Inequality |y||y+1|

Question

Hello,

I am unable to understand how to find values that satisfy the following inequality
|y|>|y+1|

Request your help.

Rgds
SBS

Smallwonder · Answer

Hi, Could you please provide the complete question (with the options) ?

/SW

santoshbs · Answer

This is what was mentioned:
if y>=0, y+1>=0, y>y+1, no solution.
if y<0, y+1<0, -y>-(y+1), solution is y<-1
if y>=0, y+1<0, y>-(y+1), no solution.
if y<0, y+1>=0, -y>y+1, solution is -1<=y<-1/2
So your final solution is y<-1/2

Bunuel · Answer

ABSOLUTE VALUE PROPERTIES: When x \leq {0} then |x|=-x , or more generally when some \ expression\leq{0} then |some \ expression|={-(some \ expression)} . For example: |-5|=5=-(-5) ; When x \geq {0} then |x|=x , or more generally when some \ expression\geq{0} then |some \ expression|={some \ expression} . For example: |5|=5 . BACK TO THE QUESTION: |y|>|y+1| APPROACH #1: |y|>|y+1| --> we have two transition points for 0 and -1 (transition point is the value of y for which an expression in the modulus equals to zero). Thus we have three ranges to check: 1. y<-1 --> in this case |y|=-y and |y+1|=-(y+1) , so we'd have -y>-(y+1) --> 0>-1 --> TRUE. So, when y<-1 inequality |y|>|y+1| holds true. 2. -1\leq{y}\leq{0} --> in this case |y|=-y and |y+1|=y+1 , so we'd have -y>y+1 --> y<-\frac{1}{2} . So, when -1\leq{y}<-\frac{1}{2} inequality |y|>|y+1| holds true. 3. y>0 --> in this case |y|=y and |y+1|=y+1 , so we'd have y>y+1 --> 0>1 --> FALSE. So, when y>0 inequality |y|>|y+1| has no solution. So, we have that |y|>|y+1| holds true for y<-\frac{1}{2} (combine true ranges from 1 and 2). APPROACH #2: Since both parts of the inequality are non-negative, then we can square it: y^2>y^2+2y+1 --> y<-\frac{1}{2} . Theory on Inequalities: x2-4x-94661.html#p731476 inequalities-trick-91482.html data-suff-inequalities-109078.html range-for-variable-x-in-a-given-inequality-109468.html everything-is-less-than-zero-108884.html graphic-approach-to-problems-with-inequalities-68037.html All DS Inequalities Problems to practice: search.php?search_id=tag&tag_id=184 All PS Inequalities Problems to practice: search.php?search_id=tag&tag_id=189 700+ Inequalities problems: inequality-and-absolute-value-questions-from-my-collection-86939.html Theory on Abolute Values: math-absolute-value-modulus-86462.html DS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=37 PS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=58 Hard set on Abolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html Hope this helps. P.S. Please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Thank you.

santoshbs · Answer

Thank you for the awesome explanation!

Inequality |y|>|y+1|

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