12 ## ABSOLUTE VALUEThe geometrical meaning of | THIS SYMBOL | Here is the purely algebraic definition of | If if Geometrically, | Both 3 and −3 are a Problem 1. Evaluate the following. To see the answer, pass your mouse over the colored area.
Problem 2. Explain the following rules. a) |−
Both − b) |2 −
2 − Therefore, according to part a), they are equal. c) | We may remove the absolute value bars because the left-hand side is never negative, and neither is the right-hand side. Absolute value equations | What values could
And so any equation that | -- has the
We call whatever appears within the vertical bars --
Example 1. Solve for |
We must solve these
The second implies
These are the two solutions: Problem 3. a) An absolute value equation has how many solutions? Two. b) Write them for this equation: |
Problem 4. Solve for | Solve these two equations:
Problem 5. Solve for |1 −
Problem 6. Solve for |2
Absolute value inequalities There are two forms of absolute value inequalities. One with
Example 2. Absolute value | For that inequaltiy to be true, what values could Geometrically, Therefore, −3 < This is the solution. The inequality will be true if In general, if an inequality looks like this -- | -- then the solution will look like this: − for any argument
Example 3. For which values of |2
−5 < 2 We must isolate −5 + 1 < 2 −4 < 2 Now divide each term by 2: −2 < The inequality will be true for any value of
Problem 7. Solve this inequality for |
−7 < Subtract 2 from each term:
−7 − 2 <
−9 <
Problem 8. Solve this inequality for |3
−10 < 3 Add 5 to each term:
−5 < 3 Divide each term by 3:
Problem 9. Solve this inequality for |1 − 2
−9 < 1 − 2 Subtract 1 from each term:
−10 < −2 Divide each term by −2. The sense will change.
5 > That is,
−4 <
Example 4. Absolute value | For which values of Geometrically,
This is the form of the solution, for any argument If | then
Problem 10. Solve for |
Problem 11. For which values of |
The first equation implies
Problem 12. Solve for |2
2 Solve those two equations:
Problem 13. Solve for |1 − 2
1 − 2
Solve those two equations. On finally dividing by −2,
The geometrical meaning of | Geometrically, | | | we mean that
On the other hand, if we write | we mean This means that
Problem 14. What is the geometric meaning of |
The distance of | | then
Problem 15. What is the geometrical meaning of each of the following? And therefore what values has a) |
b) |
c) |
d) |
e) |
Problem 16. |
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