Question 73747: How do you solve and graph:
/c-4/ > 1
The c-4 is in absolute value lines.
Thank you!
Found 2 solutions by jim_thompson5910, bucky:
Answer by jim_thompson5910(35256)   (Show Source):
Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website! Given:
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solve and graph
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One way to do absolute value problems is to solve them as two separate problems. For the
first problem, remove the absolute value signs and solve:
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+(c-4)>1
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You can treat this just as you would and equation ... except for one difference. That difference
is that if you have to multiply or divide both sides by a negative number, you must then
reverse the direction of the inequality sign. Don't forget that.
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Now back to solving:
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+(c-4)>1
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The parentheses are preceded by a plus sign so you can just remove them and the inequality
becomes:
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+c - 4 > 1
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Just as you would in an equation, eliminate the -4 on the left side by adding 4 to both
sides to get:
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+c > 5
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That's the first limit on the value of c ... it must be greater than +5.
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Now, on to the second equation. We do the same thing as before ... remove the absolute
value signs, except this time we put a minus sign in front of the quantity that was inside
the absolute value signs. When you do this you have:
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-(c-4)>1
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Because the quantity is preceded by a negative sign, when you remove the parentheses
you must change the sign of every term that was inside the parentheses. When you do
that the inequality becomes:
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-c + 4 >1
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We need to solve this for +c. Let's begin by eliminating the +4 on the left side by
subtracting 4 from both sides. This causes the inequality to become:
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-c > -3
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To change the equation so that the left side is +c, multiply both sides by negative 1.
And when you do, do NOT forget to reverse the direction of the inequality sign to get:
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+c < +3
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This tells us that c must be less than +3
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Combined, the two restrictions are that c can be any number less than 3 and it can also
be any number greater than 5. But it cannot be either 3 or 5 or any number between those
two values.
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Let's check that out with a trio of trials. Start with the problem:
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Let's set c equal to zero. That's a number less than +3 so it should work. When you substitute
0 for c you get:
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 which simplifies to:
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 and since  we get +4 > 1. That works!
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Now let's set c equal to 4. That should NOT work. Substitute 4 for c and get:
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 this obviously becomes:
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 and the absolute value of 0 is 0. Obviously 0 > 1 is NOT true so numbers
between 3 and 4 probably all need to be excluded.
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Finally try c = 6. That is greater than +5 and should work. Substitute +5 for c and get:
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 which becomes and simplifies to  .
At this point it is obvious that 2 > 1 is true and that adds to the likelihood that numbers
greater than 5 will work.
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We have a good solution!
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To graph, just make a number line and put dots at +3 and +5. Exclude those dots and shade
the number line from just to the left of +3 all the way to the left and from just to
the right of +5 all the way to the right. c can take any value in the shaded region.
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Hope this shows you a way to work absolute value inequalities. Remember the two different
equations to solve (one by using the absolute value quantity with a + sign and the other
by using that quantity with a minus sign preceding it), the need to reverse the direction of
the inequality sign if multiplying or dividing both sides by a negative number, and to
always solve for the positive value of the variable. The rest is just algebraic manipulation.
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Hope this works for you.
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