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#1 2012-05-06 22:15:29

mttal24
Member
Registered: 2012-05-01
Posts: 23

Linear Inequalities/Inequations Question

My question,

   
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I understood that, all terms have to go on LHS, to solve in form:

1. But how to proceed ahead? Please explain me step-by-step.
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2. When you reach the end, you get two values(here, fractions), say c and d. Then how do you decide whether:


or

Similarly,

or

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#2 2012-05-06 22:48:20

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Linear Inequalities/Inequations Question

Hi;

http://www.mathsisfun.com/algebra/inequ … lving.html

Here is some help:

x = 3 / 2 is a zero of the numerator and x =1 is undefined.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2012-05-07 07:20:19

Bob
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Posts: 10,058

Re: Linear Inequalities/Inequations Question

hi mttal24

Welcome to the forum!

If the question was

it would be easy to multiply by 3 and add 1 to both sides to isolate the 'x'.

The rules of algebra for equations would work fine for this inequality.

But with inequalities you always need to be on the alert for situations where you are multiplying (or dividing) by a negative.

Consider an easy example:

Add x to each side and take 5 from each side and this becomes

But look what happens if I try to do it by multiplying both sides by -1 and leave the < sign unchanged

This is incorrect! It happens because I have ignored the extra rule for inequalities

"If you multiply or divide by a negative, the inequality sign must be reversed."

So how does that effect your question?

You can make this

only if x-1 is not negative.

So let's assume x-1 > 0 and proceed

So, in this case, we want x > 1 AND x ≥ 1.5  Together these two are satisfied by x ≥  1.5

But now consider the case where we are multiplying by a negative

ie x - 1 < 0 => x < 1

The two restrictions on x here may be replaced by the tougher condition x < 1

So there are two regions that make solution sets for the inequality

x < 1 and x ≥ 1.5

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#4 2012-05-07 19:33:53

mttal24
Member
Registered: 2012-05-01
Posts: 23

Re: Linear Inequalities/Inequations Question

Thanks...
One thing more,
Can you show me a detailed procedure on how to solve inequations with the moduluses like | x-1 |

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#5 2012-05-07 20:10:54

Bob
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Posts: 10,058

Re: Linear Inequalities/Inequations Question

hi mttal24,

I find it helps to have a graph first.  Let's say you are trying to solve

Start with the graph of y = x - 1

Now convert this into the graph of    y = |x - 1|

Where y = x - 1 crosses the x axis there will be a change of direction as |x - 1| has no negative values.

You always end up with a V shaped graph.  (see diagram below)

Then draw in the line y = 3 and you can see the solution range.

To do this analytically solve two separate equations

x - 1 = 3 gives the right end limit.

1 - x = 3 gives the left end limit.

So we end up with

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#6 2012-05-07 21:13:56

mttal24
Member
Registered: 2012-05-01
Posts: 23

Re: Linear Inequalities/Inequations Question

^^ thanks. But how to do a little faster without a graph? I am seeing books which use zeroes of the linear equations given in numerator and denominator enclosed in the mods and then proceed.

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#7 2012-05-07 22:08:18

Bob
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Re: Linear Inequalities/Inequations Question

Sketch of graph is usually sufficient.  It doesn't have to be an accurate plot.

Post an example of a question and I'll say how I'd do it.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#8 2012-05-08 01:20:07

anonimnystefy
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From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Linear Inequalities/Inequations Question

You can also see intervals in which the expession inside the modulus brackets changes sign.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#9 2012-05-08 16:50:51

mttal24
Member
Registered: 2012-05-01
Posts: 23

Re: Linear Inequalities/Inequations Question

@bob bundy
I have two for you:

1. |x-1|/x+2<1

2. |x-1|+|x-2| ≥4

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#10 2012-05-08 18:50:00

Bob
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Re: Linear Inequalities/Inequations Question

hi mttal24

LATER EDIT: WHOOPS. I'VE SWITCHED A SIGN HERE AND ENDED UP DOING A SLIGHTLY DIFFERENT QUESTION.

(1)  I'll consider three separate cases.

     a) If x = 1.  |x -1| = 0 so the left hand side is certainly < 1

     b) If x > 1   |x -1| > 0 so we may ignore the || signs.

Between x = 1 and x = 2, x-2 is negative, so I need to split this into two cases.

     b1) x > 1 and x < 2 (the inequality sign changes as I multiply by the negative)

This is true so there are solutions here.

     b2) x > 2.  the inequality sign stays as it is

This cannot happen so there are no solutions for x > 2

     c) If x < 1 |x -1| < 0 so we can replace this expression with 1 - x and drop the || signs

         furthermore if x < 1 then x - 2 < 0 so when I multiply by x - 2 I must reverse the inequality sign.

I now have two restrictions on x with x < 1 automatically causing the second to be true, so x < 1 is the range.

Taking a) b1) and c) together we get x < 2

The function is undefined at x = 2

Bob

ps.  I've got to go out now for a while.  I'll look at the other one when I get back.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#11 2012-05-08 20:23:08

mttal24
Member
Registered: 2012-05-01
Posts: 23

Re: Linear Inequalities/Inequations Question

^^Thanks.
In the book, the answer given is
soln. set of the inequation is (-infinity,-2) union (-1/2, +infinity).
But, you have given x<2 which would mean (-infinity, 2)

Last edited by mttal24 (2012-05-08 22:10:28)

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#12 2012-05-08 23:41:08

Bob
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Posts: 10,058

Re: Linear Inequalities/Inequations Question

hi mtall24

you have given x<2 which would mean (-infinity, 2)

Yes, these are the same.  Brackets like this () mean the end points are not included.  If they are included you could do [].

(2)  |x-1|+|x-2| ≥4

The key points here are at x = 1 and x = 2.

So I'd consider 5 situations.

a) x = 1    You get 0 + 1 which is NOT ≥  4

b) x = 2    You get 1 + 0 so again this is NOT part of the solution.

c) x < 1 Both parts of the || are negative so you can replace the || like this

             1 - x + 2 - x  ≥ 4
              3 - 2x          ≥ 4
               -1               ≥ 2x
               -1/2            ≥ x

d) 1 < x < 2  The first || is Ok; the second must be switched

             x - 1 + 2 - x ≥ 4
                       1       ≥ 4   is NOT TRUE so no solutions here.

e)  x > 2   Both expressions are positive so may be left unchanged.

             x - 1 + x - 2 ≥ 4
                2x - 3       ≥ 4
                           2x ≥ 7
                             x ≥ 3.5

To summarise x ≤ -1/2 AND  x ≥ 3.5

Using brackets you could write this as

(-  ∞, -0.5]   AND [3.5, ∞)

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#13 2012-05-09 00:28:17

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Linear Inequalities/Inequations Question

Hi Bob

I think he is trying to say that the book got -2 and you got 2. Btw,the denominator of the expressiom is x+2.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#14 2012-05-09 00:33:26

Bob
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Registered: 2010-06-20
Posts: 10,058

Re: Linear Inequalities/Inequations Question

hi Stefy,

I see what you mean. 

I keep mis-reading questions.  Glasses Ok;  just the brain going screwy.  dizzy

Hopefully the method is OK and, with a model answer, he can do the correct one himself.

smile

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#15 2012-05-09 00:40:47

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Linear Inequalities/Inequations Question

bob bundy wrote:

just the brain going screwy.  dizzy

The mouth on you! They should wash your tongue with a soap! lolroflol

I think the method is okay. I don't,however,understand how they got one end to be at -1/2.

Stefy

Last edited by anonimnystefy (2012-05-09 00:41:05)


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#16 2012-05-09 02:37:36

Bob
Administrator
Registered: 2010-06-20
Posts: 10,058

Re: Linear Inequalities/Inequations Question

I've re-worked (1) with the plus sign and now I get

x < -2 and x > -1/2

Graph checks out as well.

And with post #11 tick.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#17 2012-05-09 17:08:07

mttal24
Member
Registered: 2012-05-01
Posts: 23

Re: Linear Inequalities/Inequations Question

Thanks. Finally I understood something...

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#18 2012-05-09 18:04:13

mttal24
Member
Registered: 2012-05-01
Posts: 23

Re: Linear Inequalities/Inequations Question

One last thing please,
Can you explain how to solve
|x-2|/x-2 ≥ 0
My ans is:x ≥2
But book's ans is: x>2

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#19 2012-05-09 19:35:04

Bob
Administrator
Registered: 2010-06-20
Posts: 10,058

Re: Linear Inequalities/Inequations Question

hi mttal24

If you got that it looks like you are getting the hang of it.  Well done!  smile

The reason for x > 2 rather than x ≥ 2 is because the function is undefined at x = 2.

At this value, both the numerator and denominator are zero so it is not possible to fix a value.

So it is necessary to exclude x = 2 from the solution.

If you look at the graph, you can see that there is no clear value at x = 2

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#20 2012-05-09 21:00:06

mttal24
Member
Registered: 2012-05-01
Posts: 23

Re: Linear Inequalities/Inequations Question

That means that the fx is undefined for denominator=0 because anything/0=undefined

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#21 2012-05-09 21:13:59

Bob
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Registered: 2010-06-20
Posts: 10,058

Re: Linear Inequalities/Inequations Question

Yes, that's it

If it's (non zero number) / zero, then this 'zooms' to infinity and that's not regarded as a 'value'.

If it's zero / zero then it could be anything because of the basic definition of division as the inverse of multiplication.

ie.  If a x b = c , then a = c / b.  this defines how division behaves.

But what if b and c are both zero?

Then we have

a x 0 = 0

There is no way you can say what 'a' is, so the full definition for division has to include " provided b is not zero "

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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