Plan:
Find the solution set for equal to 5;
then use end behavior to determine when the inequality is true.
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--> (1) OR (2)
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(1):
--> (1a) OR (1b)
(1a): -->
(1b): -->
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(2):
--> (1a) OR (1b)
(1a): -->
(1b): -->
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We are done with the first step: the values of x for which the expression is equal to 5 are -3, 2, 4, and 9. Now for the second step.
1) The value of the expression is always 0 or positive, because it is an absolute value.
2) For large positive or large negative values of x, the value of the expression will be large.
3) Every segment of the graph is linear.
Given those three facts and the four values of x for which the expression is equal to 5, we can conclude that the inequality (value of the expression is less than 5) is true between x = -3 and x = 2, and between x = 4 and x = 9.
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ANSWER: The solution set for the inequality is (-3,2) U (4,9)
| 2*(|x-3|)-7 | < 5
is equivalent to
-5 < 2*|x-3| - 7 < 5
is equivalent (after adding 7 to each of 3 parts of the inequality)
-5 + 7 < 2*|x-3| < 5 + 7
is the same as
2 < 2*|x-3| < 12
is equivalent to (after dividing all the three terms of the inequality by 2)
1 < |x-3| < 6.
The solutions to the last inequality are those values of x (those numbers or points in the number line)
that are remoted from the point x= 3 farther than 1 unit and closer than 6 units.
Obviously, these points (numbers, solutions) are
-3 < x < 2 and/or 4 < x < 9.
ANSWER. The solution set to the original inequality is the union of two intervals {-3,2) and (4,9): (-3,2) U (4,9).
