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You can put this solution on YOUR website!
I imagine that you would know how to solve this if if was something like:
\n" ); document.write( "|3 + 2x| > 9
\n" ); document.write( "You would rewrite this as:
\n" ); document.write( "3 + 2x > 9 or 3 + 2x < -9 (\"or\" because the absolute value is \"greater than\")
\n" ); document.write( "and then solve the compound inequality.
\n" ); document.write( "Well that's exactly how we're going to start
\n" ); document.write( "|3 + 2x| > |4 - x|
\n" ); document.write( "The fact that the right side is an absolute value does not significantly change things. So we start with:
\n" ); document.write( "3 + 2x > |4 - x| or 3 + 2x < -|4 - x|
\n" ); document.write( "Now we repeat the process for the remaining absolute values! But first, let's get them on the left side. (To save time I'm not going to show you how. I hope you can figure out what I did.)
\n" ); document.write( "|4 - x| < 3 + 2x or |4 - x| < -3 + (-2x)
\n" ); document.write( "Now we rewrite each of these (using \"and\" since both inequalities say an absolute value is \"less than\"):
\n" ); document.write( "(4 - x < 3 + 2x and 4 - x > -(3 + 2x)) or (4 - x < -3 + (-2x) and 4 - x > -(-3 + -2x))
\n" ); document.write( "Now we solve this big mess. Each of the 4 inequalities are not that hard. I'll solve each one separately and then put them back together with the and's and or.
\n" ); document.write( "4 - x < 3 + 2x
\n" ); document.write( "Add x
\n" ); document.write( "4 < 3 + 3x
\n" ); document.write( "Subtract 3
\n" ); document.write( "1 < 3x
\n" ); document.write( "Divide by 3
\n" ); document.write( "1/3 < x
\n" ); document.write( "4 - x > -(3 + 2x))
\n" ); document.write( "4 - x > -3 + (-2x)
\n" ); document.write( "Add 2x:
\n" ); document.write( "4 + x > -3
\n" ); document.write( "Subtract 4:
\n" ); document.write( "x > -7
\n" ); document.write( "4 - x < -3 + (-2x)
\n" ); document.write( "Add 2x:
\n" ); document.write( "4 + x < -3
\n" ); document.write( "Subtract 4L
\n" ); document.write( "x < -7
\n" ); document.write( "4 - x > 3 + 2x
\n" ); document.write( "Add x:
\n" ); document.write( "4 > 3 + 3x
\n" ); document.write( "Subtract 3:
\n" ); document.write( "1 > 3x
\n" ); document.write( "Divide by 3:
\n" ); document.write( "1/3 > x
\n" ); document.write( "Replacing each of the 4 inequalities of
\n" ); document.write( "(4 - x < 3 + 2x and 4 - x > -(3 + 2x)) or (4 - x < -3 + (-2x) and 4 - x > -(-3 + -2x))
\n" ); document.write( "with their solutions we have:
\n" ); document.write( "(1/3 < x and x > -7) or (x < -7 and 1/3 > x)
\n" ); document.write( "Now let's analyze the results. As I hope you know, you should always read inequalities from where the variable is. (This means reading it right-to-left if the variable is on the right.) The two inequalities
\n" ); document.write( "(1/3 < x and x > -7)
\n" ); document.write( "say x is greater that something. The \"and\" between them means that both must be true. The only way x and be greater than both 1/3 and -7 is if x is greater than 1/3. So we can replace
\n" ); document.write( "(1/3 < x and x > -7)
\n" ); document.write( "with
\n" ); document.write( "(1/3 < x)
\n" ); document.write( "Analyzing (x < -7 and 1/3 > x) in the same way we find that both inequalities say x is less than something. And the only way x can be less than both 1/3 and -7 is if x is less than -7. So we can replace
\n" ); document.write( "(x < -7 and 1/3 > x)
\n" ); document.write( "with
\n" ); document.write( "(x < -7)
\n" ); document.write( "Our simplified solution is now
\n" ); document.write( "(1/3 < x)
\n" ); document.write( "Analyzing (x < -7 and 1/3 > x) in the same way we find that both inequalities say x is less than something. And the only way x can be less than both 1/3 and -7 is if x is less than -7. So we can replace
\n" ); document.write( "(x < -7 and 1/3 > x)
\n" ); document.write( "with just
\n" ); document.write( "(x < -7)
\n" ); document.write( "So our simplified solution is
\n" ); document.write( "(1/3 < x) or (x < -7)
\n" ); document.write( "Unfortunately I can't figure out how to get algebra.com's graphing software to show you a number line graph of inequalities. As you Might know, with \"or\" graphs you just plot the two inequalities onto the same number line. So just plot both x > 1/3 and x < -7 on the same number line and that will be the proper graph of (1/3 < x) or (x < -7).