Question 1003262
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You have received two responses showing very similar formal algebraic solutions.<br>
For many problems involving absolute values, it is easier to solve the problem by interpreting the statement<br>
{{{abs(x-a)=b}}}<br>
as meaning the difference between x and a is equal to b.  Then the problem is easily solved on a number line, using the difference as a distance in either of the two directions.<br>
For solving absolute value inequalities, you can think first of the corresponding equation and then use common sense to find the solution to the inequality.<br>
We first need to get the equation with |x-a| alone on one side.  For your problem...<br>
{{{abs(9-3g)=12}}}
{{{abs(3g-9)=12}}}
{{{3*abs(g-3)=12}}}
{{{abs(g-3)=4}}}<br>
Interpret that to say "the distance between g and 3 is 4".  Then it is easy to determine that 4 either side of 3 on a number line is either -1 or 7.<br>
So -1 and 7 are the solutions to the absolute value EQUATION; now use common sense to see that the solution the inequality is everything between -1 and 7 -- including the end points, since the inequality is less than or equal to.<br>
This way of looking at and solving absolute value inequalities works especially well if the inequality is "greater than or equal to".  Having solved the corresponding equation to find that -1 and 7 are the two points that are exactly u units from 3 on a number line, it is easy to see that the solution for "distance between x and 3 is GREATER than or equal to 4" will be all the numbers less than or equal to -1 OR all the numbers greater than of equal to 7.<br>