Inequalities and Absolute Value, with Emphasis on When Solutions ApplyDate: 8/28/96 at 20:50:45 From: Brian Hicks Subject: Inequalities and Absolute Value Hi Dr Math! I have a few problems with inequalities and absolute values... What would the solution set be to something like this... | X | > X For this inequality, I had X > X or X < -X How could this result be possible? And... | X + 2 | - X >= 0 On this problem, my solution set was { X : 2>_0 or X < -X } 48 | 7K-30 | Where K = 14. My solution to this one was 48 | 68 | But, I cannot distribute regularly right? How would I solve this? Thanks! I appreciate your time and advice. Brian Date: 8/29/96 at 14:21:57 From: Doctor Jodi Subject: Re: Inequalities and Absolute Value Hi there! Let me go through each of your problems and try to give you some suggestions. > | X | > X Since x > x is NEVER true, you can eliminate that as a "solution": no matter what the value of x, it will never be greater than itself. So this problem could also be written x < -x. For any value less than zero, x will be less than its opposite. Does that make sense? > | X + 2 | - X >= 0 On this problem, my solution set was > { X : 2>_0 or X < -X } When you're doing problems like this one, you'll want to remember when the solutions apply. Here's how I'd do this problem: |x + 2| - x >= 0 |x + 2| >= x Now you need to divide this problem into two parts: |x + 2| is positive or zero: ---------------------------- Since |x + 2| is the same as x + 2, you can remove the abs. value sign without changing the rest of the problem: x + 2 >= x 2 >= 0 Since 2 is ALWAYS greater than zero, this means that if |x + 2| is non-negative the statement |x + 2| - x >= 0 is true. So the final answer set must include all numbers greater than or equal to -2. Now there's the second part of the problem: |x + 2| is negative: -------------------- This happens when x is less than -2. The absolute value sign will change |x + 2| to its opposite, - (x + 2) (aka -x -2 ), so |x + 2| >= x -(x + 2) >= x -x - 2 >= x -2 >= 2x -1 >= x Thus, this statement is true for any x that's less than or equal to -1 and less than -2. This means it's true for all numbers less than -2. Combining the first and the second parts of the problem, we see that the equation holds for all real numbers. ------------------------------- > 48 | 7K-30 | The absolute value of 68 is 68, so now multiply 48 * 68 to get the answer. -Doctor Jodi, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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