Question

Answer the following questions

Answer 1

Number 7: Solution
`[-5,-(1)/(9)]`U
`(3,\infty)`

Number 8: solution
`(-6,-5) U(-1,5)`

Explanation:

Number 7: Given inequality is
`(x-6)/(x+5) - (x+4)/(x-3)\leq 0`

First we simplify the equation and then we find critical points.

`((x-6)(x-3)-(x+4)(x+5))/((x+5)(x-3)) \leq 0`

`(x^2-9x+18-x^2-9x-20)/((x+5)(x-3)) \leq 0`

`(-18x-2)/((x+5)(x-3)) \leq 0`

Critical points are :
`x=-(1)/(9),3,-5`

Now we plot the points on number line and find the true and false solution. Please see the attach file fig 1 for number line.

We will check each interval for solution.

Let we take any interval
`(-(1)/(9),3)` and take any point between the interval and substitute into equation. We get
`(2)/(15)` >0 which is greater than 0.

This interval would be false.

Similarly, we will check each interval .

We get solution,
`[-5,-(1)/(9)]`U
`(3,\infty)`

Number 8: Given inequality is
`(x^2-4x-5)/(x^2+11x+30) < 0`

First we simplify the equation and then we find critical points. Factor numerator and denominator. We get

`((x-5)(x+1))/((x+5)(x+6)) < 0`

Critical points are :
`x=-6,-5,-1,5`

Now we plot the points on number line and find the true and false solution. Please see the attach file fig 2 for number line.

We will check each interval for solution.

Let we take any interval
`(-1,5)` and take any point x=0 between the interval and substitute into equation. We get
`-(1)/(6)` <0 which is less than 0.

This interval would be true.

Similarly, we will check each interval .

We get solution,
`(-6,-5) U(-1,5)`