3. y + 6 < –2x or y + 6 < x(–2). Use parentheses if the –2 follows the x.
4.
Taking on algebraic tasks
Algebra involves symbols, such as variables and operation signs, which are the tools that you can use to make algebraic expressions more usable and readable. These things go hand in hand with simplifying, factoring, and solving problems, which are easier to solve if broken down into basic parts. Using symbols is actually much easier than wading through a bunch of words.
✓ To simplify means to combine all that can be combined, cut down on the number of terms, and put an expression in an easily understandable form.
✓ To factor means to change two or more terms to just one term using multiplication. (See Chapters 11 through 13 for more on factoring.)
✓ To solve means to find the answer. In algebra, it means to figure out what the variable stands for. (You see solving equations and inequalities in Chapters 14 through 19.)
Equation solving is fun because there’s a point to it. You solve for something (often a variable, such as x) and get an answer that you can check to see whether you’re right or wrong. It’s like a puzzle. It’s enough for some people to say, “Give me an x.” What more could you want? But solving these equations is just a means to an end. The real beauty of algebra shines when you solve some problem in real life – a practical application. Are you ready for these two words: story problems? Story problems are the whole point of doing algebra. Why do algebra unless there’s a good reason? Oh, I’m sorry – you may just like to solve algebra equations for the fun alone. (Yes, some folks are like that.) But other folks love to see the way a complicated paragraph in the English language can be turned into a neat, concise expression, such as, “The answer is three bananas.”
Going through each step and using each tool to play this game is entirely possible. Simplify, factor, solve, check. That’s good! Lucky you. It’s time to dig in!
Chapter 2
Deciphering Signs in Expressions
In This Chapter
Numbers have many characteristics: They can be big, little, even, odd, whole, fraction, positive, negative, and sometimes cold and indifferent. (I’m kidding about that last one.) Chapter 1 describes numbers’ different names and categories. But this chapter concentrates mainly on the positive and negative characteristics of numbers and how a number’s sign reacts to different manipulations. This chapter tells you how to add, subtract, multiply, and divide signed numbers, no matter whether all the numbers are all the same sign or a combination of positive and negative.
Positive numbers are greater than 0. They’re on the opposite side of 0 from the negative numbers. If you were to arrange a tug-of-war between positive and negative numbers, the positive numbers would line up on the right side of 0. Negative numbers get smaller and smaller, the farther they are from 0. This situation can get confusing because you may think that –400 is bigger than –12. But just think of –400°F and –12°F. Neither is anything pleasant to think about, but –400°F is definitely less pleasant – colder, lower, smaller.
Remember: When comparing negative numbers, the number closer to 0 is the bigger or greater number. You may think that identifying that 16 is bigger than 10 is an easy concept. But what about –1.6 and –1.04? Which of these numbers is bigger?
Remember: The easiest way to compare numbers and to tell which is bigger or has a greater value is to find each number’s position on the number line. The number line goes from negatives on the left to positives on the right (see Figure 2-1). Whichever number is farther to the right has the greater value, meaning it’s bigger.
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Figure 2-1: A number line.
Examples
Q. Using the number line in Figure 2-1, determine which is larger, –16 or –10.
A. The number –10 is to the right of –16, so it’s the bigger of the two numbers.
Q. Which is larger, –1.6 or –1.04?
A. The number –1.04 is to the right of –1.6, so it’s larger. A nice way to compare decimals is to write them with the same number of decimal places. So rewrite –1.6 as –1.60; it’s easier to compare to –1.04 in this format.
Comparing Positives and Negatives with Symbols
Although my mom always told me not to compare myself to other people, comparing numbers to other numbers is often useful. And, when you compare numbers, the greater-than sign (>) and less-than sign (<) come in handy, which is why I use them in Table 2-1, where I put some positive- and negative-signed numbers in perspective.
Table 2-1 Comparing Positive and Negative Numbers
Two other signs related to the greater-than and less-than signs are the greater-than-or-equal-to sign (≥) and the less-than-or-equal-to sign (≤).
So, putting the numbers 6, –2, –18, 3, 16, and –11 in order from smallest to biggest gives you: –18, –11, –2, 3, 6, and 16, which are shown as dots on a number line in Figure 2-2.
Figure 2-2: Positive and negative numbers on a number line.
Zeroing in on zero
But what about 0? I keep comparing numbers to see how far they are from 0. Is 0 positive or negative? The answer is that it’s neither. Zero has the unique distinction of being neither positive nor negative. Zero separates the positive numbers from the negative ones – what a job!
Practice Questions
1. Which is larger, –2 or –8?
2. Which has the greater value, –13 or 2?
3. Which is bigger, –0.003 or –0.03?
4. Which is larger,
Practice Answers
1. -2. The following number line shows that the number –2 is to the right of –8. So –2 is bigger