|
Question 1074756: Any help is truly appreciated...
suppose x-2 divided by x+3 is less than or equal to 4...
-final all the critical values of the related equation.
-use a table of intervals, subdivided by critical values, to test the inequality.
-write the solution set using the interval notation.
-make a graph of the related equation to verify your solution.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your equation is (x-2) / (x+3) <= 4
there will be a vertical asymptote at x = -3, so that becomes one of your critical values.
if you set (x-2) / (x+3 = 4. then x = -14/3 and that becomes another one of your critical values.
it looks like you have 2 critical values.
x = -14/3 and x = -3
you want to look at the intervals to the left of and to the right of and in between these critical values for x.
to check the interval to the left of x = -14/3, set x to some value less than -14/3.
any value less than -14/3 will do.
since -14/3 = -4.66667, then -5 will do.
when x = -5, (x-2/(x+3) = -7/-2 = 3.5 which is less than 4 so the original inequality is true within that interval.
your next interval is between -14/3 and -3.
pick a value for x within that interval.
-4 will do.
when x = -4, (x-2)/(x+3) = (-6)/(-1) which is equal to 6 which is greater than -14/3, so the original equation is false within that interval.
your next interval is between -3 and plus infinity.
pick a value for x within that interval.
x = -1 will do.
when x = -1, (x-2)/(x+3) = (-3)/(2) = -3/2 which is less than 4 so the original inequality is true within that interval.
your equation appears to be true when x <= -14/3 and when x > -3.
it appears to be false when x > 14/3 and when x < -3.
if you graph the equation, you will see that this is so.
here's the graph.
i was able to solve this without setting the right side equal to 0, but the recommended method is to get the right side equal to 0 and then solve.
start with (x-2)/(x+3) <=4
subtract 4 from both sides of the equation to get (x-2)/(x+3) - 4 <= 0
put everything under a common denominator to get (x-2)/(x+3) - 4(x+3)/(x+3) <= 0
simplify to get [(x-2) - 4(x+3)] / (x+3) <= 0
simplify further to get (x-2 - 4x - 12) / (x+3) <= 0
combine like terms to get (-3x-14) / (x+3) <= 0
now you want to find the zero points and the asymptotes, if any.
you will have an asymptote at x = -3
you find the 0 point by setting the numerator of (-3x-14) = 0 and solving for x.
you will get x = -14/3.
same critical values as i found before, but doing it this way is the recommended way that is also used to solve higher level ration inequalities, such as quadratics as well.
some reference that might help are shown below:
http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_rational_inequalities.xml
http://www.purplemath.com/modules/ineqrtnl.htm
http://tutorial.math.lamar.edu/Classes/Alg/SolveRationalInequalities.aspx
|
|
|
| |
