Question

Answer the following questions:(1.1) (1.2) Please make sure you use a Method, do not just set parts of the function greater than or less than 0 to find answers

Answer 1

(1.1) When we multiply both sides of an inequality by a negative number, the inequality flips oven and instead of "≥" becomes "≤"

(1.2) When we multiply both sides of an inequality by a possitive number, the inequality sign remains unaltered.

(1.3) We need to consider 2 cases because the term (x + 1) could be possitive or negative. If it's possitive the sing remains unaltered, but if (x + 1) < 0, then the sign must change.

(1.4) To solve this, first let's leave 0 in the right hand side of teh inequality:

`\begin{gathered} (x-5)/(x+1)\ge2 \\ (x-5)/(x+1)-2\ge0 \end{gathered}`

Now add the terms:

`(x-5)/(x+1)-(2(x+1))/(x+1)=(x-5-2x-2)/(x+1)=(-x-7)/(x-1)\ge0`

Now we need to calculate the critical points, where the numerator and denominator are 0:

numerator:

-x - 7 = 0

x = -7

Denominator:

x + 1 = 0

x = -1

Now, we have three intervals:

Now we need to test in which intervals teh inequality is true.

For the interval (-∞, -7) Lets take -10:

`\begin{gathered} x=-10 \\ (-10-5)/(-10+1)=(-15)/(-9)=(5)/(3)\ge2 \end{gathered}`

Which is false, 5/3 ≈ 1.666667 which is less than 2.

Next let's test the interval [-7, -1). Let's grab x = -5

`\begin{gathered} x=-5 \\ (-5-5)/(-5+1)=(-10)/(-4)=(5)/(2)=2.5\ge2 \end{gathered}`

Which is TRUE

Finally, for (-1, ∞) Let's take x = 0:

`(0-5)/(0+1)=-(5)/(1)=-5`

And (-5) is less than 2.

The interval of solutions is:

`\lbrack-7,-1)`

Or:

`-7\le x<-1`