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Question 121840: I need help with solving this absolute problem
|7-x|<8
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! You can work this problem in two steps just as you would work an ordinary inequality problem.
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First work it just using the quantity inside the absolute value signs. So work it as:
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7 - x < 8
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Subtract 7 from both sides and you have:
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-x < 1
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But you are trying to solve for +x. So multiply both sides of the inequality by -1. And most
important, don't forget that if you multiply or divide both sides of an inequality by a negative
quantity, you must reverse the direction of the inequality sign. So in this case, multiplying
by -1 results in:
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x > -1
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That's one limit. x must lie to the right of -1 on the number line. Now to the second part
of the problem.
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Take the entire quantity inside the absolute value signs and precede it by a minus sign.
Then solve the problem again just as you would an inequality. So this time the problem is:
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-(7 - x) < 8
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Since the parentheses are preceded by a negative sign you can remove the negative sign
and the parentheses by changing the signs of the two terms inside the parentheses. When
you do that the inequality becomes:
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-7 + x < 8
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Solve this by adding +7 to both sides to get rid of the -7 on the left side and end up with:
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x < 15
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This is the second limit on the value of x. x must lie to the left (be less than) 15 on
the number line. So the two limits are x is greater than -1 and less than 15.
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You can write this in the form of:
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-1 < x < 15
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Hope this gives you some insight into how to solve inequality problems that involve absolute
values.
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Just to build our confidence level, let's assume a couple of values for x in this range and
see if the given absolute inequality is satisfied.
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Let's assume x = 14. Then is |7-14| < 8 ??? Inside the absolute value sign the terms 7 and -14
combine to -7 ... so we have |-7| < 8. And since the absolute value of -7 is just 7 the
inequality is 7 < 8 and that is correct.
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Let's assume x = 0. Then is |7-0| < 8. Sure, because |7| = 7 and again we have reduced the
inequality to 7 < 8 which is correct.
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Just to be sure, you may also want to try setting x = -2 and x = 16. These are outside the
range of our answer so they both should not satisfy the inequality. When x = -2 you should
get 9 < 8 which does not work. And when x = 16 you should also get 9 < 8 and this also is correct.
So it appears as if the answer is probably correct.
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