QR Question

[cougarblue84]

To respect copyright I won't post the exact question, but problem 101 in Destroyer looks similar to the following and it asks to solve:

[(X-2)/3] >_ 5 "Absolute value of [(x-2)/3] is greater than or equal to 5"

The answer key said the way to solve this problem is to rewrite the inequality without the absolute value expression and change the sign on one of the inequalities. Basically, you end up with (x-2)/3 >_ 5 and (x-2)/3 _< -5. "(x-2)/3 is greater than or equal to 5, and (x-2)/3 is less than or equal to negative 5." Then you solve for x. Made sense, until I came to problem 150, which looks similar and asks for the solution set to a problem similar to the following:

[3x-5] >_ 9 "Absolute value of 3x-5 is greater than or equal to 9"

The answer key said the way to solve this problem, since it is greater than, is to use the formula [x]>_ a or [x]>_ -a. "Absolute value of x is greater than or equal to 'a' or absolute value of x is greater than or equal to negative 'a'."

These two problems look like the exact same set up to me, so how come you rewrite the first problem without absolute values, change the sign on one, and do two 'less than or equal to/greater than or equal to' inequalities, but on the second problem you keep the absolute values, change the sign on one, and do two 'greater than or equal to' inequalities?

Thanks for bearing with me. Sorry, I just can't figure this one out. Does anyone understand and would be willing to explain? It might be a simple answer, but then again most of the difficult problems have simple answers in the end. Thanks a ton!!

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[osimsDDS]

is the absolute value also on the denomenator 3 or only on the numberator x-2??? thanks

 

[cougarblue84]

Yeah, sorry about that! I'm not the best at writing equations out on the keyboard. It's just x-2/3 without any absolute values in the solution key.

 

[Streetwolf]

Typo. Should be different signs.

Just think about it. Absolute value measures distance from zero. If you want the absolute value to be LESS THAN x, you want all values 0 through x either positive or negative. That means less than +x and greater than -x.

If you want the absolute value to be GREATER THAN x, you want all values greater than x, either positive sign or negative sign in front. That means greater than +x and less than -x.

For example if I want the absolute value of x (written as |x| by the way) to be less than 7, that means I want all values between 0 and 7 with either a positive or a negative sign. So 1 works and so does -1. 2 works and so does -2. Also fractions like 3.6 and -3.6 work. So you want x < 7 and x > -7.

If I want the absolute value of x to be greater than 7, that means I want all values greater than 7 with either a positive sign or negative sign in front. So first consider the non-signed numbers. What works? 8 would work, 9 would work, 65.52 would work. Now include the signs. +8 and -8 work. +9 and -9 work. +65.52 and -65.52 work. So you want x > 7 and x < -7.