# Format of the scenario data file: Multi-record data file.
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# 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.
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:Absolute value II
XIAO, Gang
xiao@unice.fr


html
&lt;, $m_le, &gt;, $m_ge@&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| $r1 |x+$r4|.
:Replace the absolute values by squares: (x-$r3)<sup>2</sup> $r1 (x+$r4)<sup>2</sup>.
 Then we develop: x<sup>2</sup>-$[2*$r3]x+$[$r3^2] $r1 x<sup>2</sup>+$[2*$r4]x+$[$r4^2].
 One can delete the terms x<sup>2</sup> from both sides of the inequality: -$[2*$r3]x+$[$r3^2] $r1 $[2*$r4]x+$[$r4^2].
 Ranging the terms of x to the left and constant terms to the right, we get -$[2*$r3+2*$r4]x $r1 $[$r4^2-$r3^2].
 Finally, divide by -$[2*$r3+2*$r4]&nbsp;: x $r2 $[-($r4^2-$r3^2)/(2*($r3+$r4))].
:abs,square, alg_err
 alg_err, illegal
 illegal, add_neg2
 add_neg2, alg_err
 div_zero, div_neg2, illegal
:Whatever the real values of a and b, |a|$r1|b| is equivalent to a<sup>2</sup>$(r1)b<sup>2</sup>.


:

:2, alg_err

 Then we develop: x<sup>2</sup>-$[$r3]x+$[$r3^2] $r1 x<sup>2</sup>+$[$r4]x+$[$r4^2].
 One can delete the terms x<sup>2</sup> from both sides of the inequality: -$[$r3]x+$[$r3^2] $r1 $[$r4]x+$[$r4^2].
 Ranging the terms of x to the left and constant terms to the right, we get -$[$r3+$r4]x $r1 $[$r4^2-$r3^2].
 Finally, divide by -$[$r3+$r4]&nbsp;: x $r2 $[-($r4^2-$r3^2)/($r3+$r4)].
:2, alg_err, 5, div_neg

 Then we develop: x<sup>2</sup>-$[$r3]x+$[$r3^2] $r1 x<sup>2</sup>+$[$r4]x+$[$r4^2].
 One can delete the terms x<sup>2</sup> from both sides of the inequality: -$[$r3]x+$[$r3^2] $r1 $[$r4]x+$[$r4^2].
 Ranging the terms of x to the left and constant terms to the right, we get -$[$r3+$r4]x $r1 $[$r4^2-$r3^2].
 Finally, divide by -$[$r3+$r4]&nbsp;: x $r1 $[-($r4^2-$r3^2)/($r3+$r4)].
:3, add_neg

 Subtract x<sup>2</sup> from both sides: -$[2*$r3]x+$[$r3^2] $r2 $[2*$r4]x+$[$r4^2].
 Ranging the terms of x to the left and constant terms to the right, we get -$[2*$r3+2*$r4]x $r2 $[$r4^2-$r3^2].
 Finally, divide by -$[2*$r3+2*$r4]&nbsp;: x $r1 $[-($r4^2-$r3^2)/(2*($r3+$r4))].
:3, add_neg, 5, div_neg

 Subtract x<sup>2</sup> from both sides: -$[2*$r3]x+$[$r3^2] $r2 $[2*$r4]x+$[$r4^2].
 Ranging the terms of x to the left and constant terms to the right, we get -$[2*$r3+2*$r4]x $r2 $[$r4^2-$r3^2].
 Finally, divide by -$[2*$r3+2*$r4]&nbsp;: x $r2 $[-($r4^2-$r3^2)/(2*($r3+$r4))].
:4, add_sign

 Ranging the terms of x to the left and constant terms to the right, we get -$[2*$r3+2*$r4]x $r1 $[$r4^2+$r3^2].
 Finally, divide by -$[2*$r3+2*$r4]&nbsp;: x $r2 $[-($r4^2+$r3^2)/(2*($r3+$r4))].
:4, add_sign

 Ranging the terms of x to the left and constant terms to the right, we get $[2*$r3+2*$r4]x $r1 $[$r4^2-$r3^2].
 Finally, divide by $[2*$r3+2*$r4]&nbsp;: x $r1 $[($r4^2-$r3^2)/(2*($r3+$r4))].
:4, add_sign, 5, div_neg

 Ranging the terms of x to the left and constant terms to the right, we get -$[2*$r3+2*$r4]x $r1 $[$r4^2+$r3^2].
 Finally, divide by -$[2*$r3+2*$r4]&nbsp;: x $r1 $[-($r4^2+$r3^2)/(2*($r3+$r4))].
:5, div_neg

 Finally, divide by -$[2*$r3+2*$r4]&nbsp;: x $r1 $[-($r4^2-$r3^2)/(2*($r3+$r4))].
