3c)+Solving+absolute+value+inequalities

http://www.purplemath.com/modules/absineq.htm 

Let's first return to the original definition of [|absolute value] : "|  //x //  | is the distance of  //x //  from zero." For instance, since both –2   and  2   are two units from zero, we have  | –2 | = | 2 | = 2  :  With this definition and picture in mind, let's look at some absolute value inequalities. Suppose you're asked to graph the solution to <span style="font-family: Arial, Helvetica, sans-serif;">| //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> | < 3  <span style="font-family: Arial, Helvetica, sans-serif;">. The solution is going to be all the points that are less than three units away from zero. Look at the number line: <span style="font-family: Arial, Helvetica, sans-serif;"> <span style="font-family: Arial, Helvetica, sans-serif;">The number <span style="font-family: Arial, Helvetica, sans-serif;">1  <span style="font-family: Arial, Helvetica, sans-serif;"> will work, as will  <span style="font-family: Arial, Helvetica, sans-serif;">–1  <span style="font-family: Arial, Helvetica, sans-serif;">; the number  <span style="font-family: Arial, Helvetica, sans-serif;">2  <span style="font-family: Arial, Helvetica, sans-serif;"> will work, as will  <span style="font-family: Arial, Helvetica, sans-serif;">–2  <span style="font-family: Arial, Helvetica, sans-serif;">. But <span style="font-family: Arial, Helvetica, sans-serif;">4  <span style="font-family: Arial, Helvetica, sans-serif;"> will not work, and neither will  <span style="font-family: Arial, Helvetica, sans-serif;">–4  <span style="font-family: Arial, Helvetica, sans-serif;">, because they are too far away. Even <span style="font-family: Arial, Helvetica, sans-serif;">3  <span style="font-family: Arial, Helvetica, sans-serif;"> and  <span style="font-family: Arial, Helvetica, sans-serif;">–3  <span style="font-family: Arial, Helvetica, sans-serif;"> won't work (though they're right on the edge). But <span style="font-family: Arial, Helvetica, sans-serif;">2.99  <span style="font-family: Arial, Helvetica, sans-serif;"> will work, as will  <span style="font-family: Arial, Helvetica, sans-serif;">–2.99  <span style="font-family: Arial, Helvetica, sans-serif;">. In other words, all the points between <span style="font-family: Arial, Helvetica, sans-serif;"> –3  <span style="font-family: Arial, Helvetica, sans-serif;"> and  <span style="font-family: Arial, Helvetica, sans-serif;">3  <span style="font-family: Arial, Helvetica, sans-serif;">, but not actually including  <span style="font-family: Arial, Helvetica, sans-serif;">–3  <span style="font-family: Arial, Helvetica, sans-serif;"> or  <span style="font-family: Arial, Helvetica, sans-serif;">3  <span style="font-family: Arial, Helvetica, sans-serif;">, will work in this inequality. Then the solution looks like this: <span style="font-family: Arial, Helvetica, sans-serif;"> <span style="font-family: Arial, Helvetica, sans-serif;">The open circles at the ends of the blue line [|indicate] <span style="font-family: Arial, Helvetica, sans-serif;"> "up to, but not including, these points." Your book might use parentheses instead of circles. Translating this picture into algebraic symbols, you find that the solution is <span style="font-family: Arial, Helvetica, sans-serif;">–3 < //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> < 3  <span style="font-family: Arial, Helvetica, sans-serif;">. <span style="font-family: Arial, Helvetica, sans-serif;">This pattern for "less than" absolute-value inequalities always holds: Given the inequality <span style="font-family: Arial, Helvetica, sans-serif;">| //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> | < //<span style="font-family: Arial, Helvetica, sans-serif;">a // <span style="font-family: Arial, Helvetica, sans-serif;">, the solution is always of the form  <span style="font-family: Arial, Helvetica, sans-serif;">– //<span style="font-family: Arial, Helvetica, sans-serif;">a //<span style="font-family: Arial, Helvetica, sans-serif;"> < //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> < //<span style="font-family: Arial, Helvetica, sans-serif;">a // <span style="font-family: Arial, Helvetica, sans-serif;">. Even when the exercises get more complicated, the pattern still holds. <span style="font-family: Arial, Helvetica, sans-serif;"> <span style="font-family: Arial, Helvetica, sans-serif;"> Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved <span style="font-family: Arial, Helvetica, sans-serif;">Since this is a "less than" absolute-value inequality, the first step is to clear the absolute value according to the pattern. Then I'll solve the [|linear inequality] <span style="font-family: Arial, Helvetica, sans-serif;">. <span style="font-family: Arial, Helvetica, sans-serif;">| 2 //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> + 3 | < 6 <span style="font-family: Arial, Helvetica, sans-serif;"> <span style="font-family: Arial, Helvetica, sans-serif;">–6 < 2 //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> + 3 < 6 <span style="font-family: Arial, Helvetica, sans-serif;"> [this is the pattern for "less than"]  <span style="font-family: Arial, Helvetica, sans-serif;"> <span style="font-family: Arial, Helvetica, sans-serif;">–6 – 3 < 2 //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> + 3 – 3 < 6 – 3 <span style="font-family: Arial, Helvetica, sans-serif;"> <span style="font-family: Arial, Helvetica, sans-serif;">–9 < 2 //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> < 3 <span style="font-family: Arial, Helvetica, sans-serif;"> <span style="font-family: Arial, Helvetica, sans-serif;">–9 <span style="font-family: Arial, Helvetica, sans-serif;">/ <span style="font-family: Arial, Helvetica, sans-serif;">2 <span style="font-family: Arial, Helvetica, sans-serif;"> < //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> < <span style="font-family: Arial, Helvetica, sans-serif;">3 <span style="font-family: Arial, Helvetica, sans-serif;">/ <span style="font-family: Arial, Helvetica, sans-serif;">2 <span style="font-family: Arial, Helvetica, sans-serif;"> <span style="font-family: Arial, Helvetica, sans-serif;">Then ** <span style="font-family: Arial, Helvetica, sans-serif;">the solution to  <span style="font-family: Arial, Helvetica, sans-serif;">| 2 //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> + 3 | < 6  <span style="font-family: Arial, Helvetica, sans-serif;"> is the interval  <span style="font-family: Arial, Helvetica, sans-serif;"> <span style="font-family: Arial, Helvetica, sans-serif;">–9 <span style="font-family: Arial, Helvetica, sans-serif;">/ <span style="font-family: Arial, Helvetica, sans-serif;">2 <span style="font-family: Arial, Helvetica, sans-serif;"> < //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> < <span style="font-family: Arial, Helvetica, sans-serif;">3 <span style="font-family: Arial, Helvetica, sans-serif;">/ <span style="font-family: Arial, Helvetica, sans-serif;">2  <span style="font-family: Arial, Helvetica, sans-serif;">. **
 * ** <span style="font-family: Arial, Helvetica, sans-serif;">Solve <span style="font-family: Arial, Helvetica, sans-serif;"> | 2 //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> + 3 | < 6.  **

<span style="font-family: Arial, Helvetica, sans-serif;">The other case for absolute value inequalities is the "greater than" case. Let's first return to the number line, and consider the inequality <span style="font-family: Arial, Helvetica, sans-serif;">| //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> | > 2  <span style="font-family: Arial, Helvetica, sans-serif;">. <span style="font-family: Arial, Helvetica, sans-serif;"> <span style="font-family: Arial, Helvetica, sans-serif;">The solution will be all points that are more than two units away from zero. For instance, <span style="font-family: Arial, Helvetica, sans-serif;">–3  <span style="font-family: Arial, Helvetica, sans-serif;"> will work, as will  <span style="font-family: Arial, Helvetica, sans-serif;">3  <span style="font-family: Arial, Helvetica, sans-serif;">;  <span style="font-family: Arial, Helvetica, sans-serif;">–4  <span style="font-family: Arial, Helvetica, sans-serif;"> will work, as will  <span style="font-family: Arial, Helvetica, sans-serif;">4  <span style="font-family: Arial, Helvetica, sans-serif;">. But <span style="font-family: Arial, Helvetica, sans-serif;">–1  <span style="font-family: Arial, Helvetica, sans-serif;"> will not work, and neither will  <span style="font-family: Arial, Helvetica, sans-serif;">1  <span style="font-family: Arial, Helvetica, sans-serif;">, because they're too close. Even <span style="font-family: Arial, Helvetica, sans-serif;">–2  <span style="font-family: Arial, Helvetica, sans-serif;"> will not work, and neither will  <span style="font-family: Arial, Helvetica, sans-serif;">2  <span style="font-family: Arial, Helvetica, sans-serif;"> (although they're right on the edge). But <span style="font-family: Arial, Helvetica, sans-serif;">2.01  <span style="font-family: Arial, Helvetica, sans-serif;"> will work, as will  <span style="font-family: Arial, Helvetica, sans-serif;">–2.01  <span style="font-family: Arial, Helvetica, sans-serif;">. In other words, the solution will be two separate sections: one being all the points more than two units from zero off to the left, and the other being all the points more than two units from zero off to the right. The solution looks like this: <span style="font-family: Arial, Helvetica, sans-serif;"> <span style="font-family: Arial, Helvetica, sans-serif;">Translating this solution into symbols, we get " //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> < –2  <span style="font-family: Arial, Helvetica, sans-serif;"> or  //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> > 2  <span style="font-family: Arial, Helvetica, sans-serif;">". That is, the solution is TWO inequalities, not one. DO NOT try to write this as one inequality. If you try to write this solution as " <span style="font-family: Arial, Helvetica, sans-serif;">–2 > //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> > 2  <span style="font-family: Arial, Helvetica, sans-serif;">", you will probably be counted wrong: if you take out the  //<span style="font-family: Arial, Helvetica, sans-serif;">x // <span style="font-family: Arial, Helvetica, sans-serif;"> in the middle, you'll see that you would be saying "  <span style="font-family: Arial, Helvetica, sans-serif;">–2 > 2  <span style="font-family: Arial, Helvetica, sans-serif;">", which certainly isn't true. Take the extra half a second, and write the solution correctly. <span style="font-family: Arial, Helvetica, sans-serif;">The pattern for "greater than" absolute-value inequalities always holds: the solution is always in two parts. Given the inequality <span style="font-family: Arial, Helvetica, sans-serif;">| //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> | > //<span style="font-family: Arial, Helvetica, sans-serif;">a // <span style="font-family: Arial, Helvetica, sans-serif;">, the solution always starts by splitting the inequality into two pieces:  //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> < – //<span style="font-family: Arial, Helvetica, sans-serif;">a // <span style="font-family: Arial, Helvetica, sans-serif;"> or  //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> > //<span style="font-family: Arial, Helvetica, sans-serif;">a // <span style="font-family: Arial, Helvetica, sans-serif;">. For instance: <span style="font-family: Arial, Helvetica, sans-serif;">The first thing to do is clear the absolute value bars by splitting the inequality into two. Then I'll solve the two linear inequalities. <span style="font-family: Arial, Helvetica, sans-serif;">| 2 //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> – 3 | > 5 <span style="font-family: Arial, Helvetica, sans-serif;"> <span style="font-family: Arial, Helvetica, sans-serif;">2 //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> – 3 < –5 <span style="font-family: Arial, Helvetica, sans-serif;"> or  <span style="font-family: Arial, Helvetica, sans-serif;">2 //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> – 3 > 5  <span style="font-family: Arial, Helvetica, sans-serif;"> [this is the pattern for "greater than"]  <span style="font-family: Arial, Helvetica, sans-serif;"> <span style="font-family: Arial, Helvetica, sans-serif;">2 //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> < –2 <span style="font-family: Arial, Helvetica, sans-serif;"> or  <span style="font-family: Arial, Helvetica, sans-serif;">2 //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> > 8  <span style="font-family: Arial, Helvetica, sans-serif;"> //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> < –1 <span style="font-family: Arial, Helvetica, sans-serif;"> or  //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> > 4  <span style="font-family: Arial, Helvetica, sans-serif;"> <span style="font-family: Arial, Helvetica, sans-serif;">This PAIR of inequalities is the solution; ** <span style="font-family: Arial, Helvetica, sans-serif;">the solution to <span style="font-family: Arial, Helvetica, sans-serif;">| 2 //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> – 3 | > 5  <span style="font-family: Arial, Helvetica, sans-serif;"> consists of the two intervals  //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> < –1  <span style="font-family: Arial, Helvetica, sans-serif;"> and  <span style="font-family: Arial, Helvetica, sans-serif;"> //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> > 4  <span style="font-family: Arial, Helvetica, sans-serif;">. **
 * ** <span style="font-family: Arial, Helvetica, sans-serif;">Solve <span style="font-family: Arial, Helvetica, sans-serif;"> | 2 //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> – 3 | > 5  <span style="font-family: Arial, Helvetica, sans-serif;">.  **

<span style="font-family: Arial, Helvetica, sans-serif;">There is another situation you might encounter: You'll be given a pair of inequalities, and you'll be asked to find the corresponding absolute-value inequality. This process can feel a bit weird, so I'll give a couple examples of how it works. > <span style="font-family: Arial, Helvetica, sans-serif;">–2 < //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> < 4. ** <span style="font-family: Arial, Helvetica, sans-serif;">I first look at the endpoints. Negative two and four are six units apart. Half of six is three. So I want to adjust this inequality so that it relates to <span style="font-family: Arial, Helvetica, sans-serif;">–3  <span style="font-family: Arial, Helvetica, sans-serif;"> and  <span style="font-family: Arial, Helvetica, sans-serif;">3  <span style="font-family: Arial, Helvetica, sans-serif;">, instead of to  <span style="font-family: Arial, Helvetica, sans-serif;">–2  <span style="font-family: Arial, Helvetica, sans-serif;"> and  <span style="font-family: Arial, Helvetica, sans-serif;">4  <span style="font-family: Arial, Helvetica, sans-serif;">. To accomplish this, I will adjust the ends by subtracting <span style="font-family: Arial, Helvetica, sans-serif;">1  <span style="font-family: Arial, Helvetica, sans-serif;"> from all three "sides": <span style="font-family: Arial, Helvetica, sans-serif;">–2 < //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> < 4 <span style="font-family: Arial, Helvetica, sans-serif;"> <span style="font-family: Arial, Helvetica, sans-serif;">–2 – 1 < //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> – 1 < 4 – 1 <span style="font-family: Arial, Helvetica, sans-serif;"> <span style="font-family: Arial, Helvetica, sans-serif;">–3 < //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> – 1 < 3 <span style="font-family: Arial, Helvetica, sans-serif;"> <span style="font-family: Arial, Helvetica, sans-serif;">Since the last line above is in the "less than" format, the absolute-value inequality will be of the form "absolute value of something is less than <span style="font-family: Arial, Helvetica, sans-serif;">3  <span style="font-family: Arial, Helvetica, sans-serif;">". I can convert this nicely to  **<span style="font-family: Arial, Helvetica, sans-serif;">| //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> – 1 | < 3 ** > ** //** <span style="font-family: Arial, Helvetica, sans-serif;">x **// <span style="font-family: Arial, Helvetica, sans-serif;"> __<span style="font-family: Arial, Helvetica, sans-serif;">< __<span style="font-family: Arial, Helvetica, sans-serif;">  ** <span style="font-family: Arial, Helvetica, sans-serif;">19  ** <span style="font-family: Arial, Helvetica, sans-serif;"> **<span style="font-family: Arial, Helvetica, sans-serif;">or **<span style="font-family: Arial, Helvetica, sans-serif;">  //** <span style="font-family: Arial, Helvetica, sans-serif;">x  **// <span style="font-family: Arial, Helvetica, sans-serif;"> __<span style="font-family: Arial, Helvetica, sans-serif;">> __<span style="font-family: Arial, Helvetica, sans-serif;">  ** <span style="font-family: Arial, Helvetica, sans-serif;">24  ** <span style="font-family: Arial, Helvetica, sans-serif;">. <span style="font-family: Arial, Helvetica, sans-serif;">I first look at the endpoints. Nineteen and <span style="font-family: Arial, Helvetica, sans-serif;">24  <span style="font-family: Arial, Helvetica, sans-serif;"> are five units apart. Half of five is <span style="font-family: Arial, Helvetica, sans-serif;">2.5  <span style="font-family: Arial, Helvetica, sans-serif;">. So I want to adjust the inequality so it relates to <span style="font-family: Arial, Helvetica, sans-serif;">–2.5  <span style="font-family: Arial, Helvetica, sans-serif;"> and  <span style="font-family: Arial, Helvetica, sans-serif;">2.5  <span style="font-family: Arial, Helvetica, sans-serif;">, instead of relating to  <span style="font-family: Arial, Helvetica, sans-serif;">19  <span style="font-family: Arial, Helvetica, sans-serif;"> and  <span style="font-family: Arial, Helvetica, sans-serif;">24  <span style="font-family: Arial, Helvetica, sans-serif;">. Since <span style="font-family: Arial, Helvetica, sans-serif;">19 – (–2.5) =21.5 <span style="font-family: Arial, Helvetica, sans-serif;"> and  <span style="font-family: Arial, Helvetica, sans-serif;">24 – 2.5 = 21.5  <span style="font-family: Arial, Helvetica, sans-serif;">, I need to subtract  <span style="font-family: Arial, Helvetica, sans-serif;">21.5  <span style="font-family: Arial, Helvetica, sans-serif;"> all around: //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> __<span style="font-family: Arial, Helvetica, sans-serif;">< __<span style="font-family: Arial, Helvetica, sans-serif;"> 19 <span style="font-family: Arial, Helvetica, sans-serif;"> or  //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> __<span style="font-family: Arial, Helvetica, sans-serif;">> __<span style="font-family: Arial, Helvetica, sans-serif;"> 24  <span style="font-family: Arial, Helvetica, sans-serif;"> //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> – 21.5 __<span style="font-family: Arial, Helvetica, sans-serif;">< __<span style="font-family: Arial, Helvetica, sans-serif;"> 19 – 21.5 <span style="font-family: Arial, Helvetica, sans-serif;"> or  //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> – 21.5 __<span style="font-family: Arial, Helvetica, sans-serif;">> __<span style="font-family: Arial, Helvetica, sans-serif;"> 24 – 21.5  <span style="font-family: Arial, Helvetica, sans-serif;"> //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> – 21.5 __<span style="font-family: Arial, Helvetica, sans-serif;">< __<span style="font-family: Arial, Helvetica, sans-serif;"> –2.5 <span style="font-family: Arial, Helvetica, sans-serif;"> or  //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> – 21.5 __<span style="font-family: Arial, Helvetica, sans-serif;">> __<span style="font-family: Arial, Helvetica, sans-serif;"> 2.5  <span style="font-family: Arial, Helvetica, sans-serif;"> <span style="font-family: Arial, Helvetica, sans-serif;">Since the last line above is the "greater than" format, the absolute-value inequality will be of the form "absolute value of something is greater than or equal to <span style="font-family: Arial, Helvetica, sans-serif;">2.5  <span style="font-family: Arial, Helvetica, sans-serif;">". I can convert this nicely to: **<span style="font-family: Arial, Helvetica, sans-serif;">| //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> – 21.5 | **__<span style="font-family: Arial, Helvetica, sans-serif;">> __**<span style="font-family: Arial, Helvetica, sans-serif;"> 2.5 **
 * ** <span style="font-family: Arial, Helvetica, sans-serif;">Find the absolute-value inequality statement that corresponds to the inequality
 * **<span style="font-family: Arial, Helvetica, sans-serif;">Find the absolute-value inequality statement that corresponds to the inequalities

<span style="font-family: Arial, Helvetica, sans-serif;">Warning: There is one "trick" type of question for this kind of problem, where they'll try to trip you up on homework or tests. They'll ask you to solve something like " <span style="font-family: Arial, Helvetica, sans-serif;">| //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> + 2 | < –1  <span style="font-family: Arial, Helvetica, sans-serif;">". But can an absolute value ever be negative, let alone be //<span style="font-family: Arial, Helvetica, sans-serif;">less than //<span style="font-family: Arial, Helvetica, sans-serif;"> a negative? No! So there is no solution to this inequality; it doesn't even make sense. Don't waste a lot of time trying to "solve" this; just write down "no solution". <span style="font-family: Arial, Helvetica, sans-serif;">Similarly, if you're given something like " <span style="font-family: Arial, Helvetica, sans-serif;">| //<span style="font-family: Arial, Helvetica, sans-serif;">x //<span style="font-family: Arial, Helvetica, sans-serif;"> – 2 | > –3  <span style="font-family: Arial, Helvetica, sans-serif;">", the solution would, naturally, be "all  //<span style="font-family: Arial, Helvetica, sans-serif;">x // <span style="font-family: Arial, Helvetica, sans-serif;">" or "all real numbers", since the absolute value, being itself positive or zero, must then always be greater than any negative number.