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\n" ); document.write( "One method of solving this involves letting the entire quantity within the absolute
\n" ); document.write( "value signs first have a positive sign and then have a negative sign. Solve these two cases.
\n" ); document.write( "Perhaps it is best explained by using this problem as a typical example.
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\n" ); document.write( "The quantity inside the absolute value signs is
\n" ); document.write( "equation using +(x+2) for the left side. In this case the inequality becomes.
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\n" ); document.write( "Solve this for x just as you would an ordinary equation ... by subtracting +2 from both
\n" ); document.write( "sides to get the answer as
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\n" ); document.write( "But that's only the first part of the solution. The second part says to give the entire
\n" ); document.write( "quantity inside the absolute value signs a negative sign and solve the inequality again.
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\n" ); document.write( "So this time the inequality becomes:
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\n" ); document.write( "You can remove the parentheses preceded by a negative sign by changing the sign of all the
\n" ); document.write( "terms inside the parentheses and then just erasing the parentheses. If you do this you
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\n" ); document.write( "Begin the solution by adding 2 to both sides to eliminate the -2 on the left side. When
\n" ); document.write( "you do that, the inequality becomes:
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\n" ); document.write( "Next comes a tricky part. You want to solve for positive x, so you are going to multiply both
\n" ); document.write( "sides by -1. However, if you multiply or divide both sides of an inequality by a negative
\n" ); document.write( "number you must reverse the direction of the inequality sign. No big deal, just something to
\n" ); document.write( "remember. So in this case after the multiplication and sign reversal the solution for x
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\n" ); document.write( "In summary, the regions on the number line that satisfy the original inequality are any
\n" ); document.write( "value of x equal to or to the left of -6 and any value of x equal to or to the right of +2.
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\n" ); document.write( "You can do a quick check by returning to the original problem and solve it for x = -7,
\n" ); document.write( "and x = +3. Or use any convenient value in the two regions ... for example x = - 10 and
\n" ); document.write( "x = + 10. You should see that the inequality is satisfied. Plus you can use any value
\n" ); document.write( "of x between -6 and +2 to prove to yourself that the original inequality is not satisfied by
\n" ); document.write( "values of x in that region. Usually x = 0 works well for this check because all the terms
\n" ); document.write( "containing x just disappear. It's usually a good idea to check the points you found on
\n" ); document.write( "the number line also. If you let x = -6 and then let x = +2, you should find that in
\n" ); document.write( "both cases the inequality of this problem is true only because of the equal sign associated
\n" ); document.write( "with the less than or greater than sign. Therefore, for this inequality x = -6
\n" ); document.write( "and x = +2
\n" ); document.write( "are included in the solution set.
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\n" ); document.write( "In summary, solve two problems ... first use the entire quantity inside the absolute value
\n" ); document.write( "signs with a plus sign and solve the inequality for +x. ... next use the entire quantity inside
\n" ); document.write( "the absolute value signs with a negative sign and solve the inequality for +x. Don't forget
\n" ); document.write( "multiplication (or division) of both sides of the inequality by a negative number,
\n" ); document.write( "requires you to reverse the inequality sign. Finally use numbers in each of the regions that
\n" ); document.write( "you identify for the answer, and plug them into the original problem just to make sure
\n" ); document.write( "they work correctly. It also pays to check the end point numbers (in this problem they
\n" ); document.write( "were -6 and +2) to validate what region they fall into (are they part of the solution set
\n" ); document.write( "or are they not). [In this problem they were in the solution set because of the equal sign.]\r
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\n" ); document.write( "Hope you find this method easy to remember and useful to you. \n" ); document.write( "