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Discussion
Drag the point around the number line below to see the distance from different points to 0.
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What do you notice about the distance function?
When you drag the point to any position x, its distance from 0 has the following properties:
It is never negative: the minimum distance is 0 (at x=0), and otherwise it’s positive.
Points equally far to the right and left of 0 have the same distance—for example, x=3 and x=−3 both give distance 3.
As you move the point either to the right or to the left, the distance grows at the same rate.
The "distance function" here is just the absolute value: d(x)=∣x∣.
Absolute value
The absolute value of x is defined as
In certain situations, it might be useful to compare two functions. In such situations, we use inequalities, like f(x)>g(x).
If asked to solve such an inequality, you have to find all of the inputs in the domain of the two functions for which the inequality is true.
Inequalities of functions
Inequalities of the form
can be solved either algebraically or with technology.
It is crucial to remember that when multiplying both sides of an inequality by a negative number, the inequality changes direction:
Example
Solve the inequality
Subtracting the RHS:
Now we make a table of signs:
So the solution is 0<x<1 or 2<x.
With a GDC, we can subtract the RHS again to get x3−3x2+2x>0. Then, graph Y_1= x3−3x2+2x and use 2nd > calc > zero to get the zeros of the graph. Finally, produce a table of signs as above or look at the graph to see when the curve is above the x-axis
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