Lionila wrote:
Dear Karishma,
First of all thank you for offering this forum!
I am struggling with inequalities that require squaring such as a>b where I cannot be sure whether a+b is <> or = 0.
One example
√(x-1) > x-3
I know that a>= 0 but I do not know whether a+b is >< or = 0
I also know that x>=1
I believe that I need to solve this question considering each case as follows:
case a+b>0 --> I square and do not flip the sign!
I get: 0 > (x-5)(x-2)
solution 2<x<5
I test a number from this range and the original inequality holds true so I conclude that this is one correct result.
Is it correct that I need to test the range? I assume this because a+b>0 is only a hypothesis and not a fact.
case a+b<0 --> I square and flip the sign because at least one side is negative
I get: 0<(x-5)(x-2)
solution: x>5 and 1=<x<2
I test a number in each range. The range x>5 does not hold the inequality true. But the second range 1=<x<2 does. How is this possible?
1) a+b can only be > or < than 0 so I assume that only the first or the second range can be valid, or not?
2) Also within the same case a+b<0, why is 1=<x<2 valid and x>5 invalid?
case a+b = 0
I get: 0=(x-5)(x-2)
solution x = 2 or 5
I test both roots but only 2 is valid.
Overall I conclude that 1=<x<5 is the solution.
Thanks for commenting/clarifying on my approach, solution, assumptions and inferences.
Thanks upfront!
Lionila
When you have inequalities such as
x > y
in which x or y may be positive or negative, you cannot square both sides. You do not know whether the inequality will flip or not.
e.g.
4 > 3
You can square without flipping the inequality sign.
4 > -3
You can square without flipping the inequality sign.
4 > -6
When you square, the inequality sign flips.
-4 > -5
When you square, the inequality sign flips.
When both sides are positive, squaring does not change the inequality sign.
When both sides are negative, squaring flips the inequality sign.
When one is positive, one is negative, you do not know whether the inequality sign will be flipped or not.
As for this question, send me the link of the actual question and I will let you know how to solve it.
I know this is true from the values but doesn't this violate the basic rule in inequalities "If multiplying with a negative number, you must flip the sign"? The sign does not flip here even though we multiply the inequality with (-3).
and I cannot find the source anymore. I understand if you don't want to go through it then but if you find the time, I would greatly appreciate it.