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Question 92582: Solve each absolute value inequality and graph the solution set.
|x-2|> = 3
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Given to solve and graph:
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You can solve this by setting up two inequalities. The inequalities come from taking the
quantity inside the absolute value signs and preceding it first with a + sign for one inequality
and then preceding it with a minus sign.
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And since you have an inequality, remember the basic rule that if you multiply or divide both
sides by a negative, then you reverse the direction of the inequality sign.
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With this in mind, let's begin by writing the inequality using the quantity inside the
absolute value signs preceded by a + sign.
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and the parentheses can be eliminated without affecting the inequality because the parentheses
are preceded by an implied + sign. This makes the inequality become:
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As you would do in an equation, you can get rid of the -2 on the left side by adding +2
to both sides to get:
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This is one solution. It says that x must equal or be greater than 5.
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Now to the second part of the solution. Take the quantity inside the absolute value signs
and preceded it by a minus sign to get:
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Since the parentheses are preceded by a negative sign, when you remove them you change the
sign of each of the terms inside and the equation becomes:
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Next you can get rid of the +2 on the left side by adding -2 to both sides to
get:
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We are trying to solve for +x so we need to multiply both sides by -1 to make the x
switch from -x to +x. Don't forget the rule that when you multiply or divide both sides
of an inequality by a negative quantity you reverse the direction of the inequality sign.
So when you multiply both sides by a -1 you get:
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This tells you that x must equal or be less than -1.
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To graph the two solutions to this problem, create a number line, and on that line put
dots at two points ... at -1 and at +5. Then you make the number line heavy or bold
from minus infinity all the way up to (and including) the point at -1. Next you make the
number line heavy all the way from plus infinity down to (and including) the point at +5.
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The heavy parts of the number line then tell you where values of x will satisfy the original
problem. Note that the number is not heavy between the values of x= -1 and x = +5, so that
is the region in which any value of x will not satisfy the inequality.
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Let's do a few checkpoints to see whether our answer is correct.
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How about letting x = 0, a value that is in the region we said should not work. When we
substitute 0 for x in the original problem it becomes:
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This simplifies to
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and the absolute value of -2 is +2. That means that the inequality becomes
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and this is not true. So we know that at one value between x=-1 and x= +5 the inequality
will not work.
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Next let's try letting x= +6. That is in a region that we said x has values that should
work. If you substitute +6 for x in the original problem you get:
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and this simplifies to
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and the absolute value of +4 is +4 so the inequality becomes:
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That works ... so one value in a region that we said should work, does work.
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Finally, let's let x = -2. That also is in a region that we said should work. Substituting
-2 for x in the original problem makes it become:
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This simplifies to
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and the absolute value of -4 is +4. So the inequality simplifies to:
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This is true, so we know that when x equals -2, the original inequality holds true. It
helps to verify that our original equation is true.
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You can try additional values of x to help you confirm to your self that if:
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or
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Then the original inequality will be true.
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Hope this helps you to understand how you can work inequalities of this type.
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