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Question 1144582: Solve. 1.) |x-3|-5≥2
2.) |2x+12|=-6
Could someone step by step explain how to solve these two problems? Thank you.
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! problem 1.
|x-3| - 5 >= 2
add 5 to both sides of the equation to get:
|x-3| >= 7
by definition, this means that:
(x-3) >= 7 and -(x-3) >= 7
(x-3) >= 7 becomes x-3 >= 7.
add 3 to both sides of that equation to get:
x >= 10
-(x-3) >= 7 becomes -x + 3 >= 7.
subtract 3 from both sides of that equation to get:
-x >= 4
multiply both sides of that equation by -1 to get:
x <= -4
that's because multiplying both sides of an inequality by a negative number reverses the inequality.
you get:
x >= 10 and x <= -4
when x = 10, |x-3| - 5 >= 2 becomes 2 >= 2 which is true.
when x = 12, |x-3| - 5 >= 2 becomes 4 >= 2 which is true.
when x = -4, |x-3| - 5 >= 2 becomes |-7| - 5 >= 2 becomes 7 - 5 >= 2 which becomes 2 >= 2 which is true.
when x = -6, |x-3| - 5>= 2 becomes |-9| - 5 >= 2 which becomes 9 - 5 >= 2 which becomes 4 >= 2 which is true.
when x is greater than -4 and less than 10, like x = 0, then:
|x-3| - 5 >= 2 becomes |-3| - 5 >= 2 which becomes 3 - 5 >= 2 which becomes -2 >= 2 which is NOT true.
the inequality is only true when x <= -4 and x >= 10.
this can be seen grephically below.
for your second problem, a similar procedure is taken.
start with |2x + 12| = -6
since the absolute value can never be negative, there is no solution to this problem.
if you tried to solve it, you would get the following.
this is true if and only if (2x + 12) = -6 and -(2x + 12) = -6
when (2x + 12) = -6, you get:
2x + 12 = -6 becomes 2x = -18 which becomes x = -9.
when x = -9, |2x + 12| = -6 becomes |-18 + 12| = -6 which becomes|-6| = -6 which becomes 6 = -6 which is NOT true.
when -(2x + 12) = -6, you get:
-2x - 12 = -6 which becomes -2x = 6 which becomes -x = 3 which becomes x = -3.
when x = -3, |2x + 12| = -6 which becomes |-6 + 12| = -6 which becomes |6| = -6 which becomes 6 = -6 which is NOT true.
there is no solution because the absolute value of an expression has to be greater than or equal to 0, which it is not when the absolute value is equal to -6.
Answer by ikleyn(52834)  (Show Source):
You can put this solution on YOUR website! .
I want to show you the short way to solve these problems.
(1) | x - 3 | - 5 >= 2
It is equivalent to this inequality
| x - 3 | >= 7.
The solutions to the last inequality are all numbers in the number line that are remoted from the number 3 to 7 or more units.
Obviously, these numbers are x >= 3 + 7 = 10 OR x <= 3 - 7 = -4.
So, the solution to the original inequality is the union of two sets { x <= -4 } and { x >= 10 }. ANSWER
Notice that all absolute value inequalities OF THIS FORM can be easily solved using this chain of arguments.
See the lesson
- Solving absolute value inequalities
in this site.
(2) | 2x + 12 | = -6.
An absolute value of a number is ALWAYS non-negative.
It can not be negative.
So, the given equation HAS NO solutions in real number. <<<---=== At this point, the solution is just COMPLETED.
You can make your conclusion even without writing this absolute value equation in any other form.
Moreover, when you make this conclusion without writing this absolute value equation in any other form,
you demonstrate that you FIRMLY KNOW what the absolute value is.
If, in opposite, you try to transform such an equation to any other form, you demonstrate that you are not firm in your knowledge.
Happy learning (!)
Come again soon to the forum to learn something new (!)
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