Advanced Math/Inequality

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Expert: Sherry Wallin - 12/7/2008


Question
The problem I was given was to solve for "x" algebraically when

(x+5) divided by (x-3) is greater than or equal to zero.


I know that the denominator cannot equal zero, therefore x must be greater than 3, but beyond that I have no idea how to approach the problem, or what steps to take to solve for all values of x

Answer

number line
Hi Steph~
   This is a problem with reasoning and math. What do you know about when a number (a rational number, a fraction) = 0? Right, when the numerator is 0, correct? And what do you know about the sum of two numbers over a denominator? Isn't 1/7 + 3/7 = (1+3)/7? and vice versa?
So write (x+5)/(x-3) as x/(x-3) + 5/(x-3)>= 0

So x/(x-3)>= -5/(x-3)
Now use cross multiplication to get rid of the denominators.
x(x-3) >= -5(x-3)
x^2-3x >=-5x + 15
x^2-3x +5x - 15 >= 0
x^2 +2x-15 >= 0
(x+5)(x-3)>=0 remembering as you stated above that x != 3
so possible solutions are x >= -5 and x >= 3. But you can not include 3 so it is x>3. So take your "critical values" and put them on a number line and check those values in your factors. [Please see image in Word document attached]. You don't need to get actual values just if they are positive or negative or 0 since you are looking for all numbers >= 0.(x-3)(x+5)
So my "check points" are -6 -> (-)(-)= +
                        1: -> (-)(+) = -
                        4: -> (+)(+) = +
So the intervals that our rational expression is >= 0 is
(- infinity,-5]union(3,infinity). Notice the square bracket next to the -5 to indicate the solution could be -5 and the curved parenthesis next to 3 to indicate you can't include 3.

I hope this clearly helps you solve the problem and gain new understanding.

Math Prof  

Sherry Wallin

Expertise

I can answer most questions up through Calculus and some in Number Theory and Abstract Algebra.

Experience

I have had my Bachelor's Degree since 1987 and have been a teacher since 1988. I earned my Masters Degree in Mathematics May 2010. I have been teaching at the same community college since 2002.

Education/Credentials
I have taught 12 years at the community college level, medical college, and technical college as well as a high school instructor and alternative education instructor and charter school instructor.

Awards and Honors
Master's GPA 3.56 Bachelor's GPA 3.34 Post grad work not degree related GPA 4.0

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