Solve the following inequality and write your answer in interval notation. `|x-2|<=-7`Is the answer to this no solution??

Expert Answers
txmedteach eNotes educator| Certified Educator

Well, the answer to this one is pretty short intuitively, but I'll show you how to solve it if you didn't recognize off-the-bat that there's no way for this to have a solution.

Now, we're given


So, we have two cases:

1) if x-2 is positive

2) if x-2 is negative

Let's suppose that x-2 is positive (or zero for completeness), then

`|x-2| = x-2`

Our inequality then becomes:

`x-2 <=-7`

Now, we solve by adding two to both sides:

`x <=-5`

Of course, this contradicts our earlier assumption that x-2 is positive, so this cannot be part of the solution.

Now, let's suppose the second case, where `x-2<0` :

This would mean our absolute value becomes:

`|x-2| = -(x-2) = 2-x`

So our inequality would become:

`2-x <=-7`

Solving by subtracting 2:

`-x <= -9`

Now, we divide by -1 (remember, when you multiply or divide by a negative number you flip the inequality sign):

`x >= 9`

However, we hit a contradiction again! We assumed in doing this part that x-2 is negative. This does not hold for the solution set that we found. Therefore, this cannot be part of the solution.

So, in both of our cases, there was no solution. Therefore, there is no solution for the overall case.

The easy way to think of this? You're looking at an absolute value. The range of the absolute value function is `[0,oo)`. The problem asks you to find what number gives you a negative absolute value, which can't exist! Therefore, there is no solution!

I hope that helps!