solve (x + 1)/(x - 2) > 3?

Answer 1

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

(x + 1)/(x - 2) > 0

(x + 1)/(x - 2) - 3 > 0

(x + 1)/(x - 2) - 3(x - 2)/(x - 2) >0

(x + 1 - 3(x - 2))/(x - 2) > 0

(-2x + 7)/(x - 2) > 0

The ratio is undefined at x = 2 and the ratio is equal to zero at x = 7/2. The ratio is negative for x < 2. The ratio is positive for 2 < x < 7/2. The ratio is negative for x > 7/2.

So, (x + 1)(/(x - 2) > 0 for 2 < x < 7/2 .

Answer 2

You want to multiply by x-2 to get that value out of the denominator, and the equation should then be very simple.

However, we have to be careful when multiplying to solve inequalities, because multiplying by a negative number flips the '>' sign. (i.e., if "x > y" then "-2 * x < -2 * y"). This gives us two cases:

(a) when x-2 is positive (i.e., x > 2):
(x + 1) > 3 * (x - 2)
(x + 1) > 3x - 6
2x < 7
x < 7/2

(b) when x-2 is negative (i.e., x < 2):
(x + 1) < 3 * (x - 2) [Note the flipped '>' to '<'!]
(x + 1) < 3x - 6
2x > 7
x > 7/2

There are no values of x which are both greater than 7/2 (3.5) AND less than 2 (which result in negative x-2), so we can ignore (b). However, while the (b) case may have been ruled out for this particular problem, it might be important to other problems (and the splitting into positive vs negative x-2 also provides the lower bound on the range).

The answer is what we found in part (a): the quotient is greater than three where x-2 is positive (i.e., x>2) AND x<7/2

The range of values of x where that quotient is greater than three is:
2 < x < 7/2

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We can check this out by trying some numbers just outside that range:

when x=1, (x+1)/(x-2) = 2/-1 = -2, which is not greater than three

when x=4, (x+1)/(x-2) = 5/2 = 2.5, which is not greater than three

And we can check a number inside that range:

when x=3, (x+1)/(x-2) = 4/1 = 4, which IS greater than three.

Answer 3

First multiply each side by (x-2). NOTE THAT THIS IS ONLY VALID IF X-2 > 0.

x + 1 > 3(x - 2) ...or...
x + 1 > 3x - 6

Now subtract x from each side:

1 > 2x - 6

Now add 6 to each side, then divide by 2:

7/2 > x

or x < 7/2 provided that x -2 > 0. More precisely,

2 < x < 7/2

Answer 4

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

Multiply both sides by (x-2) to get rid of the fraction.

(x-2) times (x+1)/(x-2) > 3(x-2)
x + 1 > 3x - 6
1 > 2x - 6
7 > 2x
7/2 > x or x < 7/2

Answer 5

If x > 2
then x+1>3x-6
-2x > -7
x<3.5
The first solution interval ] 2 , 3.5 [
If x < 2
The x+1 < 3x -6
-2x < -7
x> 3.5
Ther is no solution satisfies these conditions
So the final solution is the interval ] 2 , 3.5 [

Answer 6

first case ..... x+1>3... x>4....
second case... 1/(x-2) > 3.... x-2>1/3
x>7/3...

then the answer that satisfies both is .. x>4