Question

solve the following inequality. Write your solutions in interval notation.
x-5/x+2>=1

I do not know what i am doing at all, please help!

Answer 1

you know that if you multiply by a negative number, the direction of the inequality changes. To clear the fraction, you must multiply by (x+2).

So, if (x+2)>0, the direction does not change, and

x-5 > x+2
-5 > 2
No value of x satisfies this inequality

Now, if (x+2) < 0,
x-5 < x+2
-5 < 2
Every value of x makes this true. But, w had the condition that x+2 < 0, or x < -2.

So, every x < -2 is a solution. In interval notation, (-∞,-2) is the solution set.

Or, you could do a polynomial division to see that (x-5)/(x+2) = 1 - 7/(x+2).
So, any value of x that makes 7/(x+2) negative will give a value greater than 1. That means x+2 < 0, ...