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