solve the following inequality
x^2(8+x)(x-5)/ (x+5)(x-2)> or equal too 0
this looks like a fraction
please show work
critical values:
x=0 , x=-8, x=5, x=-5 and x=2
So want to see where the graph of
y = x^2(8+x)(x-5)/ ((x+5)(x-2)) lies above or on the x-axis
investigate the following domains
1. x < -8
2. x between -8 and -5
3. x between -5 and 0
4. x between 0 and 2
5. x between 2 and 5
6. x > 5
You don't actually have to work out the calculations, all you care about is whether the answer is + or -
I will do 5.
a number between 2 and 5, I pick x = 4
then
(+)(+)(-)/((+)(+)) ≥ 0 ?? , no,
so x between 2 and 5 is not a solution
You can do the others the same way
I also graphed y = x^2(8+x)(x-5)/((x+5)(x-2))
http://www.wolframalpha.com/input/?i=y+%3D+x%5E2%288%2Bx%29%28x-5%29%2F%28%28x%2B5%29%28x-2%29%29
which confirmed that
x<-8 OR -5 < x < 2 OR x > 5
the answer has to be in interval notation, so how would that look?
I am not a big fan of "interval notation"
Back in the dark ages when I taught we used the above notation, which is much more specific
Here is a page that shows "interval notation", look about half way down the page.
It should be quite easy to convert my answer to what you need.
http://www.regentsprep.org/Regents/math/ALGEBRA/AP1/IntervalNot.htm
thank you I didn't see this post after you posted that!!! you are truly a life saver!!
To solve the inequality, we need to find the values of x that make the expression
(x^2(8+x)(x-5))/((x+5)(x-2)) ≥ 0.
First, let's find the critical points of the inequality by setting the numerator and denominator equal to zero and solving for x.
Numerator:
x^2(8+x)(x-5) = 0
This equation has three critical points: x = 0, x = -8, and x = 5.
Denominator:
(x + 5)(x - 2) = 0
This equation has two critical points: x = -5 and x = 2.
Now, we create a number line and plot the critical points on it:
-----------o---------------------o-------o-------------------------
-∞ -8 -5 0 2 5 +∞
Next, we choose test points within each interval and substitute them into the inequality expression. We will determine if the inequality is satisfied or not for each test point.
Let's test the interval (-∞, -8):
Choose x = -9:
(-9)^2(8+(-9))(x-5)/((-9)+5)((-9)-2)
= 81(-1)(-14)/(-4)(-11)
= 1134/44
= 25.77
Since 25.77 is positive, the inequality is satisfied in this interval.
Next, let's test the interval (-8, -5):
Choose x = -6:
(-6)^2(8+(-6))(x-5)/((-6)+5)((-6)-2)
= 36(2)(-11)/(-1)(-8)
= 792/8
= 99
Since 99 is positive, the inequality is satisfied in this interval.
Next, let's test the interval (-5, 0):
Choose x = -4:
(-4)^2(8+(-4))(x-5)/((-4)+5)((-4)-2)
= 16(4)(-9)/(1)(-6)
= -576/(-6)
= 96
Since 96 is positive, the inequality is satisfied in this interval.
Next, let's test the interval (0, 2):
Choose x = 1:
(1)^2(8+(1))(x-5)/((1)+5)((1)-2)
= 1(9)(-4)/(6)(-1)
= -36/(-6)
= 6
Since 6 is positive, the inequality is satisfied in this interval.
Lastly, let's test the interval (5, ∞):
Choose x = 6:
(6)^2(8+(6))(x-5)/((6)+5)((6)-2)
= 36(14)(1)/(11)(4)
= 504/44
= 11.45
Since 11.45 is positive, the inequality is satisfied in this interval.
Combining the results:
The inequality is satisfied for x values in the intervals (-∞, -8], [-5, 0], (5, ∞).
Therefore, the solution to the inequality is:
x ≤ -8, x ∈ [-5, 0], x > 5.