13. Solve completely v x E R, show the equation in set and graphical notation:  3x-5/x-2 less thn or equal to 5/x+5 14. Solve v x E R: 2^x=7 15. If f(x)=2x-3, find f(log 10). 16. Simplify v x...

13. Solve completely v x E R, show the equation in set and graphical notation: 

3x-5/x-2 less thn or equal to 5/x+5

14. Solve v x E R: 2^x=7

15. If f(x)=2x-3, find f(log 10).

16. Simplify v x E R and give parameters: x^2+x-6/x-2

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steveschoen | College Teacher | (Level 1) Associate Educator

Posted on

Hi, Steve,

For #13, we would get everything on one side of the inequality.  So, for example, subtracting 5/(x+5) from each side:

(3x-5)/(x-2)  -  5/(x+5)  <-- 0

Now, from here, this inequality doesn't exist at 2 points, when x = 2 and -5.  So, we use those as "markers" to check other numbers.  Also, we would write this as one fraction on the left, because the top would become a quadratic.  We need to find the zeros for that also.  The top would become:

(3x-5)(x+5) - 5(x-2)

3x^2 + 15x - 5x - 25 - 5x + 10

3x^2 + 5x - 15

The zeros for this are x = -3.22 and 1.55

So, we have 4 numbers that can break up or domain, 2, -5, -3.22, and 1.55.  We use these to break up our domain into 5 sections:

All  numbers less than -5

All numbers from -5 to -3.22

All numbers from -3.22 to 1.55

All numbers from 1.55 to 2

And, all numbers greater than 2

We pick a number from each section to determine if that section provides solutions, is a solution to the inequality.  For instance, we can pick x = -6.  Doing that:

(3(-6)-5)/(-6-2)  -  5/(-6+5)  <-- 0

-23/-8  -  (-5) <-- 0

23/8 + 5 <-- 0

This is obviously false.  So, this set of real numbers has no solution for the inequality.  Doing the same for the other sections, we get where the solutions are:

from -5 to -3.22

and 1.55 to 2

Or, using parenthesis and brackets, we would have

(-5,-3.22] U [1.55,2)

Parenthesis because the inequality doesn't exist at -5 and 2.  Brackets because the inequality does exist at -3.22 and 1.55, specifically being where the inequality = 0.

For 14, we can rewrite the equation as a log:

log(base 2) of 7 = x

Using the change of base formula, we can write:

(log 7)/(log 2) = x which is approx. 2.81.  Assuming the base of the log is 10.

For 15, assuming the base of the log is 10, log 10 = 1.  So, we are looking for f(1).  f(1) = 2*1 - 3 = -1

For 16, we would factor the top into (x+3)(x-2).  So, the "x-2" would cancel out, leaving x+3.  However, the condition being x can't be 2.  For, if x = 2, then we have 2-2 on the bottom = 0.  And, we can't divide by 0.

I hope these help, Steve.  Good luck.

Steve

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