SOLUTION: I HOPE U CAN HELP ME WITH THIS....SOLVE THE EQUATION AND INEQUALITIES. WRITE THE SOLUTION TO THE INEQUALITIES IN INTERVAL NOTATION. THANK U. A) 2|3-P|-4=2 B) 2|3-P|-4<2

Algebra ->  Absolute-value -> SOLUTION: I HOPE U CAN HELP ME WITH THIS....SOLVE THE EQUATION AND INEQUALITIES. WRITE THE SOLUTION TO THE INEQUALITIES IN INTERVAL NOTATION. THANK U. A) 2|3-P|-4=2 B) 2|3-P|-4<2       Log On


   

Question 134463: I HOPE U CAN HELP ME WITH THIS....SOLVE THE EQUATION AND INEQUALITIES. WRITE THE SOLUTION TO THE INEQUALITIES IN INTERVAL NOTATION. THANK U.
A) 2|3-P|-4=2
B) 2|3-P|-4<2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
A)


2%2Aabs%283-p%29-4=2 Start with the given equation


2%2Aabs%283-p%29=6 Add 4 to both sides.


abs%283-p%29=3 Divide both sides by 2



Break up the absolute value (remember, if you have abs%28x%29=a, then x=-a or x=a)

3-p=-3 or 3-p=3 Set the expression 3-p equal to the original value 3 and it's opposite -3




Now lets focus on the first equation 3-p=-3


-p=-3-3Subtract 3 from both sides


-p=-6 Combine like terms on the right side


p=%28-6%29%2F%28-1%29 Divide both sides by -1 to isolate p



p=6 Divide






Now lets focus on the second equation 3-p=3



-p=3-3Subtract 3 from both sides


-p=0 Combine like terms on the right side


p=%280%29%2F%28-1%29 Divide both sides by -1 to isolate p



p=0 Divide





So the solutions to 2%2Aabs%283-p%29-4=2 are:

p=6 or p=0



Notice if we graph y=2%2Aabs%283-x%29-4 and y=2 (just set each side equal to y and graph), we get


graph%28500%2C500%2C-2%2C8%2C-10%2C10%2C2%2Aabs%283-x%29-4%2C2%29 Graph of y=2%2Aabs%283-x%29-4 (red) and y=2(green)

and we can see the two graphs intersect at x=6 and x=0. So this verifies our answer.








2%2Aabs%283-p%29-4%3C2 Start with the given inequality


2%2Aabs%283-p%29%3C6 Add 4 to both sides.


abs%283-p%29%3C3 Divide both sides by 2



Break up the absolute value (remember, if you have abs%28x%29%3C+a, then x+%3E+-a and x+%3C+a)

3-p+%3E+-3 and 3-p+%3C+3 Break up the absolute value inequality using the given rule


-3+%3C+3-p+%3C+3 Combine the two inequalities to get a compound inequality


-6+%3C+-p+%3C+0 Subtract 3 from all sides


6+%3E+p+%3E+0 Multiply all sides by -1 to isolate p. This will flip the inequality signs

0%3Cp%3C6 Rearrange the inequality

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Answer:

So our answer is

0%3Cp%3C6



which looks like this in interval notation





if you wanted to graph the solution set, you would get
Graph of the solution set in blue and the excluded values represented by open circles