Название | Algebra II For Dummies |
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Автор произведения | Sterling Mary Jane |
Жанр | Зарубежная образовательная литература |
Серия | |
Издательство | Зарубежная образовательная литература |
Год выпуска | 0 |
isbn | 9781119090731 |
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.
A linear absolute value equation is an equation that takes the form
For example, to solve the absolute value equation
If
You have two solutions: 2 and
For instance, to solve
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
An absolute value inequality contains both an absolute value,
✔ To solve for x in
✔ To solve for x in
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
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
Adding 7 you get
Harnessing inequalities moving in opposite directions
An absolute value inequality with a greater-than sign, such as