Question 211444: |5x+2| - 10 = -3 and |5x+2| - 10 <= -3
I have to explain the difference in the answers. I was able to get with the first equation:
5x+2=7 OR -5x-2=7
5x=5 OR -5x=9
x=1 OR x=-9/5
I am not sure if I am even on the right track here. I am not sure on the second problem if I solve it the same way with the greater than/equal to sign.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! -----
|5x+2| - 10 = -3 is your first equation (I think).
Add 3 to both sides of this equation to get:
|5x+2| = -3 + 10
This becomes:
|5x+2| = 7
By the basic definition of absolute values, this means that:
(5x+2) = 7 if x >= 0
or:
-(5x+2) = 7 if x <= 0
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multiply both sides of the equation -(5x+2) = 7 by -1 and you get:
(5x+2) = -7
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your possible solutions become:
(5x+2) = 7 if x >= 0
or:
(5x+2) = -7 if x < 0
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solve 5x+2 = 7 to get:
5x = 5
x = 1
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solve 5x+2 = -7 to get:
5x = -9
x = -9/5
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your solutions are:
x = 1
or:
x = -9/5
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to confirm these answers are correct, you substitute them in the original equation to see if that equation is true.
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your original equation is:
|5x+2|-10=-3
substitute 1 for x to get:
|5+2| - 10 = -3
this becomes:
|7|-10 = -3 which becomes 7-10 = -3 which becomes -3 = -3 which is true.
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substitute -9/5 for x in the original equation to get:
|5*-9/5 + 2| - 10 = -3
this becomes:
|-9 + 2| - 10 = -3
this becomes |-7| - 10 = -3 which becomes 7 - 10 = -3 which becomes -3 = -3 which is also true.
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your answers for the first equation of |5x+2|-10=-3 are good.
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Your second equation to solve is:
|5x+2| - 10 <= -3
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This is essentially the same equation except now you are dealing with smaller than or equal to rather than just equal to.
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you solve it the same way except you may see a reversal of signs as you work your way through it.
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add 10 to both sides of this equation to get:
|5x+2| <= -3 + 10 which becomes:
|5x+2| <= 7
By the basic definition of absolute value, this equation becomes:
(5x+2) <= 7 if x is greater than or equal to 0
or:
-(5x+2) <= 7 if x is less than 0.
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multiply both sides of -(5x+2) <= 7 by -1 to get:
(5x+2) >= -7
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Note the reversal of signs.
Multiply both sides of an inequality by -1 reverses the inequality.
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your possible solutions are now:
(5x+2) <= 7
and:
(5x+2) >= -7
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Note the use of AND rather than OR.
More on this later.
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subtract 2 from both sides of (5x+2) <= 7 to get:
5x <= 5
divide both sides by 5 to get:
x <= 1
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subtract 2 from both sides of (5x+2) >= -7 to get:
5x >= -9
divide both sides by 5 to get:
x >= -9/5
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your possible solutions are x <= 1 AND x >= -9/5
Note the use of AND rather than OR
Your answer has to be BETWEEN -9/5 and 1 which requires the use of AND rather than OR.
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These solutions can be written as:
-9/5 <= x <= 1
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The difference in the answers is caused by the one equation being equal to only and the other equation being equal to or less than.
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In the first equation, your answer is a or b only.
In the second equation, your answer is a or b and anything in between.
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