# MATH HELP: SOLVING INEQUALITIES?

Update:

I have tried so many ways but am still unable to solve it. Relevance
• -5 < (x + 2) / (x - 5)

If this was an equation we would multiply both sides by (x - 5).  We can do that here but remember if we divide by a negative number we have to flip the sign.

x cannot be 5.  If x > 5, then (x - 5) will be positive.  If x < 5 then (x - 5) will be negative. We can use this to split the inequality into two so we can multiply both sides by (x - 5) and flip the sign as needed:

-5(x - 5) < x + 2 if x > 5 and -5(x - 5) > x + 2 if x < 5

Now solve for each:

-5x + 25 < x + 2 if x > 5 and -5x + 25 > x + 2 if x < 5

-6x < -23 if x > 5 and -6x > -23 if x < 5

Need to flip the signs again since we're dividing both sides by a negative value:

x > 23/6 if x > 5 and x < 23/6 if x < 5

23/6 is approximately 3.833.  So any number larger than 5 will be larger than 23/6.  The first one simplifies to:

x > 5 and x < 23/6 if x < 5

And the other one, if x is less than 23/6 it will be less than 5.  So that turns into:

x > 5 and x < 23/6

In interval notation:

(-∞, 23/6) U (5, ∞)

Your graph will have an open circle at 23/6 and an arrow going to the left and an open circle at 5 and an arrow going to the right.