[DOWNLAD] PDF Painless Algebra (Painless Series) Painless Algebra (Painless Series) Defines algebraic terms, shows how to avoid pitfalls in calculation. Painless algebra by Lynette Long; 3 editions; First published in ; Subjects: In library, Algebra, Textbooks, Accessible book, Algebra. Get Free Read & Download Files Painless Algebra PDF. PAINLESS ALGEBRA. Download: Painless Algebra. PAINLESS ALGEBRA - In this site isn`t the same.
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INTRODUCTION Painless algebra! Impossible, you think. Not really. I have been teaching math or teaching teachers how to teach math for over twenty years. Editorial Reviews. From the Inside Flap. (back cover) Really. This won't hurt at all . Painless Algebra (Painless Series) - Kindle edition by Lynette Long. New in this edition are painless approaches to understanding and graphing linear equations, solving systems of linear inequalities, and.
Think of the letters x, a, y, and z as mystery numbers. A mathematical expression is part of a mathematical sentence, just as a phrase is part of an English sentence. Here are a few examples of mathematical expressions: In each of these expressions it is impossible to know what x is. The variable x could be any number. Mathematical expressions are named based on how many terms they have.
A monomial expression has one term. A binomial expression has two unlike terms combined by an addition or subtraction sign. Binomials and trinomials are polynomial expressions.
The following are also polynomial expressions: A mathematical sentence contains two mathematical phrases joined by an equals sign or an inequality sign.
An equation is a mathematical sentence in which the two phrases are joined by an equals sign. Notice that the word equation starts the same way as the word equal. It does not have an equals sign. It is a mathematical expression. Some equations are true and some equations are false.
It could be true or it could be false. An inequality is a mathematical sentence in which two phrases are joined by an inequality symbol. Watch how these mathematical expressions are changed from Math Talk into Plain English.
Terms that consist only of numbers are like terms. Terms that use the same variable to the same degree are like terms. Terms that use different variables are unlike terms. You can add any numbers. To add like variables, just add the coefficients. The coefficient is the number in front of the variable. In the expression 7a, 7 is the coefficient and a is the variable. In the expression 21 y, 21 is the coefficient and y is the variable.
In the expression x, 1 is the coefficient and x is the variable. Now note how like terms are added to simplify the following expressions. Caution—Major Mistake Territory! When adding expressions with the same variable, just add the coefficients and attach the variable to the new coefficient. Do not put two variables at the end. You can subtract one number from another number.
Just subtract the coefficients and keep the variable the same. When subtracting expressions with the same variables, just subtract the coefficients and attach the variable to the new coefficient.
Do not subtract the variables. Multiplication An is seldom used to indicate multiplication. To avoid this problem, mathematicians use other ways to indicate multiplication. You can multiply any two numbers. To multiply these expressions requires two painless steps. Multiply the coefficients. Attach the variable at the end of the answer. Here are two examples: To multiply these expressions requires three steps. Multiply the variables. Combine the two answers.
Here are three examples. Multiply 3x times 2y. First multiply the coefficients. First multiply the coefficients, 6 and 1. In this book, we will use the horizontal fraction bar instead of the slash mark. You can divide any two numbers.
To divide these expressions requires three steps. Divide the coefficients. Divide the variables. Multiply the two answers. The division is painless. Divide 3x by 4x.
First divide the coefficients. First divide the coefficients, 12 and 1. It is neither positive nor negative. There are some rules about zero you should know. If zero is added to any number or variable, the answer is that number or variable. Division by zero is undefined.
You can never divide by zero. Solve them. The order in which you solve a problem may affect the answer. Look at the following problem: Mathematicians have agreed on a certain sequence, called the Order of Operations, to be used in solving mathematical problems. Without the Order of Operations, several different answers would be possible when computing mathematical expressions.
The Order of Operations tells you how to simplify any mathematical expression in four easy steps. Step 1: Do everything in parentheses. Step 2: Compute the value of any exponential expressions. In the problem 5 32, square the three first and then multiply by five.
Step 3: Start at the left and go to the right. Subtract last. Start on the left and go to the right. There is one trick to keep in mind: Watch as these mathematical expressions are changed from Math Talk into Plain English. Do what is inside the parentheses. Compute the values of any exponential expressions.
There are no exponential expressions. Multiply three times three and then six times one. Square 3. Keep in mind the Order of Operations. In other words, three plus four is equal to four plus three.
Six plus two is equal to two plus six. Given any two numbers a and b, a plus b is equal to b plus a. Subtraction is not commutative. The order of the numbers in a subtraction problem does make a difference. Given any two numbers, a and b, a times b is equal to b times a. Division is not commutative. The order of the numbers in a division problem does make a difference.
If you first add a and b and then add c to the total, the answer is the same as if you first add b and c and then add the total to a. If you first multiply a and b and then multiply the product by c, the answer is the same as if you first multiply b and c and then multiply a by the product.
Multiplying a by the quantity b plus c is equal to a times b plus a times c. Be careful—some of the problems are tricky. The natural numbers The whole numbers The integers The rational numbers The irrational numbers The real numbers The natural numbers The natural numbers are 1, 2, 3, 4, 5,. The three dots,. The whole numbers The whole numbers are 0, 1, 2, 3, 4, 5, 6,. The whole numbers are the natural numbers plus zero.
All of the natural numbers are whole numbers. The integers The integers are the natural numbers, their opposites, and zero.
The integers are. All of the whole numbers are integers. All of the natural numbers are integers. The rational numbers The rational numbers are any numbers that can be expressed as the ratios of two whole numbers.
All of the integers are rational numbers. All of the whole numbers are rational numbers. All of the natural numbers are rational numbers. The irrational numbers The irrational numbers are numbers that cannot be expressed as the ratios of two whole numbers. The rational numbers are not irrational numbers.
The integers are not irrational numbers. The whole numbers are not irrational numbers. The natural numbers are not irrational numbers. The real numbers The real numbers are a combination of all the number systems.
The real numbers are the natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Every point on the number line is a real number. All of the irrational numbers are real numbers. All of the rational numbers are real numbers.
All of the integers are real numbers. All of the whole numbers are real numbers. All of the natural numbers are real numbers. Is five a whole number or a natural number? Five is both a whole number and a natural number. A number can belong to more than one number system at the same time. Is six a whole number or a rational number? Six is both a whole number and a rational number. Six can be written as 6 or as Some numbers belong to more than one number system. Set 2, page 15 1.
CA Set 6, page 32 1. N, W, In, Ra, Re 6. N, W, In, Ra, Re 2. In, Ra, Re 7. In, Ra, Re 3. W, In, Ra, Re 8. N, W, In, Ra, Re 4. Ra, Re 9. Ir, Re 5. Ra, Re 34 Tonight the temperature is expected to drop to negative twelve degrees.
When you first learned to count, you counted with positive numbers, the numbers greater than zero: You probably first learned to count to 10, then to , and maybe even to 1, Finally, you learned that you could continue counting forever, because after every number there is a number that is one larger than the number before it. Well, there are negative numbers, too. Negative numbers are used to express cold temperatures, money owed, feet below sea level, and lots of other things.
You count from negative one to negative ten with these numbers: After every negative number there is another negative number that is one less than the number before it. When you place all these positive numbers, negative numbers, and zero together, you have what mathematicians call the integers. The integers are made up of three groups of numbers: Sometimes the positive integers are written like this: Here is a graph of the positive integers: A number between any two counting numbers is not an integer.
If there is no sign in front of a number, you can assume that the number is positive. Sometimes they are written like this: Here is a graph of the negative integers: Here is a graph of zero: Here is a graph representing all the integers: Just keep going!
Sometimes mathematicians want to compare two numbers and decide which is larger and which is smaller. But instead of larger and smaller, mathematicians use the words greater than and less than.
Notice that the arrow always points to the smaller number. Positive numbers are always greater than negative numbers. Positive numbers are always greater than zero. Negative numbers are always less than zero. The larger a negative number looks, the smaller it actually is.
Stop complaining. Zero is always less than a positive number. Zero is always greater than a negative number. When comparing two negative numbers, remember that the number that looks larger is actually smaller. Negative six looks larger than negative two, but it is actually smaller. Still confused? Graph both numbers on a number line. The negative number closer to zero is always larger.
Five of these statements are true. The sum of the problem numbers of the true statements is Case 1: Both numbers are positive.
Both numbers are negative. One of the two numbers is positive and one is negative. One of the two numbers is zero. Painless Solution: Add the numbers just as you would add any two numbers. The answer is always positive. Take this Painless Solution and call me in the morning.
Pretend both numbers are positive. Add them. Place a negative sign in front of the answer. One number is positive and one number is negative. Subtract the smaller number from the larger number.
Give the answer the sign of the number that would be larger if both numbers were positive. Pretend both numbers are positive and subtract the smaller number from the larger number.
Eight is larger than three and it is positive, so the answer is positive.
Four is larger than two and it is negative, so the answer is negative. One of the numbers is zero. Zero plus any number is that number. Use a number line to help in adding integers.
Start at the first number. Then, if the second number is positive, move to the right. If the second number is negative, move to the left. The first number is positive and the second is negative. The first number is negative and the second number is positive.
The first number is zero. The second number is zero. Six possible cases. How do you remember how to subtract one number from another? Just use the Painless Solution. Change the subtraction problem into an addition problem by taking the opposite of the number being subtracted.
Now, solve the problem just as you would any other addition problem. To take the opposite of a number, reverse the sign. The opposite of 0 is always 0. Watch how easy it is to solve these subtraction problems with the Painless Solution. Make sure you change both signs, not just one. Change the sign of the problem and the sign of the number being subtracted.
Visual Clue: Use a number line to solve subtraction problems. Find the first number on the number line. If the second number is positive, move to the left. If the second number is negative, move to the right. Here are two examples.
Solve 4 — —4. Start at 4 and then move to the right four spaces. Start at 0 and then move to the left three spaces. In the lists below, match each subtraction problem on the left to the correct addition problem. If you get all six correct, you will spell a word.
Use the addition problems in Brain Ticklers 3 for help. Just change the subtraction problem into an addition problem, and take the opposite of the number being subtracted. Then solve the new addition problem. When you multiply two integers, there are four possible cases.
One number is positive and the other is negative. Just multiply the numbers. Just pretend the numbers are positive. Multiply the numbers. One number is positive and the other negative. Multiply the numbers together.
The answer is always negative. The answer is always zero. Zero times any number or any number times zero is always zero. A positive number times a positive number is a positive number. A negative number times a negative number is a positive number.
A positive number times a negative number is a negative number. One number is negative and one number is positive. The dividend is zero. The divisor is zero. Divide the numbers. Case 3: Pretend the numbers are positive.
Zero divided by any number except 0 is zero. Division by zero is always undefined. How can you divide something into zero part? The rules for division are the same as those for multiplication. A negative number divided by a negative number is positive.
A positive number divided by a negative number is negative. A negative number divided by a positive number is negative. When you add, subtract, multiply, or divide signed numbers, how do you figure out the sign of the answer? First, decide the type of problem. Is it an addition, subtraction, multiplication, or division problem? What is the sign of the problem? Now you know the sign of the answer. Then circle the letter next to the correct answer and spell a phrase.
Word problems have a bad reputation, but actually they are easy. To solve a word problem, all you have to do is change Plain English into Math Talk.
Here is how to solve a few word problems that use integers. An elevator went up three floors and down two floors. How much higher or lower was the elevator than when it started? The elevator was one floor higher than when it started. What was the change in temperature? There was an eight-degree change in temperature. The temperature dropped two degrees every hour. How many degrees did it drop in six hours?
The temperature dropped 12 degrees. For how many days has Bob bought lunch? Type of problem: Bob bought lunch for four days. A variable is a letter that is used to represent a number. Some equations have variables in them, and some do not. Think of the letter x as a mystery number. Here is how to change the following equations from Math Talk into Plain English. One of the major goals of algebra is to figure out what this mystery number is.
When you figure out the value of the mystery number and insert it into the equation, the mathematical sentence will be true. Sometimes you can look at an equation and guess what the mystery number is.
What do you think the mystery number is? Sometimes you can look at an equation and figure out the correct answer, but most of the time you have to solve an equation using the principles of algebra. Probably not. By the time you finish this chapter, however, you will consider the equation as easy as pie. Easy as pie! There are three steps to solving an equation with one variable. Simplify each side of the equation. Multiply or divide.
Here is how to do each step. Simplifying each side of the equation To simplify an equation, first simplify the left side. Next, simplify the right side. When you simplify each side of the equation, use the Order of Operations.
On each side, do what is inside the parentheses first. Use the Distributive Property of Multiplication over Addition to get rid of the parentheses. Multiply and divide. Add and subtract. There is nothing to simplify on the left side, so simplify the right side of the equation.
First add what is inside the parentheses. Multiply 5 times x and 5 times 2. Look how easy it is! Combine all the terms with the same variable. Subtract x from 5x. In simplifying an equation so that it can be solved: Whatever is on the left side of the equals sign stays on the left side. Whatever is on the right side of the equals sign stays on the right side. Using addition and subtraction to solve equations with one variable Once an equation is simplified, the next step is to get all the variables on one side of the equation and all the numbers on the other side.
If an equation is a true sentence, what is on one side of the equals sign is equal to what is on the other side of the equals sign. You can add the same number or variable to both sides of the equation—the new equation is also true. You can also subtract the same number or variable from both sides of the equation to obtain a true equation. Start by getting all the variables on the left side of the equation and all the numbers on the right side.
To get the two numbers on the right side, add four to both sides of the equation. Why 4? The only variable will be on the left side of the equation, and the two numbers will be on the right side. To get all the numbers on the right side, subtract five from both sides of the equation.
Why 5? When you subtract five from the left side, there will be no numbers on that side of the equation. To do this, subtract x from both sides of the equation.
Why x? To get the two variables on the left side, subtract 3x from both sides of the equation.
Why 3x? To get the two numbers on the right side of the equation, add five to both sides. Whatever you do to one side of an equation, you must do to the other side. You must treat both sides equally. Keep the variable x on the left side of the equation and the numbers on the right side. Both sides of the equation will still be equal. You can also divide one side of an equation by any number as long as you divide the other side by the same number.
Once you have all the variables on one side of the equation and all the numbers on the other side, how do you decide what number to multiply or divide by?
You pick the number that will give you only one x. If the equation has 5x, divide both sides by 5. If the equation has 3x, divide both sides by 3. I divided by three. Compute by multiplying. Remember the three steps for solving an equation. Use the Order of Operations. Multiply or divide both sides of the equation by the same number. Here is an example of an equation to solve where you must use all three steps.
Simplify the equation. Combine like terms. Subtract 3 from both sides. Divide both sides of the equation by 2. Remember to use the three steps. Subtract 15 from both sides. Divide both sides of the equation by 3. Simplify by combining like terms. Subtract 2 from both sides. Multiply both sides by 2. Checking your work Once you solve an equation, it is important to check your work. Substitute the answer in the original equation wherever you see an x or other variable. Then compute the value of the sentence with the number in place of the variable.
If the two sides of the equation are equal, the answer is correct. To check, substitute 3 for x. Multiply 3 3. To check, substitute 2 for x. Multiply 3 times 7. Rosa made a mistake. To check, substitute 4 for x. When you check a problem, you find out whether your answer is right or wrong, but you do not learn the correct answer if your answer is wrong.
To find the correct answer, solve the problem again. To find out, check each problem by substituting the answer for the variable. Once you translate a problem correctly, solving it is easy. These simple rules should help you change Plain English into Math Talk.
Rule 1: Change the word equals or any of the words is, are, was, and were into an equals sign. Rule 2: A number is twice twelve. John is two inches shorter than Quan. Quan is 62 inches tall.
How tall is John? Here is how you can begin to solve word problems. A number plus three is twelve. Find the number.
Change this sentence into Math Talk. Subtract three from both sides of the equation. Four times a number plus two is eighteen. Subtract two from both sides of the equation. Two times the larger of two consecutive integers is three more than three times the smaller integer. Find both integers. This problem is solved incorrectly. Set 18, page 83 1. Inequalities have symbols for four different expressions.
Here is how you change the following inequalities from Math Talk into Plain English. An inequality can be true or false. Here are some examples of true inequalities. Since negative six is equal to negative six, this sentence is true.
The inequality states whether the variable is larger, smaller, or maybe even equal to a specific number. If an inequality has a variable in it, some numbers will make this inequality true while others will make it false. If x is equal to 3, 4, or 5, this inequality is true.
The graph gives you a quick picture of all the mystery numbers on the number line that will work. There are three steps to graphing an inequality on the number line. Locate the number in the inequality on the number line. Two is marked on the number line. Circle the number two. Circling the number means that it is not included in the graph. Notice that all the numbers are shaded, not just all the whole numbers. All the numbers to the right of two are greater than two.
Zero is already marked on the number line. Circle and shade zero, since it is included in the graph. Here is how you read the following graphs and change them from Math Talk into Plain English. There are a couple of important differences, however, so pay close attention.
Follow the same three steps when solving inequalities that you followed when solving equations, and add a fourth step. Simplify each side of the inequality using the Order of Operations.
Simplifying each side of the inequality is a two-step process. First simplify the left side. Next simplify the right side. Move all the variables to one side of the inequality and all the numbers to the other side. Multiply or divide both sides of the inequality by the same number. Now here is the key difference: If you multiply or divide by a negative number, reverse the direction of the inequality. Graph the answer on the number line. Now you can solve an inequality.
Simplify the left side of the inequality. Add or subtract the same number to or from both sides of the inequality. Add 2 to both sides. Why 2? Because then all the numbers will be on the right side of the inequality.
Divide both sides by 2. Because you want to have one x on one side of the inequality. Circle the number 3 on the number line. Circling 3 indicates that 3 is not included in the graph.
Shade the number line to the right of the number 3, since the inequality states that x is greater than 3. Simplify each side of the inequality. Combine the like terms on the left side of the inequality.
Subtract 4 from both sides. Because you want to have only one x on the left side of the inequality. Because you are dividing by a negative number, you must reverse the direction of the inequality. Shade the circle. Whenever you multiply or divide an inequality by a negative number, you must remember to change the direction of that inequality. Simplify the left side. Subtract 5 from both sides.
Remember to reverse the direction of the inequality, since you are multiplying both sides by a negative number. When solving an inequality with one variable, follow these steps to success. Simplify both sides of the inequality. Add or subtract the same number or variable on both sides of the inequality. Make sure all of the variables are on one side of the inequality and all the numbers are on the other side. The objective is to get x on one side of the inequality by itself.
If you multiply or divide by a negative number, you must reverse the direction of the inequality. Checking your work To check whether you solved an inequality correctly, follow these simple steps. Change the inequality sign in the problem to an equals sign.
Substitute the number in the answer for the variable. If the sentence is true, continue to the next step. If the sentence is not true, STOP.
The answer is wrong.
Substitute zero in the inequality to check the direction of the inequality. Check to see whether Allison is right or wrong. First change the inequality into an equation. Substitute the answer for x. Substitute 1 for x. Now check the direction of the inequality by substituting 0 for x.
Allison was correct. Much of an inequality problem is changed the same way you change a word problem into an equation.
The trick is deciding which inequality to use and which way it should point. Here are some tips that should help. Look for these phrases when solving inequalities. Watch how these mathematical expressions are changed from Plain English into Math Talk. Twice a number is at most six.
Study each of them. The product of five and a number is less than zero. What could the number be? First change the problem from Plain English into Math Talk. Divide both sides by 5. The sum of two consecutive numbers is at least thirteen.
What could the first number be? Solve the problem. First simplify. Subtract 1 from both sides of the equation. Three times a number plus one is at most ten. Now solve the inequality. Subtract 1 from both sides of the inequality.
Divide both sides by 3. False 4. True 2. True 5. True 3. Before you can learn how to graph equations and inequalities, you need to learn how to plot points and graph a line. Imagine two number lines that intersect each other. One of these lines is horizontal and the other line is vertical. The horizontal line is called the x-axis. The vertical line is called the y-axis. The point where the lines intersect is called the origin. The origin is the point 0, 0. Each of the two number lines has numbers on it.
It is just like a number line. The numbers to the right of the origin are positive. The numbers to the left of the origin are negative. It is like a number line that is standing straight up. The numbers on the top half of the number line are positive.
The numbers on the bottom half are negative. Each point to be graphed is written as two numbers, such as 3, 2. The first number is the x value. The second number is the y value.
The first number tells you how far to the left or the right of the origin the point is. The second number tells you how far above or below the origin the point is. To graph a point, follow these four painless steps. Put your pencil at the origin. Start with the x term. Successfully reported this slideshow. We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads.
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