document.write( "Question 1184097: Solve the inequality. (Enter your answer using interval notation.)
\n" ); document.write( "|x| < 6 \n" ); document.write( "

Algebra.Com's Answer #814627 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "The absolute value inequality means x is between -6 and +6:

\n" ); document.write( " $\"abs%28x%29%3C6\"$ ==> $\"-6%3Cx%3C6\"$ ==> (-6,6)

\n" ); document.write( "It is helpful to solve absolute value inequalities by interpreting $\"abs%28x-a%29%3Cb\"$ to mean \"the distance between x and a is less than b\".

\n" ); document.write( "Your example is then

\n" ); document.write( " $\"abs%28x%29%3C6\"$ ==> $\"abs%28x-0%29%3C6\"$ ==> the distance between x and 0 is less than 6.

\n" ); document.write( "That means x can be up to 6 units to the left of 0 or up to 6 units to the right of 0; and that gives the answer (-6,6).

\n" ); document.write( "Let's look at another slightly more complicated example using a formal algebraic solution and a solution using the interpretation of absolute value as the distance between points.

\n" ); document.write( "Solve $\"abs%28x-4%29%3C3\"$

\n" ); document.write( "(1) Algebraically....

\n" ); document.write( "\"x-4 is between -3 and +3\":

\n" ); document.write( " $\"-3+%3C+x-4+%3C+3\"$
\n" ); document.write( " $\"-3%2B4+%3C+x+%3C+3%2B4\"$
\n" ); document.write( " $\"1+%3C+x+%3C+7\"$

\n" ); document.write( "Solution: (1,7)

\n" ); document.write( "(2) Using distances....

\n" ); document.write( "\"the distance between x and 4 is less than 3\"

\n" ); document.write( "3 to the left of 4 is 1; 3 to the right is 7

\n" ); document.write( "Solution: (1,7)

\n" ); document.write( "The solutions by both methods are relatively simple. But for more complicated absolute value problems, using the interpretation of absolute value as the distance between points is usually easier.

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