SOLUTION: | 9-3g | less than or equal to 12

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Question 1003262: | 9-3g | less than or equal to 12
Found 3 solutions by CubeyThePenguin, josgarithmetic, greenestamps:
Answer by CubeyThePenguin(3113) About Me  (Show Source):
You can put this solution on YOUR website!
|9 - 3g| <= 12

Case 1: 9 - 3g <= 12
-3g <= 3
g >= -1

Case 2: -9 + 3g <= 12
3g <= 21
g <= 7

Combine: -1 <= g <= 7

Answer by josgarithmetic(39625) About Me  (Show Source):
You can put this solution on YOUR website!
abs%289-3g%29%3C=12

3%2Aabs%283-g%29%3C=12

abs%283-g%29%3C=4


If 3-g is not negative, then
3-g%3C=4
-g%3C=4-3
-g%3C=1
g%3E=-1

If 3-g is negative, then
-3%2Bg%3C=4
g%3C=7


answer: highlight%28-1%3C=g%3C=7%29

Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


You have received two responses showing very similar formal algebraic solutions.

For many problems involving absolute values, it is easier to solve the problem by interpreting the statement

abs%28x-a%29=b

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.

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.

We first need to get the equation with |x-a| alone on one side. For your problem...

abs%289-3g%29=12
abs%283g-9%29=12
3%2Aabs%28g-3%29=12
abs%28g-3%29=4

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.

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.

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.