document.write( "Question 817129: absolute value of 3x+2 - absolute value of x-3=5 \n" ); document.write( "

Algebra.Com's Answer #491948 by jsmallt9(3758) $\"\"$ $\"About$

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$\"abs%283x%2B2%29+-+abs%28x-3%29=5\"$
\n" ); document.write( "First let's review how to solve a simple absolute value equation or inequality:

Isolate an absolute value.
Rewrite the equation/inequality as two equations/inequalities without an absolute value:
- One equation or inequality will be exactly the same except it will not have an absolute value.
- If it is a less than (or less than or equal to) inequality then write the word \"and\". If it is an equation or a greater than inequality, write the word \"or\".
- The second equation or inequality comes from
  - Writing the absolute value expression without its absolute value.
  - Reversing the equation or inequality symbol.
  - Rewriting the other side as its opposite/negative.
Solve the two equations/inequalities.

For example:
\n" ); document.write( " $\"abs%28x-3%29+%2B+1+%3C+6\"$
\n" ); document.write( "Subtract 1 to isolate the absolute value:
\n" ); document.write( " $\"abs%28x-3%29+%3C+5\"$
\n" ); document.write( "Rewrite without absolute values:
\n" ); document.write( " $\"x-3+%3C+5\"$ [same as above without absolute value]
\n" ); document.write( "and [since it is a less than inequality]
\n" ); document.write( " $\"x-3+%3E+-5\"$ [reverse the inequality and make the other side its opposite]
\n" ); document.write( "In short:
\n" ); document.write( " $\"x-3+%3C+5\"$ and $\"x-3+%3E+-5\"$
\n" ); document.write( "Then we solve these.

\n" ); document.write( "Now let's try it on your problem:
\n" ); document.write( " $\"abs%283x%2B2%29+-+abs%28x-3%29=5\"$
\n" ); document.write( "Isolate. Adding the second absolute value to each side:
\n" ); document.write( " $\"abs%283x%2B2%29+=+abs%28x-3%29+%2B+5\"$
\n" ); document.write( "Rewrite without the (isolated) absolute value. (Note: We can only eliminate one absolute value at a time, the isolated one.)
\n" ); document.write( " $\"3x%2B2+=+abs%28x-3%29+%2B+5\"$ or $\"3x%2B2+=+-%28abs%28x-3%29+%2B+5%29\"$
\n" ); document.write( "Note the minus and the parentheses on the right side of the second equation. We need to express the opposite of the whole side!
\n" ); document.write( "Simplifying the right side of the second equation:
\n" ); document.write( " $\"3x%2B2+=+abs%28x-3%29+%2B+5\"$ or $\"3x%2B2+=+-abs%28x-3%29+-+5%29\"$

\n" ); document.write( "Now comes the hard part. We have to repeat the previous steps, on each equation, to eliminate the remaining absolute values.
\n" ); document.write( "Isolate. Subtracting 5 from both sides of the first equation and adding 5 to both sides of the second:
\n" ); document.write( " $\"3x-3+=+abs%28x-3%29\"$ or $\"3x%2B7+=+-abs%28x-3%29\"$
\n" ); document.write( "Multiplying both sides of the second equation by -1:
\n" ); document.write( " $\"3x-3+=+abs%28x-3%29\"$ or $\"-3x-7+=+abs%28x-3%29\"$
\n" ); document.write( "Now we're ready to rewrite. (If you have trouble following this then you might want to first rewrite these with the absolute value on the left side.) Since we have two equations and since each of these will become two equations when we rewrite them, we will end up with 4 equations!
\n" ); document.write( " $\"3x-3+=+x-3\"$ or $\"-%283x-3%29+=+x-3\"$ or $\"-3x-7+=+x-3\"$ or $\"-%28-3x-7%29+=+x-3\"$
\n" ); document.write( "Simplifying...
\n" ); document.write( " $\"3x-3+=+x-3\"$ or $\"-3x%2B3+=+x-3\"$ or $\"-3x-7+=+x-3\"$ or $\"3x%2B7+=+x-3\"$
\n" ); document.write( "Solving. (I hope doing all four at once is not too much.)
\n" ); document.write( " $\"2x-3+=+-3\"$ or $\"3+=+4x-3\"$ or $\"-7+=+4x-3\"$ or $\"2x%2B7+=+-3\"$
\n" ); document.write( " $\"2x+=+0\"$ or $\"6+=+4x\"$ or $\"-4+=+4x\"$ or $\"2x+=+-10\"$
\n" ); document.write( " $\"x+=+0\"$ or $\"6%2F4+=+x\"$ or $\"-4%2F4+=+4x\"$ or $\"x+=+-10%2F2\"$
\n" ); document.write( "Simplifying:
\n" ); document.write( " $\"x+=+0\"$ or $\"3%2F2+=+x\"$ or $\"-1+=+x\"$ or $\"x+=+-5\"$

\n" ); document.write( "Last we check. Use the original equation to check:
\n" ); document.write( " $\"abs%283x%2B2%29+-+abs%28x-3%29=5\"$
\n" ); document.write( "We should find that only 3/2 and -5 actually work. Zero and -1 do not.

\n" ); document.write( "P.S. If you want to understand why we got solutions that didn't work, then look at the original equation:
\n" ); document.write( " $\"abs%283x%2B2%29+-+abs%28x-3%29=5\"$
\n" ); document.write( "If you understand that absolute values are never negative then with a little thought we can figure out that
\n" ); document.write( " $\"abs%283x%2B2%29+%3E=5\"$
\n" ); document.write( "must be true because we have to subtract the other absolute value from this one to get 5. (If the second absolute value is zero (its lowest possible value) then the first one would have to be 5. If the second absolute value is positive then the first one has to be that much more than 5.)

\n" ); document.write( "Let's solve $\"abs%283x%2B2%29+%3E=5\"$ :
\n" ); document.write( " $\"3x%2B2+%3E=5\"$ or $\"3x%2B2+%3E=-%285%29\"$
\n" ); document.write( "Solving...
\n" ); document.write( " $\"3x+%3E=3\"$ or $\"3x+%3E=-7\"$
\n" ); document.write( " $\"x+%3E=1\"$ or $\"x+%3E=-7%2F3\"$
\n" ); document.write( "Since $\"abs%283x%2B2%29+%3E=5\"$ must be true, then the full solution to your equation is:
\n" ); document.write( "( $\"x+%3E=1\"$ or $\"x+%3E=-7%2F3\"$ ) and ( $\"x+=+0\"$ or $\"3%2F2+=+x\"$ or $\"-1+=+x\"$ or $\"x+=+-5\"$ )
\n" ); document.write( "Now we can see why 0 and -1 do not work. They do not fit the ( $\"x+%3E=1\"$ or $\"x+%3E=-7%2F3\"$ ) part of the solution. \n" ); document.write( "

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