Question #b8a92

#"there is no solution to the equation"#

#x=-9" is an extraneous solution"#

#"that is a solution obtained by the simplification of the "#
#"process that does not satisfy the original equation"#

#color(blue)"As a check"#

#"left side "=3|-18+7|=3|-11|=33#

#"right side " =(3xx-9)-6=-33#

#"since left side " !=" right side"#

#rArrx=-9" is an extraneous solution"#

#"this is a valuable lesson in the need to check solutions"#

One way to see how the #-9# is not a solution is to plug into both sides of the original

#3abs(2xx-9+7)=3xx-9-6#

#3abs(-18+7)=-27-6#

Before we go any further, you should now see these two sides cannot be equal. The left hand side will result in a positive number, but the right hand side will result in a negative number!

#3abs(-11)=-33#

Remember, absolute value makes the number inside positive.

#3xx11=-33#

#33!=-33#

If you try to solve this algebraically, you get

#3abs(2x+7)=3x-6#

Divide by #3#

#abs(2x+7)=x-6#

The trick to solving this any more depends on the sign (positive or negative) of what's inside the absolute value brackets.

If #2x+7# is a negative number, then #2x+7 < 0# which means that #2x < -7#, which means that #x < -7/2#. So for any value of #x# less than #-7/2#, the inside will become POSITIVE on the left. But plugging a number less then #-7/2# into the right side at this point will only give NEGATIVE numbers. So the expression is false for the infinite number of values less then #-7/2#.