Leslie R. answered • 03/02/25
Experienced High School math and Spanish language tutor
An inequality written in this form can be solved by evaluating the values of x in each of the intervals between each of the zeros of the expression. Therefore, the first step is to identify those zeros (values of x) that will make the statement equal to zero.
The statement turns zero when x = -9, -4, 0, and 7. For example if x= -9 then the parenthesis -9+ 9= 0.
Let´s evaluate the expression for each of the following intervals of x: (-infinity, -9), (-9, -4), (-4, 0), (0, 7), and (7, + infinitive). The result for each interval must be greater than zero for each interval to be part of the solution.
(-infinitive, -9). We take any value in this interval, for example, x= -10
Evaluate -10 in the statement: (-10)2(-10+9)(-10-9)/(-10+4)(-10-7) is greater than zero?
+ - - / - - (if we multiply these signs we get a +)
This indicates that when x is less than -9, the statement is true and, therefore is part of the solution.
*If you notice, we are not interested in the concrete value of the expression, it´s enough to determine if the value is positive or negative.
Now the interval (-9, -4). We can take any value between these numbers. For example: -5.
(-5)2(-5+9)(-5-9)/(-5+4)(-5-7) is greater than zero?
+ + - / - - (if we multiply these signs we get a -) Not a solution.
Now the interval (-4, 0). We take any value here, for instance, x= -2
(-2)2(-2+9)(-2-9)/(-2+4)(-2-7) is greater than zero?
+ + - / + - (if we multiply these signs we get a +) Is a solution.
Now the interval (0,7) and apply the same process as before. We can take x= 2:
(2)2(2+9)(2-9)/(2+4)(2-7) is greater than zero?
+ + - / + - (if we multiply these signs we get a +) Is a solution.
And the last interval is (7, +infinitive). Let´s take x = 8.
(8)2(8+9)(8-9)/(8+4)(8-7) is greater than zero?
+ + - / + + (if we multiply these signs we get a -) Not a solution
The intervals of x in which the statement is true are: (-infinitive, -9), (-4, 0), and (0,7).
