SOLUTION: I have been struggling with this problem for a while now and I was wondering if anyone could help me? I would deeply appreciate it!! Please and Thank You!! Combining Open Sen

Algebra ->  Inequalities -> SOLUTION: I have been struggling with this problem for a while now and I was wondering if anyone could help me? I would deeply appreciate it!! Please and Thank You!! Combining Open Sen      Log On


   

Question 126360This question is from textbook Algebra Structure and Method
: I have been struggling with this problem for a while now and I was wondering if anyone could help me? I would deeply appreciate it!! Please and Thank You!!

Combining Open Sentences
solve each open sentence. Graph the solution set, if there is one.
6-c is less than 2c+3 is less than or equal to 8+c This question is from textbook Algebra Structure and Method

Found 2 solutions by stanbon, jim_thompson5910:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
6-c is less than 2c+3 is less than or equal to 8+c
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6-c <= 2c+3
3c>=3
c> -1
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AND
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2c+3<=8+c
c<=5
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Final: -1 < c <=5
=================
Cheers,
Stan H.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
6-c%3C2c%2B3%3C=8%2Bc Start with the given inequality



Break up the inequality. Remember a%3Cb%3Cc is equivalent to the group a%3Cb and b%3Cc


6-c+%3C+2c%2B3+ and 2c%2B3+%3C+=8%2Bc+



Now let's solve the first inequality



6-c%3C2c%2B3 Start with the given inequality



-c%3C2c%2B3-6Subtract 6 from both sides


-c-2c%3C3-6 Subtract 2c from both sides


-3c%3C3-6 Combine like terms on the left side


-3c%3C-3 Combine like terms on the right side


c%3E%28-3%29%2F%28-3%29 Divide both sides by -3 to isolate c (note: Remember, dividing both sides by a negative number flips the inequality sign)



c%3E1 Divide

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Answer:
So the first part of our answer is c%3E1








2c%2B3%3C=8%2Bc Start with the given inequality



2c%3C=8%2Bc-3Subtract 3 from both sides


2c-c%3C=8-3 Subtract c from both sides


c%3C=8-3 Combine like terms on the left side


c%3C=5 Combine like terms on the right side

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Answer:
So the second part of our answer is c%3C=5





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Putting the the two answers together, we get:
c%3E1 and c%3C=5


Also, the two solutions form the compound inequality

1%3Cc%3C=5





So the solution in interval notation is: (1,5]


Now let's graph the solution set



Note: at c=1 there is a open circle (which means this point is excluded) and at c=5 there is a closed circle (which means this point is included)