# Question regarding absolute equalities vs. absolute inequalities

If I have $$|3x| < x + 4$$, I break it into two cases and get that $$x<2$$ and $$x>-1$$ and the question is done (I think).

In my solutions I have another problem that asks me to solve $$|2x+3| = 2-x$$,

and it is solved in the same way, except once the two solutions are found, they are plugged back into the equation to see if L.S. = R.S., and only one is valid.

My question is why is this necessary for that second problem but not for the first one? I don't understand. Thanks in advance!

## 1 Answer

The extraneous solution are common when you square both sides an equation or an inequality to get rid of square roots or absolute values. In either case it is a good practice to check the result to see if all the solutions or intervals of solutions are satisfactory.

In you first example $$|3x| < x+4$$, we consider two cases,

1) $$x\ge 0$$ where the inequality is equivalent to $$3x, or $$0\le x <2$$

2)$$x<0$$ where the inequality is equivalent to $$-1

As the result the solution interval is $$-1

Now we test the intervals by plugging a test value in our inequality to see if we did it right.

For example for $$x=-2$$ we get $$6<2$$ which is false.

For your second example $$|2x+3|=2-x$$

We consider two cases.

1) $$2x+3<0$$ where the equation is equivalent to x=-5

2) $$2x+3\ge 0$$ where the equation is equivalent to $$x=-1/3$$

We check our results and fortunately both both solutions work.