Mth 271
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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Applied Calculus Book pg. 0-12. Assignment 2, Homework problem #27 in exercises 0.2. absolute value of 10-x is greater than 4.
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When I solved this problem I got 14>x<6. I do not understand how the book answer is x<6 or x>14. I got the x<6, but not the x>14. When I solved this I first had -4<10-x>4. Then I subtracted 10 from both sides. Leaving me with -14 <-x>-6. To get rid of the -x I had to divide by -1 which changes the signs and leaves the answer as 14>x<6.
| a | < b means that a < b and also that a > -b, where b is assumed to be a positive number.
For example, | x | < 5 is not satisfied for all x values less than 5. For example, -10 is less than 5 but | -10 | = 10 , which is not less than 5.
In fact for this inequality x must be less than 5, but it cannot be less than -5. In fact, x satisfies | x | < 5 if, and only if, -5 < x < 5.
Now consider | x | > 5. This is certainly satisfied if x > 5. However it is also satisfied if x < -5. Whereas in the preceding example both inequalities x < 5 and -5 < x can be satisfied by numbers in a certain range, x > 5 and x < -5 cannot both be satisfied by any number. The inequality | x | > 5 is satisfied if x > 5, or if x < -5.
These examples and others of this type lead to the following rules:
| x | < b means -b < x < b (i.e., -b < x AND x < b)
| x | > b means x < -b OR b < x.
Applying this concept to the present problem:
| 10 - x | > 4 means
10 - x > 4 OR 10 - x < -4.
10 - x > 4 has solution x < 6, which agrees with the solution you obtained. All numbers x such that x < 6 are solutions to the original equation.
10 - x < -4 has solution x > 14, which was not part of your solution. All numbers x such that x > 14 are solutions to the original equation.
The solution to the equation | 10 - x | > 4 is therefore:
x < 6 OR x > 14.