are some examples of inequality notation and the corresponding interval notation:
Notice that the second example has a bracket by the –2, because the “greater than or equal to” indicates that you include the –2 also. The same is true of the 4 in the third example. The last example shows you why interval notation can be a problem at times. Taken out of context, how do you know if (–3, 7) represents the interval containing all the numbers between –3 and 7 or if it represents the point (–3, 7) on the coordinate plane? You can’t tell. You consider the context. A problem containing such notation has to give you some sort of hint as to what it’s trying to tell you.
A compound inequality is an inequality with more than one comparison or inequality symbol – for instance, –2 < x < 5. To solve compound inequalities for the value of the variables, you use the same inequality rules (see the intro to this section), and you expand the rules to apply to each section (intervals separated by inequality symbols).
To solve the inequality
Many ancient cultures used their own symbols for mathematical operations, and the cultures that followed altered or modernized the symbols for their own use. You can see one of the first symbols used for addition in the following figure, located on the far left – a version of the Italian capital P for the word piu, meaning plus. Tartaglia, a self-taught 16th century Italian mathematician, used this symbol for addition regularly. The modern plus symbol, +, is probably a shortened form of the Latin word et, meaning and.
The second figure from the left is what Greek mathematician Diophantes liked to use in ancient Greek times for subtraction. The modern subtraction symbol, –, may be a leftover from what the traders in medieval times used to indicate differences in product weights.
Leibniz, a child prodigy from the 17th century who taught himself Latin, preferred the third symbol from the left for multiplication. One modern multiplication symbol, × or
The symbol on the far right is a somewhat backward D, used in the 18th century by French mathematician Gallimard for division. The modern division symbol,÷ , may come from a fraction line with dots added above and below.
You write the answer,
Here’s a more complicated example. You solve the problem
You write the answer,
Absolute Value: Keeping Everything in Line
When you perform an absolute value operation, you’re not performing surgery at bargain-basement prices; you’re taking a number inserted between the absolute value bars,
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
