# Format of the scenario data file: Multi-record data file.
#
# Record 1: header.
#	Line 1: title
#	Line 2: author
#	Line 3: email
#	Line 4: translator
#	Line 5: email
#	Line 6: format (html,tex; default html)
#	Line 7 and up: random data.
# Record 2: presentation of the problem.
# Record 3: Good scenario. One step per line.
# Record 4: Seemingly bad reason(s) for each step, one line per step.
# Record 5: Remarks. One line per step.
# Record 6: Reserved.
# Record 7 and up: Bad scenarios.
#	Line 1: starting step, bad reason.
#	Line 2: remark.
#	Line 3 and up: one step per line.
#

:Absolute value I
XIAO, Gang
xiao@unice.fr


html
&gt;, $m_ge@&lt;, $m_le
1,2,3,4,5,6,7,8,9
1,2,3,4,5,6,7,8,9
:Here is an argument to solve the inequality
 |x-$r3| $r1 $r4.
:Following the cases where x-$r3 is positive or negative, one gets the equivalent condition: x-$r3 $r2 -$r4 or x-$r3 $r1 $r4.
 Adding $r3 to all the sides of the chain of inequalities, x $r2 $[-$r4+$r3] or x $r1 $[$r4+$r3].
:bad_chain, abs
 add_sign, illegal
:  
We indeed have the right to add a same term to all the sides of a chain of inequalities.
:

:1, abs
 The case x-$r3<0 should not be forgotten!
 This inequality implies x-$r3 $r1 $r4.
 Adding $r3 to both sides, x $r1 $[$r4+$r3].
:1, abs, 2, add_sign
 The case x-$r3<0 should not be forgotten!
 This inequality implies x-$r3 $r1 $r4.
 Moving the term -$r3 to the right, x $r1 $[$r4-$r3].
:2, add_sign

 Moving the term -$r3 to the other sides, x $r2 $[-$r4-$r3] or x $r1 $[$r4-$r3].
:1, abs

 Following the cases where x-$r3 is positive or negative, one gets the equivalent condition: -$r4 $r1 x-$r3 $r1 $r4.
 Adding $r3 to all the sides of the chain of inequalities, $[-$r4+$r3] $r1 x $r1 $[$r4+$r3].
:1, abs, 2, add_sign

 Following the cases where x-$r3 is positive or negative, one gets the equivalent condition: -$r4 $r1 x-$r3 $r1 $r4.
 Adding $r3 to all the sides of the chain of inequalities, $[-$r4-$r3] $r1 x $r1 $[$r4-$r3].
