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is positive or zero; the absolute value of a is equal to the opposite of a if a is negative.”

      Cracking the ISBN check code

      Have you ever noticed the bar codes and ISBN numbers that appear on the backs of the books you buy (or, ahem, borrow from friends)? Actually, the International Standard Book Number (ISBN) has been around for only about 40 years. The individual numbers tell those in the know what the number as a whole means: the language the book is printed in, who the publisher is, and what specific number was assigned to that particular book. You can imagine how easy it is to miscopy this long string of numbers – just try it with this book’s ISBN. If you write the numbers down, you could reverse a pair of numbers, skip a number, or just write the number down wrong. For this reason, publishers assign a check digit for the ISBN – the last digit. UPC codes and bank checks have the same feature: a check digit to try to help catch most errors.

      To form the check digit on ISBNs, you take the first digit of the ISBN number and multiply it by 10, the second by 9, the third by 8, and so on until you multiply the last digit by 2. (Don’t do anything with the check digit.) You then add up all the products and change the sum to its opposite – you should now have a negative number. Next, you add 11 to the negative number, and add 11 again, and again, and again, until you finally get a positive number. That number should be the same as the check digit.

      For instance, the ISBN for Algebra For Dummies (Wiley), my original masterpiece, is 0-7645-5325-9. Here’s the sum you get by performing all the multiplication: 10(0) + 9(7) + 8(6) + 7(4) + 6(5) + 5(5) + 4(3) + 3(2) + 2 (5) = 222.

      You change 222 to its opposite, –222. Add 11 to get –211; add 11 again to get –200; add 11 again, and again. Actually, you add the number 21 times – 11(21) = 231. So, the first positive number you come to after repeatedly adding 11s is 9. That’s the check digit! Because the check digit is the same as the number you get by using the process, you wrote down the number correctly. Of course, this checking method isn’t foolproof. You could make an error that gives you the same check digit, but this method finds most of the errors.

Solving absolute value equations

      A linear absolute value equation is an equation that takes the form

. You don’t know, taking the equation at face value, if you should change what’s in between the bars to its opposite, because you don’t know if the expression is positive or negative. The sign of the expression inside the absolute value bars all depends on the size and sign of the variable x. To solve an absolute value equation in this linear form, you have to consider both possibilities: ax + b may be positive, or it may be negative.

      

, you solve both ax + b = c and ax + b = –c.

      For example, to solve the absolute value equation

, you write the two linear equations and solve each for x by subtracting 5 and dividing by 4: If 4x + 5 = 13, then 4x = 8 and x = 2.

      If

, then

and

.

      You have two solutions: 2 and

. Both solutions work when you replace the x in the original equation with their values.

      

One restriction you should be aware of when applying the rule for changing from absolute value to individual linear equations is that the absolute value term has to be alone on one side of the equation.

      For instance, to solve

, you have to subtract 7 from each side of the equation and then divide each side by 3. Subtracting 7, you have ; then, when you divide by 3, the problem becomes .

      Now you can write the two linear equations and solve them for x:

      If 4 – 3x = 6, then –3x = 2 and

.

      If 4 – 3x = –6, then –3x = –10 and

.

Seeing through absolute value inequality

      An absolute value inequality contains both an absolute value,

, and an inequality: <, >, < , or >. But, then, you knew that was coming.

      

To solve an absolute value inequality, you have to change from absolute value inequality form to just plain inequality form. The way to handle the change from absolute value notation to inequality notation depends on which direction the inequality points with respect to the absolute value term. The methods, depending on the direction, are quite different:

      ✔ To solve for x in

you solve –c < ax + b < c.

      ✔ To solve for x in

|, you solve both ax + b > c and ax + b < –c.

      The first change sandwiches the ax + b between c and its opposite. The second change considers values greater than c (toward positive infinity) and smaller than –c (toward negative infinity).

       Sandwiching the values in inequalities

      You apply the first rule of solving absolute value inequalities to the inequality

, because of the less-than direction of the inequality. You rewrite the inequality, using the rule for changing the format: . Next, you add one to each section to isolate the variable; you get the inequality . Divide each section by two to get . You can write the solution in interval notation as [–2, 3].

      Be sure that the absolute value inequality is in the correct format before you apply the rule. The absolute value portion should be alone on its side of the inequality sign. If you have

, for example, you need to add 7 to each side and divide each side by 2 before changing the form:

      Adding 7 you get

. Applying the first rule the problem becomes –9 < 3x + 5 < 9. Subtracting 5 from each interval gives you –14 < 3x < 4. Then, dividing each interval by 3 you have . In interval notation, the answer is written .

       Harnessing inequalities moving in opposite directions

      An absolute value inequality with a greater-than sign, such as

> 11, has solutions that go infinitely high to the right and infinitely low to the left on the number line. To solve for the values that work, you rewrite the absolute value, using the rule for greater-than inequalities; you get two completely separate inequalities to solve. The solutions relate to the inequality 7 – 2x > 11 or to the inequality 7 – 2x < –11. Notice that when the sign of the value

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