SOLUTION: 2. Solve by graphical means: x^3 – x^2 – 4x + 4 < 0 (Hint: factor by grouping) Label the vertex and axis. Denote the left-hand side by f(x). Factor t

Algebra ->  Functions -> SOLUTION: 2. Solve by graphical means: x^3 – x^2 – 4x + 4 < 0 (Hint: factor by grouping) Label the vertex and axis. Denote the left-hand side by f(x). Factor t      Log On


   

Question 129111: 2. Solve by graphical means:

x^3 – x^2 – 4x + 4 < 0
(Hint: factor by grouping) Label the vertex and axis.
Denote the left-hand side by f(x). Factor to find the x intercepts, and sketch the graph with the aid of a few additional points.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let's factor x%5E3-x%5E2-4x%2B4

x%5E3-x%5E2-4x%2B4 Start with the given expression

%28x%5E3-x%5E2%29%2B%28-4x%2B4%29 Group like terms


x%5E2%28x-1%29-4%28x-1%29 Factor out the GCF x%5E2 out of the first group. Factor out the GCF -4 out of the second group


%28x%5E2-4%29%28x-1%29 Since we have the common term x-1, we can combine like terms

%28x%2B2%29%28x-2%29%28x-1%29 Now factor x%5E2-4 to get %28x%2B2%29%28x-2%29


So x%5E3-x%5E2-4x%2B4 factors to %28x%2B2%29%28x-2%29%28x-1%29


Notice if we solve %28x%2B2%29%28x-2%29%28x-1%29=0 we find the zeros x=-2, x=1 and x=2


In order to solve x%5E3-x%5E2-4x%2B4%3C0 we need to test some points. So let's pick a point that is less than x=-2

So let's test x=-3


x%5E3-x%5E2-4x%2B4%3C0 Start with the given inequality


%28-3%29%5E3-%28-3%29%5E2-4%28-3%29%2B4%3C0 Plug in x=-3

-20%3C0 Simplify. Since this inequality is true, any value that is less than x=-2 will satisfy the inequality.



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Now let's test a value that is in between x=-2 and x=1


So let's test x=0


x%5E3-x%5E2-4x%2B4%3C0 Start with the given inequality


%280%29%5E3-%280%29%5E2-4%280%29%2B4%3C0 Plug in x=0

4%3C0 Simplify. Since this inequality is not true, this means that the interval [-2,1] is not in the solution set.



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Now let's test a value that is in between x=1 and x=2


So let's test x=1.5


x%5E3-x%5E2-4x%2B4%3C0 Start with the given inequality


%281.5%29%5E3-%281.5%29%5E2-4%281.5%29%2B4%3C0 Plug in x=1.5

-0.875%3C0 Simplify. Since this inequality is true, any value that is in between x=1 and x=2 will satisfy the inequality.


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Now let's test a value that is greater than x=2


So let's test x=3


x%5E3-x%5E2-4x%2B4%3C0 Start with the given inequality


%283%29%5E3-%283%29%5E2-4%283%29%2B4%3C0 Plug in x=3

10%3C0 Simplify. Since this inequality is not true, this means that any value greater than x=2 will not satisfy the inequality



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Answer:


So the solution set is





Notice if we graph y=x%5E3-x%5E2-4x%2B4, we can visually verify our answer.


+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E3-x%5E2-4x%2B4%29+