Need some help solving this inequality?

Question

x - 1
------ < 2
x + 2

Thanks.

Answer 1

Multiplying both sides by x+2 can be tricksy. One never knows if x+2 is positive or negative, and therein lies the risk of not knowing whether the inequality symbol should stay the same or be switched backwards.

Instead, subtract 2 from both sides, and use common denominators to rewrite the left hand side as one, single ratio:

x - 1
------ - 2 < 0
x + 2

(x - 1) - 2(x+2)
------------------- < 0
x + 2

x - 1 -2x - 4
------------------ < 0
x + 2

-x - 5
-------- < 0
x + 2

Next find the critical points for x that set the top = 0 and the bottom = 0, namely x = -2 and x = 5

Test any number to the left of -2, and the result is a false statement
Test any number inbetween -2 and 5, and the result is a true statement
Test any number to the right of 5, and the result is a false statement.

solution: -2 < x < 5

Answer 2

First, multiply by (x+2) to get rid of the denominator. However, multiplying by a negative number flips the inequality, so you end up with two cases.

Case 1: x+2 is positive:

(x - 1) / (x + 2) < 2
x - 1 < 2(x + 2)
x - 1 < 2x + 4
-5 < x
x > -5

Since x+2 is positive, we also know:
x + 2 > 0
x > -2

... the values of x BOTH greater than -5 AND greater than -2, are simply greater than -2.

Case 2: x+2 is negative:

(x - 1) / (x + 2) < 2
x - 1 > 2(x + 2)
x - 1 > 2x + 4
-5 > x
x < -5

Since x+2 is negative, we also know:
x + 2 < 0
x < -2

... the values of x BOTH less than -5 AND less than -2, are simply less than -5.

So the answer is two ranges:
(1) x > -2
(2) x < -5
... it's everything other than -2 to -5 inclusive.

===================
To check the answer, let's try some numbers for x around those values:

x = -3 is in the excluded range and should not be a solution:
(-3 - 1) / (-3 + 2) = -4/-1 = 4, which is not < 2.

x = -6 should be a solution:
(-6 - 1) / (-6 + 2) = -7/-4 = 1.75 which is < 2

x = -1 should be a solution:
(-1 -1) / (-1 + 2) = -2/1 = -2 which is < 2

x = 1 should be a solution:
(1 - 1) / (1 + 2) = 0/3 = 0 which is < 2

x = 100 should be a solution:
(100 - 1) / (100 + 2) = 99 / 102 which is just under 1, and definitely < 2