SOLUTION: Hi there, I'm having difficulty solving this inequality: 3(t-1)(t-3) > 0 So at first I tried isolating it looks like: 3 > 0 , (t-1) > 0 , (t-3) > 0 In this sense, I

Algebra ->  Inequalities -> SOLUTION: Hi there, I'm having difficulty solving this inequality: 3(t-1)(t-3) > 0 So at first I tried isolating it looks like: 3 > 0 , (t-1) > 0 , (t-3) > 0 In this sense, I       Log On


   

Question 98471: Hi there,
I'm having difficulty solving this inequality:
3(t-1)(t-3) > 0
So at first I tried isolating it looks like:
3 > 0 , (t-1) > 0 , (t-3) > 0
In this sense, I would get:
t > 1 or t > 3
But the right answer is supposed to be t > 3 (Ok, I get this one I think), and t < 1. How did they get the last inequality?
Thank you very much!
Found 2 solutions by edjones, jim_thompson5910:
Answer by edjones(8007) About Me  (Show Source):
You can put this solution on YOUR website!
3(t-1)(t-3) > 0
It should be t>1 and t>3
Ed

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
3%28t-1%29%28t-3%29+%3E+0 Start with the given inequality

3%28t-1%29%28t-3%29+=+0 Set the left side equal to zero


Now set each factor equal to zero:

t-3=0 or t-1=0

t=3 or t=1 Now solve for t in each case


So our solutions are t=3 or t=1


These values make the entire inequality equal to zero. So that means that at t=3 or t=1 the graph will go from positive to negative (or vice versa). So we must use these points to test the inequality


So now pick any value less than t=1 and test the inequality 3%28t-1%29%28t-3%29+%3E+0. So lets pick t=0


3%280-1%29%280-3%29+%3E+0 Plug in t=0

3%28-1%29%28-3%29+%3E+0 Subtract

9%3E+0 Multiply

Since t=0 makes the inequality true, that means everything less than 1 will make the inequality true. So one part of our answer is t%3C1.


----------------------

Now pick any point in between t=1 and t=3. Let's pick t=2




3%282-1%29%282-3%29+%3E+0 Plug in t=2

3%281%29%28-1%29+%3E+0 Subtract

-3%3E+0 Multiply

Since t=2 makes the inequality false, that means the possible answer t%3E1 false. So this shows why t%3E1 is not an answer.

----------------------

Now pick any point greater than t=3. Let's pick t=4



3%284-1%29%284-3%29+%3E+0 Plug in t=0

3%283%29%281%29+%3E+0 Subtract

9%3E+0 Multiply

Since t=4 makes the inequality true, that means one part of our answer is t%3E3


That last part is somewhat unnecessary since we already eliminated everything else. However, it is important to see everything that is going on




So our solution is t%3C1 or t%3E3



Notice if we graph the equation we get



+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+3%28x-1%29%28x-3%29%29+ Graph of 3%28t-1%29%28t-3%29


and we can clearly see that everything to the left of 1 is greater than zero. Also, we can see that everything to the right of 3 is greater than zero. So our answer is verified.