Algebra I Workbook For Dummies
Page 23
A) B)
Okay. I made it too easy for you. Choice A is just fine, but choice B isn’t correct. A common error is to mishandle the constant multiplier in the denominator. The correct way to write choice B is .
Chapter 4 has more information on negative exponents.
Making Sense of Reversing the Sense
Because reversing the sense in an inequality doesn’t come up as often as other maneuvers, people frequently overlook the process. The inequality 3 < 4 is a true statement. When do you reverse the sense (switch the direction of the inequality sign)? When you multiply or divide each side by a negative number. Which of the two following manipulations is correct for multiplying each side of the inequality by –2?
A) Multiplying by –2, –5 < 7 → 10 > –14 B) Multiplying by –2, 5 < 12 → –10 > –24
Choice A is correct. Remember that the smaller number is always to the left of a larger number on a number line.
Chapter 16 describes how to deal with inequalities.
Using the Slope Formula Correctly
You use the slope formula to determine the number that describes how steep a line is and whether it rises or falls. If you have two points, and , then you find the slope of the line by using . In Chapter 21, you find several cautions about using the formula correctly. See if you can spot the correct uses of the formula from the following choices. You’re finding the slope of the line that goes through (3, 4) and (5, 6).
A) B)
C) D)
The line through those two points has a slope of 1, so the two correct uses are A and D. The error in choice B is that the x values are on top — a common mistake is reversing the x’s and y’s. The error in choice C is mixing and matching. You have to use the two coordinates of a single point in the same position — either both first or both last.
Writing Several Fractions as One
Fractions always cause groans and moans. They’re one of the most unappreciated types of numbers. But handling them correctly helps them behave. When adding fractions, you find a common denominator and then add the numerators. See whether you can spot the correct additions.
A) B)
C) D)
This was probably too easy. You could see that A and B were terrible. In choice A, the common denominator is 30, not 10. And when you add the new numerators, you get 15 + 10 + 6 = 31. Another hint that A is wrong is the fact that the answer is smaller than either of the first two fractions in the problem — not larger. Choice B is almost as bad. Just put in some numbers for x, y and z if you want to just demonstrate that this is wrong, wrong, wrong.
Chapter 23
Ten Quick Tips to Make Algebra a Breeze
Working in algebra is a lot like taking a driver’s test. No matter what else you do, you need to know and follow all the rules. And just as you were given some hints about driving from your instructor, you find lots of helps here with your algebra. Buried in those lists of algebra rules are some maneuvers, procedures, and quick tricks that help ease the way. The helps may include eliminating fractions or decimals, or simplifying the set-up or changing the form of an equation to make it ready to combine with other expressions.
This chapter offers you ten quick tips to make your experience with algebra a little easier and a little more efficient, and to improve your opportunities for success.
Flipping Proportions
A proportion is a true statement in which one fraction is set equal to another fraction. One property of a proportion is that flipping it doesn’t change its truth. The rule is if , then . You find more on proportions in Chapter 12.
Flip the proportion to put the variable in the numerator. Here’s a situation where flipping makes the statement easier to solve. is a proportion. Now flip it to get the variable in the numerator: . Multiplying each side by 8, you get x + 1 = 24, or x = 23. You’d get the same result by just cross-multiplying, but it’s often handier to have the variable in the numerator of the fraction.
Multiplying Through to Get Rid of Fractions
As much fun as fractions are — we couldn’t do without them — they’re sometimes a nuisance when you’re trying to solve for the value of a variable. A quick trick to make things easier is to multiply both sides of the equation by the same number. The choice of multiplier is the least common multiple — all the denominators divide it evenly. In the following example where I multiply each fraction by 12, see how easy it becomes.
Multiply an equation by the least common factor to eliminate fractions.
Zeroing In on Fractions
When you’re solving an equation that involves a fraction set equal to 0, you only need to consider the fraction’s numerator for a solution.
When a fraction is equal to zero, only the numerator can be 0. For instance, has a solution only when the numerator equals 0. Solving (x – 2)(x + 11) = 0, you find that the equation is true when x = 2 or x = –11. These are the only two solutions. You can forget about the denominator for this type problem (as long as you don’t choose a solution that makes the denominator equal to 0). Chapter 12 has more on rational equations.
Finding a Common Denominator
When adding or subtracting fractions, you always write the fractions with the same common denominator so you can add (or subtract) the numerators. (See Chapter 3 for more on fractions.)
One way to find a common denominator is to divide the product of the denominators by their greatest common factor.
In the earlier section ““Multiplying Through to Get Rid of Fractions,” I provide a quick tip for multiplying through by a common denominator. Many times you can easily spot the common multiple. For example, when the denominators are 2, 3, and 6, you can see that they all divide 6 evenly — 6 is the common denominator. But what if you want to add ? You’d like a common denominator that isn’t any bigger than necessary. Just multiply the 24 × 60 to get 1,440, which is pretty large. Both 24 and 60 are divisible by 12, so divide the 1,440 by 12 to get 120, which is the least common denominator.
Dividing by 3 or 9
Reducing fractions makes them easier to work with. And the key to reducing fractions is recognizing common divisors of the numerator and denominator. For instance, do you know the common factor for the numerator and denominator in ? It’s 9!
A number is divisible by 3 if the sum of its digits is divisible by 3. A number is divisible by 9 if the sum of its digits is divisible by 9. For example, the number 1,047 is divisible by 3 because the sum of its digits (1, 0, 4, and 7) is 12, which is divisible by 3. And the number 10,071 is divisible by 9 because its digits add up to 9.
Now looking at , you can divide both numerator and denominator by 9 to reduce the fraction and get . There aren’t any more common factors, so the fraction is reduced.
Dividing by 2, 4, or 8
The rules for determining whether a number is divisible by 2, 4, or 8 are quite different from the rules for divisibility by 3 and 9.
You only have to look at the last digit to see if a number is divisible by 2. You only have to look at the number formed by the last two digits to see if a number is divisible by 4, and at the last three digits for 8.
If a number ends in 0, 2, 4, 6, or 8 (it’s an even number), the entire number is divisible by 2. The number 113,579,714 is divisible by 2, even though all the digits except the last are odd.
If the number formed by the last two digits of a number is divisible by 4, then the entire number is divisible by 4. The number 5,783,916 is divisible by 4 because the number formed by the last two digits, 16, is divisible by 4.
If the number formed by the last three digits of a number is divisible by 8, then the whole number is divisible by 8. The number 43,512,619,848 is divisible by 8, because the number formed by the last three digits, 848, is divisible by 8. You may run into some three-digit numbers that aren’t so obviously divisible by 8. In that case, just do the long division on the last three digits. Dividing the three-digit number is still quicker than dividing the entire number.
Commutin
g Back and Forth
The commutative law of addition and multiplication says that you can add or multiply numbers in any order, and you’ll get the same answer. This rule is especially useful when you couple it with the associative rule that allows you to regroup or reassociate numbers to make computations easier.
a + b = b + a and
For instance, look at how I can rearrange the numbers to my advantage for doing computations:
Factoring Quadratics
A quadratic expression has a general format of , and you frequently find it more useful to change that format and factor the quadratic into the product of two binomials. For example, the quadratic factors into . In the factored form, you can solve for what makes the expression equal to zero or you can factor if the expression is in a fraction.
To factor a quadratic expression efficiently, list the possible factors and cross them out as you eliminate the combinations that don’t work.
Make a list (and check it twice) of the multipliers of the lead coefficient (the a) and the constant (the c); you combine the factors to produce the middle coefficient (the b).
In the quadratic , you list the multipliers of 12, which are 12 × 1, 6 × 2, 4 × 3, and list the multipliers of 15, which are 15 × 1, 5 × 3. You then choose a pair from the multipliers of 12 and a pair from the multipliers of 15 that give you a difference of 11 when you multiply them together.
Making Radicals Less Rad, Baby
Radicals are symbols that indicate an operation. Whatever is inside the radical has the root operation performed on it. Many times, though, writing the radical expressions with exponents instead of radicals is more convenient — especially when you want to combine several terms or factors with the same variable under different radicals.
The general rule for changing from radical form to exponential form is .
For instance, to multiply , you change it to . Now you can multiply the factors together by adding the exponents. Of course, you need to change the fractions so they have a common denominator, but the result is worth the trouble: .
Applying Acronyms
Some of the easiest ways of remembering mathematical processes is to attach cutesy acronyms to them to help ease the way. For example:
FOIL (First, Outer, Inner, Last): FOIL is used to help you remember the different combinations of products that are necessary when multiplying two binomials together.
NOPE (Negative, Odd, Positive, Even): NOPE tells you that a product is negative when you have an odd number of negative signs and it’s positive when the number of negative signs is even.
PEMDAS (Please Excuse My Dear Aunt Sally): When simplifying expressions that have various operations and grouping symbols, you first simplify in Parentheses, then Exponents, then Multiplication or Division, and then Addition or Subtraction.
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