How should I solve the following inequality?

[TheEmojiSlayer2005]

How should I solve such an inequality that contains multiple absolute values?

 

[Dr.Peterson]

View attachment 36192
How should I solve such an inequality that contains multiple absolute values?

My first thought is to rewrite the LHS in piecewise form, and solve something like 5 cases.

What have you tried?

 

[blamocur]

View attachment 36192
How should I solve such an inequality that contains multiple absolute values?

$$|u|+|v| \geq u+v$$

 

[Dr.Peterson]

$$|u|+|v| \geq u+v$$

That's clearly what you are intended to notice; they designed the problem carefully. But, of course, most problems that look similar to this one would either need a different related inequality, like $|u|+|v| \geq u-v$, or would require a general method such as I suggested.

So my answer is perhaps better for the general question as asked, "How should I solve such an inequality that contains multiple absolute values?", but yours is far more insightful for the particular questions.