Solve 1/y>=-2: y<=-0.5 (-∞,-0.5)

[cloud360]

Homework Statement

solve

1/y>=-2

Homework Equations

The Attempt at a Solution

why is the answer y<=-0.5, with set notation: (-infinity,-0.5)

My solution:

1/y>=-2 (take y to other side)
1>=-2y (divide by -2 and flip symbol)
-1/2<=y
y>=-0.5
buy this is wrong according to my teachers solution

 

[vishal007win]

in your solution the first step you did was wrong,
from
1/y>=-2
u can't do
1>=-2y

 

[cloud360]

in your solution the first step you did was wrong,
from
1/y>=-2
u can't do
1>=-2y

why. i don't understand, can you pelase explain?


if you divide by a negative you change the symbol. but in my first part, i am not diving by a negative !

but can you explain why you can't go from

1/y>=-2

to

1>=-2y

 

[Tedjn]

Because y might be negative. If y is positive, then your reasoning works to conclude that y >= -0.5 (certainly true because we assumed y is positive to begin with). So what are the solutions to the inequality if y is positive? If y is negative, then your reasoning fails (why?)

 

[cloud360]

how come you DO NOT changing symbol when dividing 1 by -2 ! see bold part. why is my symbol changing wrong

1/y>=-2 (take y to other side)
1>=-2y (divide by -2 and flip symbol)
-1/2<=y
y>=-0.5

 

[Tedjn]

I don't understand. You are flipping the symbol, which is what you are supposed to do, because you are dividing by a negative number on both sides.

 

[cloud360]

I don't understand. You are flipping the symbol, which is what you are supposed to do, because you are dividing by a negative number on both sides.

so if my flipping symbol is correct. then answer must be

y>=-0.5

 

[Tedjn]

No, because the part vishal pointed out was incorrect was actually the jump between the two bolded statements below:

1/y>=-2 (take y to other side)
1>=-2y (divide by -2 and flip symbol)
-1/2<=y
y>=-0.5

 

[cloud360]

No, because the part vishal pointed out was incorrect was actually the jump between the two bolded statements below:

1/y>=-2 (take y to other side)
1>=-2y (divide by -2 and flip symbol)
-1/2<=y
y>=-0.5

edit, made error:

how come you cannot do this. how come you can't mutiply by y?

e.g if i had 1/x>-2...wont i be able to mutiply by sides by x

 

[Tedjn]

If you had 1/x > -2, i.e. you renamed your variable, then you aren't dealing with y any more. You cannot multiply by y and keep the sign unchanged, because you do not know whether the solutions y you are looking for are positive or negative. If they are negative, your sign would need to be switched.

The best you can do is say, if y is positive, then a series of steps leads to these (positive) solutions (in your case, all positive numbers that are greater than -0.5). Then, you can say, if y is negative, another series of steps leads to these additional (negative) solutions.

 

[cloud360]

If you had 1/x > -2, i.e. you renamed your variable, then you aren't dealing with y any more. You cannot multiply by y and keep the sign unchanged, because you do not know whether the solutions y you are looking for are positive or negative. If they are negative, your sign would need to be switched.

The best you can do is say, if y is positive, then a series of steps leads to these (positive) solutions (in your case, all positive numbers that are greater than -0.5). Then, you can say, if y is negative, another series of steps leads to these additional (negative) solutions.

so if the other side is negative. should i change the symbol.

because this question just can not be solved with the algebra i have been thought for last 7 years

 

[Mark44]

You can multiply by y, but since you don't know the sign of y, you have to look at two cases: one where y < 0 and one where y > 0.

If y < 0,
1/y >= -2
==> 1 <= -2y (multiplying by y changes the inequality direction)
==> -1/2 >= y (dividing by -2 changes the inequality direction again)
or y <= -1/2

Now, if y > 0,
1/y >= -2
==> 1 >= -2y (multiplying by y does not change the inequality direction)
==> -1/2 <= y (dividing by -2 changes the inequality direction)
or y >= -1/2, but since we stipulated that y > 0, we have
y > 0

The net result is the set {y | y <= -1/2 or y > 0}

 

[cloud360]

You can multiply by y, but since you don't know the sign of y, you have to look at two cases: one where y < 0 and one where y > 0.

If y < 0,
1/y >= -2
==> 1 <= -2y (multiplying by y changes the inequality direction)
==> -1/2 >= y (dividing by -2 changes the inequality direction again)
or y <= -1/2

Now, if y > 0,
1/y >= -2
==> 1 >= -2y (multiplying by y does not change the inequality direction)
==> -1/2 <= y (dividing by -2 changes the inequality direction)
or y >= -1/2, but since we stipulated that y > 0, we have
y > 0

The net result is the set {y | y <= -1/2 or y > 0}

EXCELLENT EXPLANATION !

ok, please can you kindly tell me...would i look at the 2 cases where the variable can be positive(+) or negative(-). if i had:

1)1/x>=-2
2)1/(x+1)>=-2
3)1/(y+1)>=positive 2
4)1/(y)>=positive 2

 

[Mark44]

1. yes
2. yes
3. yes
4. yes

In short, whenever you multiply both sides of an inequality by a variable, you need to look at the two cases.

 

[cloud360]

1. yes
2. yes
3. yes
4. yes

In short, whenever you multiply both sides of an inequality by a variable, you need to look at the two cases.

ok. i applied what you just thought me. but it isn't working for 1/(x+1)>=-2

my solution:

if x>0
===>1/(x+1)>=-2
===>x>=1.5


if x<0
===>1/(-x+1)>=-2
===>1>=-2x-2
===>3>=-2x
===>-1.5<=2x
===>x>=-1.5

the answer is wrong according to this website

http://www3.wolframalpha.com/input/?i=1/(x+1)>=-2

 

[cloud360]

its very strange. the answer for x>0 worked out algebraically is always the answer for x<0 but with opposite symbol

 

[Mark44]

For 1/(x + 1) >= -2, the two cases are x + 1 < 0 and x + 1 > 0. Those inequalities are equivalent to, respectively, x < - 1 and x > -1. Notice that we're not concerned with x + 1 = 0, because we would be dividing by zero.