document.write( "Question 1208939: Solve |3x - |2x + 1|| = 4\r
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Algebra.Com's Answer #847484 by math_tutor2020(3817)\"\" \"About 
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\n" ); document.write( "Answers: x = -1 and x = 5\r
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\n" ); document.write( "\n" ); document.write( "Explanation\r
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\n" ); document.write( "\n" ); document.write( "The rule we need is |x| = k breaks down into x = k or x = -k where k is nonnegative.
\n" ); document.write( "For instance, |x| = 27 means x = 27 or x = -27.
\n" ); document.write( "Both -27 and 27 are the same distance from zero on the number line.\r
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\n" ); document.write( "\n" ); document.write( "Using that rule we can break
\n" ); document.write( "|3x - |2x+1|| = 4
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\n" ); document.write( "3x - |2x+1| = 4 and 3x - |2x+1| = -4\r
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\n" ); document.write( "\n" ); document.write( "Let's solve the first piece.
\n" ); document.write( "3x - |2x+1| = 4
\n" ); document.write( "|2x+1| = 3x-4
\n" ); document.write( "2x+1 = 3x-4 or 2x+1 = -(3x-4)
\n" ); document.write( "2x+1 = 3x-4 or 2x+1 = -3x+4
\n" ); document.write( "2x-3x = -4-1 or 2x+3x = 4-1
\n" ); document.write( "-x = -5 or 5x = 3
\n" ); document.write( "x = 5 or x = 3/5
\n" ); document.write( "Those are two potential solutions so far.
\n" ); document.write( "The key term is \"potential\" since we need to verify each solution.\r
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\n" ); document.write( "\n" ); document.write( "Now solve the 2nd piece found earlier.
\n" ); document.write( "3x - |2x+1| = -4
\n" ); document.write( "|2x+1| = 3x+4
\n" ); document.write( "2x+1 = 3x+4 or 2x+1 = -(3x+4)
\n" ); document.write( "2x+1 = 3x+4 or 2x+1 = -3x-4
\n" ); document.write( "2x-3x = 4-1 or 2x+3x = -4-1
\n" ); document.write( "-x = 3 or 5x = -5
\n" ); document.write( "x = -3 or x = -5/5 = -1
\n" ); document.write( "We found two more potential solutions.\r
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\n" ); document.write( "\n" ); document.write( "The four potential solutions we need to check are
\n" ); document.write( "x = 5 or x = 3/5
\n" ); document.write( "x = -3 or x = -1\r
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\n" ); document.write( "\n" ); document.write( "Let's return to the original equation and plug in x = 5
\n" ); document.write( "|3x - |2x+1|| = 4
\n" ); document.write( "|3*5 - |2*5+1|| = 4
\n" ); document.write( "|15 - |10+1|| = 4
\n" ); document.write( "|15 - |11|| = 4
\n" ); document.write( "|15 - 11| = 4
\n" ); document.write( "|4| = 4
\n" ); document.write( "4 = 4 ..... works
\n" ); document.write( "The solution x = 5 is confirmed.\r
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\n" ); document.write( "\n" ); document.write( "Now test x = 3/5
\n" ); document.write( "|3x - |2x+1|| = 4
\n" ); document.write( "|3*3/5 - |2*3/5+1|| = 4
\n" ); document.write( "|9/5 - |6/5+1|| = 4
\n" ); document.write( "|9/5 - |6/5+5/5|| = 4
\n" ); document.write( "|9/5 - |11/5|| = 4
\n" ); document.write( "|9/5 - 11/5| = 4
\n" ); document.write( "|(9-11)/5| = 4
\n" ); document.write( "|-2/5| = 4
\n" ); document.write( "2/5 = 4 ..... this doesn't work out
\n" ); document.write( "We determine that x = 3/5 doesn't work, so it is extraneous.
\n" ); document.write( "We must cross it off the list.\r
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\n" ); document.write( "\n" ); document.write( "Now onto x = -3
\n" ); document.write( "|3x - |2x+1|| = 4
\n" ); document.write( "|3*(-3) - |2*(-3)+1|| = 4
\n" ); document.write( "|-9 - |-6+1|| = 4
\n" ); document.write( "|-9 - |-5|| = 4
\n" ); document.write( "|-9 - 5| = 4
\n" ); document.write( "|-14| = 4
\n" ); document.write( "14 = 4 ...... this doesn't work
\n" ); document.write( "Remove x = -3 from the list.\r
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\n" ); document.write( "\n" ); document.write( "Lastly we need to check x = -1
\n" ); document.write( "|3x - |2x+1|| = 4
\n" ); document.write( "|3*(-1) - |2*(-1)+1|| = 4
\n" ); document.write( "|-3 - |-2+1|| = 4
\n" ); document.write( "|-3 - |-1|| = 4
\n" ); document.write( "|-3 - 1| = 4
\n" ); document.write( "|-4| = 4
\n" ); document.write( "4 = 4 ...... this works\r
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\n" ); document.write( "\n" ); document.write( "Therefore we confirm that only x = -1 and x = 5 are the answers.\r
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\n" ); document.write( "\n" ); document.write( "Verification using a Desmos graph is shown here
\n" ); document.write( "https://www.desmos.com/calculator/lr9kd0nk5f
\n" ); document.write( "The two intersection points have x coordinates of x = -1 and x = 5.
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