Inequality homework problem

  • #1
210
1

Homework Statement


[itex]\frac{3x + 1}{2x - 6}[/itex] < 3


Homework Equations





The Attempt at a Solution



[itex]\frac{3x + 1}{2(x -3}[/itex] < 3

[itex]\frac{3x +1}{x - 3}[/itex] < 6

Assume x < 3

3x + 1 > 6(x - 3)
3x + 1 > 6x - 18
3x + 1 - 6x + 18 > 0
19 > 3x
x < 19/3

No Contradiction.


Assume x > 3
3x + 1 < 6x - 18
3x - 19 > 0
3x > 19
x > 19/3

No Contradiction




How do I know which set of values to take?

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
uart
Science Advisor
2,797
21


Homework Statement


[itex]\frac{3x + 1}{2x - 6}[/itex] < 3


Homework Equations





The Attempt at a Solution



[itex]\frac{3x + 1}{2(x -3}[/itex] < 3

[itex]\frac{3x +1}{x - 3}[/itex] < 6

Assume x < 3

3x + 1 > 6(x - 3)
3x + 1 > 6x - 18
3x + 1 - 6x + 18 > 0
19 > 3x
x < 19/3

No Contradiction.


Assume x > 3
3x + 1 < 6x - 18
3x - 19 > 0
3x > 19
x > 19/3

No Contradiction

How do I know which set of values to take?
[

Hi Darth. You need to consider both the assumption and the consequent result in determining the solution region.

For example {x < 3 and x < 19/3} means that x must be less than 19/3, but also less than 3, but this is equivalent to simply stating x < 3.
 
  • #3
uart
Science Advisor
2,797
21


For your second solution region, x must be greater than 3, but also greater than 19/3. This is equivalent to simply stating [itex] x > \ldots[/itex]. (You fill that one in. :smile:)
 
  • #4
210
1


I attempted a different method

Is the correct answer

x < 3
x > 19/3
 
  • #5
uart
Science Advisor
2,797
21


Yes that is correct.

BTW. What was your "different method". One alternative is to multiply both sides by the square of the denominator, which makes it into a single quadratic inequality.
 
  • #6
210
1


Yeah, that was what I did
Then tested the point
 
  • #7
CAF123
Gold Member
2,946
88


Could also do [tex] \frac{3x+1 - 3(2x-6)}{2x-6}<0 [/tex] simplify and then draw up an algebraic table to see when each of the quantities (3x-19), (2x-6) are negative, zero or positive (or undefined)for values of x less than 3, 3, 3<x<19/3, 19/3 and greater than 19/3.
 

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