algebra 2 question

Question

Can I get help with the inequality of this please, I can’t find the solution
X^2(6+x) (x-9)/(x+4) (x-6)>0

Answer 1

Hi Diane,

 

I'll be happy to guide you through a step-by-step procedure solution to this problem, which can then be used for similar problems.

 

Step 1 (set all factors to zero and solve for x to find all critical points where there could be a sign change):

    x² = 0; x = 0

    6 + x = 0;  x = -6

    x - 9 = 0; x = 9

    x + 4 = 0; x = -4

    x - 6 = 0; x = 6

 

Step 2 (plot all critical points found in Step 1 on number line and divide number line into regions marked off by the critical points)

    (a)  Click this link (https://dl.dropbox.com/s/2z87dyfl74dgnib/Marked_Number_Line.html?raw=1) to see our marked number line.

    (b)  The critical points are identified in red and the regions (roman numerals) are identified in blue.

 

Step 3 (number line test):

    (a)  Test each region using a representative test number in that region to evaluate the sign of each factor for that region:

              (1)  Region I (used -10 as test number).

              (2)  Region II (used -5 as test number).

              (3)  Region III (used -2 as test number).

              (4)  Region IV (used 3 as test number).

              (5)  Region V (used 7 as test number)

              (6)  Region VI (used 10 as test number).

    (b)  Test each critical point to evaluate the sign of each factor at that critical point.

    (c)  Evaluate the sign of the function in each region and at each critical point based upon the signs of the factors (also note any other function behavior described below):

              (1)  If number of negative factors is odd then sign of function is odd.

              (2)  If number of negative factors is even then sign of function is positive.

              (3)  If any factor is 0 then function is 0.

              (4)  If any factor in denominator is 0 then function is U (undefined).

              (5)  If a numerator factor is 0 and a denominator factor is 0 then function has a H (hole) in it at that point.  A hole means the function does NOT exist for the tested number.

    (d)  Click this link (https://dl.dropbox.com/s/kx3kwswbsfqcf6p/Number_Line_Test_Table.html?raw=1) to see our number line test table (regions that passed the test are highlighted in green). 

 

Step 4 (Inequality Solution Set)

    (a)  Formulate the solution set for the inequality based upon the results in number line test table from Step 3-d (for our problem we're looking at any region or critical point where our function tested positive, i.e. > 0).

    (b)  Our inequality solution set for this  problem based upon the passed regions (highlighted in green) from step 3-d:

              x < -6 or -4 < x < 0 or 0 < x < 6 or x > 9

 

The above step-by-step procedure will be useful for for solving any similar inequality problems involving factored fractions being tested against the value of zero.

 

Thanks for submitting this problem and glad to help. 

 

God bless, Jordan.