities.

Explanation:

You must remember that absolute value function definition's is:

Then, when you have a inequation to solve like this:

`11+|-2+x|=30`

you must operate like you're solving a first degree equation just to obtain a expression with the absolute value lonely in one of two members of equality. I.e.:

`11+|-2+x|=30 rArr |-2+x|=30-11 rArr`

`rArr |-2+x|=19`

Then, you must apply the absolute value definition. This means that we have two possibilities:

(1) If `-2+x < 0` (i.e. `x < 2`) then:

`|-2+x| = - (-2+x) = 2-x rArr 2-x=19 rArr`

`rArr x=-17`

(2) If `-2+x >= 0` (i.e. `x >= 2`) then:

`|-2+x|=-2+x rArr -2+x=19 rArr`

`rArr x=21`

Once you've calculated the two possibilities, you have to check the answers. They must fit the conditions we have imposed to eliminate the absolute value and, of course, should also verify the original inequation.

What means that? In our problem, this means that the two expressions:

`x < 2` and `x=-17`

must match each other, as they do it. In addition, if we replace the value `x=-17` in the original inequation, we obtain:

`11+|-2-17|=30`,

a expression that becomes true. Then, the answer `x=-17` is a valid solution of the problem.

In the same way, we must check if

`x >= 2` and `x=21`

mach each other. They do it, again, then `x=21` must be checked in the original inequation:

`11+|-2+21|=30`

In this case, we also obtain a true expression, so that the solution is valid. In short, after resolving and check solutions, we can say that there are two values of `x` that satisfy the inequality, the values `x = -17` and `x = 19`.