addition and multiplication principles

tina4545

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Aug 11, 2010
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I need to solve this by Friday 8-13-10

2/3-x/5<4/15 using the addition and multiplication principles

Any help would be greatly appreciated.
Thank You
 
tina4545 said:
I need to solve this by Friday 8-13-10

2/3-x/5<4/15 using the addition and multiplication principles

Any help would be greatly appreciated.
Thank You


Hi tina4545,

23x5<415\displaystyle \frac{2}{3}-\frac{x}{5}<\frac{4}{15}

First, multiply all the terms by the least common denominator which is 15.

15(23)15(x5)<15(415)\displaystyle 15\left(\frac{2}{3}\right)-15\left(\frac{x}{5}\right)<15\left(\frac{4}{15}\right)



 
23x5 < 415, solve for x.\displaystyle \frac{2}{3}-\frac{x}{5} \ < \ \frac{4}{15}, \ solve \ for \ x.

x5 < 41523\displaystyle -\frac{x}{5} \ < \ \frac{4}{15}-\frac{2}{3}

x5 < 25\displaystyle -\frac{x}{5} \ < \ -\frac{2}{5}

x < 2\displaystyle -x \ < \ -2

\(\displaystyle Hence, \ x \ > \ 2, \ note, \ when \ we \ divide \ or \ multiply \ by \ -1 \ (or \ any \ negative \ number), \ we \ change \\)

the sense of the inequality, why?\displaystyle the \ sense \ of \ the \ inequality, \ why?

Ill leave the check up to you.\displaystyle I'll \ leave \ the \ check \ up \ to \ you.
 
To continue:\displaystyle To \ continue:

We have x < 2 and we want to rid ourselves of that pesky negative sign on x.\displaystyle We \ have \ -x \ < \ -2 \ and \ we \ want \ to \ rid \ ourselves \ of \ that \ pesky \ negative \ sign \ on \ x.

What we really have is (1)x < (1)2, the one being understood.\displaystyle What \ we \ really \ have \ is \ (-1)x \ < \ (-1)2, \ the \ one \ being \ understood.

Proposition One: Aha, I know, Ill multiply both sides by 1, I can do that.\displaystyle Proposition \ One: \ Aha, \ I \ know, \ I'll \ multiply \ both \ sides \ by \ -1, \ I \ can \ do \ that.

Hence, (1)(1)x < (1)(1)2      x < 2, problem solved, no more pesky  sign.\displaystyle Hence, \ (-1)(-1)x \ < \ (-1)(-1)2 \ \implies \ x \ < \ 2, \ problem \ solved, \ no \ more \ pesky \ - \ sign.

But wait a minute if x = 3, then 3 < 2, but 3 < 2, I dont think so.\displaystyle But \ wait \ a \ minute \ if \ x \ = \ 3, \ then \ -3 \ < \ -2, \ but \ 3 \ < \ 2, \ I \ don't \ think \ so.

Naughty, naughty, you are a bad person, get thee hence to bed with no supper.\displaystyle Naughty, \ naughty, \ you \ are \ a \ bad \ person, \ get \ thee \ hence \ to \ bed \ with \ no \ supper.

Proposition Two: x < 2, x + 2 < 2 +2      x+2 < 0, and\displaystyle Proposition \ Two: \ -x \ < \ -2, \ -x \ + \ 2 \ < \ -2 \ +2 \ \implies \ -x+2 \ < \ 0, \ and

xx+2 < 0+x      2 < x or x > 2.\displaystyle x-x+2 \ < \ 0+x \ \implies \ 2 \ < \ x \ or \ x \ > \ 2.

Now, if x = 3, then 3 < 2 true and 3 > 2 true.\displaystyle Now, \ if \ x \ = \ 3, \ then \ -3 \ < \ -2 \ true \ and \ 3 \ > \ 2 \ true.

Hence, the second proposition is true and the first is false.\displaystyle Hence, \ the \ second \ proposition \ is \ true \ and \ the \ first \ is \ false.

However, mathematicians, realizing this use a shorthand method, knowing that if you\displaystyle However, \ mathematicians, \ realizing \ this \ use \ a \ shorthand \ method, \ knowing \ that \ if \ you

divide or multiply an inequality by a negative number, you change the sense of the\displaystyle divide \ or \ multiply \ an \ inequality \ by \ a \ negative \ number, \ you \ change \ the \ sense \ of \ the

inequality, thereby avoiding the unecessary (necessary to the novice) grunt work.\displaystyle inequality, \ thereby \ avoiding \ the \ unecessary \ (necessary \ to \ the \ novice) \ grunt \ work.

In other words, if you have x < 2, multiply both sides by 1 and change the sense.\displaystyle In \ other \ words, \ if \ you \ have \ -x \ < \ -2, \ multiply \ both \ sides \ by \ -1 \ and \ change \ the \ sense.

x < 2,      x > 2 and x < 2      x > 2.\displaystyle -x \ < \ -2, \ \implies \ x \ > \ 2 \ and \ -x \ < \ 2 \ \implies \ x \ > \ -2.
 
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