solve for x if (2x-1)/(3x+1)<2?

Hopefully you aren't going with the other answers because none of them are correct.

Let's look at the denominator first.

3x + 1 = 0
3x = -1
x = -1/3

So this is a vertical asymptote. We have to be careful to split this into two cases. You can't just blindly multiply both sides by 3x + 1, because you need to know if it is positive or negative.

Case 1:
3x + 1 is positive (when x > -1/3)
If you multiply both sides by (3x + 1) where it is positive, you don't have to change the sign.

2x - 1 < 2(3x + 1)
2x - 1 < 6x + 2
-4x < 3
Now you are multiplying by a negative, so reverse the sign:
x > -3/4

So you need x > -1/3 and x > -3/4. If you draw this on a number line, you'll see this is equivalent to just saying:
x > -1/3

Case 2:
3x + 1 is negative (x < -1/3)

If you multiply both sides by (3x + 1) where it is negative, you must change the sign.

2x - 1 > 2(3x + 1)
2x - 1 > 6x + 2
-4x > 3
Again change the direction of the sign.
x < -3/4

So now you have:
x < -1/3 and x < -3/4
Again this simplifies to just:
x < -3/4

Combining the cases you have:
x < -3/4
or
x > -1/3

Graphing on a number line it looks like this:
<=== -1 ==( -3/4 ) --- -1/2 --- (-1/3)======= 0 ======>

Edit:
Okay, it's going to be much easier to show if I graph it. Here's the graph of the function in purple. The line y = 2 is shown in green going horizontally. Then there is a vertical asymptote at -1/3. All the places where the function is below the line y = 2 are shown in red (below -3/4 and above -1/3).

2x - 1 < 6x + 2
- 3 < 4x
x > - 3 / 4

solve for x if (2x-1)/(3x+1)<2

(2x-1)/(3x+1)<2

(2x-1)<2(3x+1)

2x - 1 < 6x + 2

-1 - 2 < 6x - 2x

-3 < 4x

-3/4 < x
or

x > -3/4

(2x-1)/(3x+1)<2 ; { 3x+1 > 0 ; x > -1/3 }
tip: we should multiply positive number in both sides

(3x+1)* [(2x-1)/(3x+1)] < (3x+1)*2 ;

2x-1 < 6x +2 ;
-4x < 3;
4x > -3;

answer:
x > -3/4 and x > -1/3
so the correct domain for x is x > -1/3

(2x -1) / (3x +1) < 2 / 1

1(2x -1) < 2(3x + 1)

2x -1 < 6x + 2

-1 -2 < 6x -2x

-3 < 4x

-3/4 <x

x > -3/4

:)

(2x-1)<2(3x+1)

2x - 1 < 6x + 2

-1 - 2 < 6x - 2x

-3 < 4x

-3/4 < x

hi,

(2x-1)/(3x+1)<2 --> multiply both sides by 3x+1.
2x-1 < 2(3x+1)
2x-1 < 6x+2 --> combine like terms.
2x-6x<2+1
-4x<3
-4x/-4 < 3/-4
x>-3/4.

NB: since we divide both sides by negative the conditional < change into >.

therefore:
x>-3/4.

([x + a million]/[3x - a million]) + ([2x + a million]/[3x - 2]) = - a million ([x + a million][3x - 2]) + ([2x + a million][3x - a million]) = - ([3x - a million][3x - 2]) 3x² - 2x + 3x - 2 + 6x² - 2x + 3x - a million = - (9x² - 6x - 3x + 2) 9x² + 2x - 3 = - 9x² + 9x - 2 18x² - 7x = a million x² - 7/36x = a million/18 + (- 7/36)² x² - 7/36x = seventy two/a million,296 + 40 9/a million,296 (x - 7/36)² = 121/a million,296 x - 7/36 = +/- 11/36 x = 11/36 + 7/36, x = 18/36, x = a million/2 x = - 11/36 + 7/36, x = - 4/36, x = - a million/9 answer: x = a million/2, - a million/9 evidence (x = a million/2 or 0.5): ([0.5 + a million]/[3{0.5} - a million]) + ([2{0.5} + a million]/[3{0.5} - 2]) = - a million (a million.5/[a million.5 - a million]) + ([a million + a million]/[a million.5 - 2]) = - a million (a million.5/0.5) + (2/- 0.5) = - a million 3 - 4 = - a million you're able to teach x = - a million/9 an identical way.

(2x - 1)/(3x + 1) < 2
2x - 1 < 6x + 2
2x - 6x < 1 + 2
-4x < 3
x > 3/-4 (-0.75)