Pre-Algebra Notes Unit Five: Rational Numbers; Solving Equations & Inequalities

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1 Pre-Algebra Notes Unit Five: Rational Numbers; Solving Equations & Inequalities Rational Numbers Rational numbers are numbers that can be written as a quotient of two integers. Since decimals are special fractions, all the rules we learn for fractions will work for decimals. The only difference is the denominators for decimals are powers of 0; i.e., 0, 0, 0, 0 4, etc... Students normally think of powers of 0 in standard form: 0, 00, 000, 0,000, etc. In a decimal, the numerator is the number to the right of the decimal point. The denominator is not written, but is implied by the number of digits to the right of the decimal point. The number of digits to the right of the decimal point is the same as the number of zeros in the power of 0: 0, 00, 000, 0,000 Therefore, one place is tenths, two places are hundredths, and three places are thousandths. Examples: ) 56 places ) 5 places ). place The correct way to say a decimal numeral is to: ) Forget the decimal point (if it is less than one). ) Say the number. ) Then say its denominator and add the suffix ths. Examples: ) 5 Fifty-three hundredths ) 70 Seven hundred two thousandths ) Two tenths 4) 5.6 Five and sixty-three hundredths When there are numbers on both sides of the decimal point, the decimal point is read as and. You say the number on the left side of the decimal point, and then the decimal point is read as and. You then say the number on the right side with its denominator. McDougal Littell, Pre-Algebra 8, Unit 05: Rational Numbers; Solving Equations & Inequalities Page of

2 Examples: ) Write 5.0 in word form. Fifteen and two hundred three thousandths ) Write in word form. Seven and four hundred eighty-three ten-thousandths ) Write in word form. Two hundred forty-seven and forty-five hundredths Converting Fractions to Decimals: Terminating and Repeating Decimals Syllabus Objective: (.) The student will write rational numbers in equivalent forms. (.) the student will translate among different forms of rational numbers. CCSS 8.NS.-: Understand informally that every number has a decimal expansion; show that the decimal expansion of a rational number repeats eventually or terminates. A rational number, a number that can be written in the form of a (quotient of two integers), will b either be a terminating or repeating decimal. A terminating decimal has a finite number of decimal places; you will obtain a remainder of zero. A repeating decimal has a digit or a block of digits that repeat without end. One way to convert fractions to decimals is by making equivalent fractions. Example: Convert to a decimal. Since a decimal is a fraction whose denominator is a power of 0, look for a power of 0 that will divide into evenly. 5 0 Since the denominator is 0, we need only one digit to the right of the decimal point, and the answer is 5. McDougal Littell, Pre-Algebra 8, Unit 05: Rational Numbers; Solving Equations & Inequalities Page of

3 Example: Convert to a decimal. 4 Again, since a decimal is a fraction whose denominator is a power of 0, we look for powers of 0 that the denominator will divide into evenly. 4 won t go into 0, but 4 will go into 00 evenly Since the denominator is 00, we need two digits to the right of the decimal point, and the answer is 75. There are denominators that will never divide into any power of 0 evenly. Since that happens, we look for an alternative way of converting fractions to decimals. Could you recognize numbers that are not factors of powers of ten Using your Rules of Divisibility, factors of powers of ten can only have prime factors of or 5. That would mean, whose prime factors are and, would not be a factor of a power of ten. That means that will never divide into a power of 0 evenly. For example, a fraction such as 5 will not terminate it will be a repeating decimal. Not all fractions can be written with a power of 0 as the denominator. We need to look at another way to convert a fraction to a decimal: divide the numerator by the denominator. Example: Convert to a decimal. 8 This could be done by equivalent fractions since the only prime factor of is However, it could also be done by division Doing this division problem, we get 75 as the equivalent decimal. Example: Convert 5 to a decimal. This could not be done by equivalent fractions since one of the factors of is. We can still convert it to a decimal by division Six is repeating, so we can write it as 46. The vinculum is written over the digit or digits that repeat. McDougal Littell, Pre-Algebra 8, Unit 05: Rational Numbers; Solving Equations & Inequalities Page of

4 Example: Convert 4 to a decimal. This would be done by division or 6 Converting Decimals to Fractions Syllabus Objective: (.) The student will write rational numbers in equivalent forms. (.) The student will translate among different forms of rational numbers. CCSS 8.NS.-: Convert a decimal expansion which repeats eventually into a rational number. To convert a decimal to a fraction: ) Determine the denominator by counting the number of digits to the right of the decimal point. ) The numerator is the number to the right of the decimal point. ) Simplify, if possible. Examples: ) Convert 5 to a fraction ) Convert 6 to a fraction ) Convert 8. to a mixed number and improper fraction or 5 5 McDougal Littell, Pre-Algebra 8, Unit 05: Rational Numbers; Solving Equations & Inequalities Page 4 of

5 But what if we have a repeating decimal While the decimals and look alike at first glance, they are different. They do not have the same value. We know is three tenths,. How can we say or write as a fraction 0 As we often do in math, we take something we don t recognize and make it look like a problem we have done before. To do this, we eliminate the repeating part the vinculum (line over the ). Example: Convert to a fraction.... Let s let x... Notice, and this is important, that only one number is repeating. If I multiply both sides of the equation above by 0 (one zero), then subtract the two equations, the repeating part disappears. 0x. x x x x is the equivalent fraction for Example: Convert 45 to a fraction. The difficulty with this problem is the decimal is repeating. So we eliminate the repeating part by letting x Note, three digits are repeating. By multiplying both sides of the equation by 000 (three zeros), the repeating parts line up. When we subtract, the repeating part disappears. 000x x x x or McDougal Littell, Pre-Algebra 8, Unit 05: Rational Numbers; Solving Equations & Inequalities Page 5 of

6 Example: Convert to a fraction. Note, one digit is repeating, but one is not. By multiplying both sides of the equation by 0, the repeating parts line up. When we subtract, the repeating part disappears. 0x. x. x. x.. x or which simplifies to 0 5 Ready for a short cut Let s look at some patterns for repeating decimals. or 0. or or 4 00 or 0 88 or or 4 It is easy to generate the missing decimals when you see the pattern! Let s continue to look at a few more repeating decimals, converting back into fractional form. Because we are concentrating on the pattern, we will choose NOT to simplify fractions where applicable. This would be a step to add later. McDougal Littell, Pre-Algebra 8, Unit 05: Rational Numbers; Solving Equations & Inequalities Page 6 of

7 The numerator of the fraction is the same numeral as the numeral under the vinculum. We can also quickly determine the denominator: it is ths for one place under the vinculum, ths for two places under the vinculum, ths for three places under the vinculum, and so on. But what if the decimal is of a form where not all the numerals are under the vinculum Let s look at a few The numerator is generated by subtracting the number not under the vinculum from the entire number (including the digits under the vinculum). We still determine the number of nines in the denominator by looking at the number of digits under the vinculum. The number of digits not under the vinculum gives us the number of zeroes Note that again we chose not to simplify fractions where applicable as we want to concentrate on the pattern. Does Do you believe it Let's look at some reasons why it's true. Using the method we just looked at: McDougal Littell, Pre-Algebra 8, Unit 05: Rational Numbers; Solving Equations & Inequalities Page 7 of

8 Surely if x, then x. But since x also equals... we get that.... But this is unconvincing to many people. So here's another argument. Most people who have trouble with this fact oddly don't have trouble with the fact that /.... Well, consider the following addition of equations then: This seems simplistic, but it's very, very convincing, isn't it Or try it with some other denominator: Which works out very nicely. Or even: It will work for any two fractions that have a repeating decimal representation and that add up to. The problem, though, is BELIEVING it is true. So, you might think of... as another name for, just as... is another name for /. McDougal Littell, Pre-Algebra 8, Unit 05: Rational Numbers; Solving Equations & Inequalities Page 8 of

9 Comparing and Ordering Rational Numbers Syllabus Objectives: (.4) The student will explain the relationship among equivalent representations of rational numbers. We will now have fractions, mixed numbers and decimals in ordering problems. Sometimes you can simply think of (or draw) a number line and place the numbers on the line. Numbers increase as you go from left to right on the number line, so this is particularly helpful when you are asked to go from least to greatest. If placement is not obvious (for instance, when values are very close together), it may be advantageous to write all the number in the same form (decimal or fractional equivalents), and then compare. Example: Order the numbers 5 5, 0., 4.,,, from least to greatest. 4 Let s first rewrite all improper fractions as mixed numbers. 5 5 ; ; Now let s place the values on the number line From least to greatest, the order would be 5,,, 0.,, Sometimes writing the numbers in the same form will assist you in ordering. Example: Order 7, 05.,, 5,. from least to greatest. 8 () Find the decimal equivalents, then compare () Line up the decimals, the order from least to greatest is: () Use the original forms: 7 05.,,,,. 8 McDougal Littell, Pre-Algebra 8, Unit 05: Rational Numbers; Solving Equations & Inequalities Page of

10 OR find the fractional equivalents and then compare Having found a common denominator, the order from least to greatest is: ,,,, Adding and Subtracting Fractions with Like Denominators Using the original forms: 7 05.,,,,. 8 Solving Equations and Inequalities Containing Fractions and Decimals Syllabus Objective: (.) The student will solve equations and inequalities with rational numbers. First Strategy for Solving: You solve equations and inequalities containing fractions and decimals the same as you do with whole numbers; the strategy does not change. To solve linear equations or inequalities, put the variable terms on one side of the equal sign, and put the constant (number) terms on the other side. To do this, use opposite (inverse) operations. Example: Solve: x x x 5 Undo adding one-third by subtracting onethird from both sides of the equation; make equivalent fractions with a common denominator of 5. McDougal Littell, Pre-Algebra 8, Unit 05: Rational Numbers; Solving Equations & Inequalities Page 0 of

11 Example: Solve: x 5 x 5 5 x x or Undo dividing by 5 by multiplying both sides by 5. Cancel. Multiply numerators and denominators, and simplify. Example: Solve: 4 0 x x x 4 4 ( ) x ( ) x 7 x 7 x 6 We could have saved a little time by recognizing that multiplying by and then dividing by could have been done in one step by multiplying by the reciprocal. x x 4 x 4 x 6 McDougal Littell, Pre-Algebra 8, Unit 05: Rational Numbers; Solving Equations & Inequalities Page of

12 Example: Solve x x x x x Second Strategy for Solving: Another way to solve an equation or inequality with fractions is to clear the fractions by multiplying both sides of the equation or inequality by the LCD of the fractions. The resulting equation/inequality is equivalent to the original. You can also clear decimals by determining the greatest number of decimal places and multiplying both sides of the equation/inequality by that power of Example: Solve x x x x + 65 ( ) 6 4x Original equation. Multiply each side by LCD of 6. Distribute. Simplify. Undo adding 0 by subtracting 0from both sides. 4x 5 4x x 4 Undo multiplying by 4 by dividing by 4. Simplify. McDougal Littell, Pre-Algebra 8, Unit 05: Rational Numbers; Solving Equations & Inequalities Page of

13 Example: Solve x x. (. +. x ) (. ) x x 84 80x x. 55 Original inequality. Since greatest number of decimals is, multiply by Distribute and simplify. Undo addition by subtracting 54 from each side. Simplify. Undo multiplication by dividing both sides by 8 Simplify. 0 or 0 Example: Solve x (. x + ) (. ) x x x x. Original inequality. Since greatest number of decimals is, multiply by Distribute and simplify. Undo addition by subtracting 000 from each side. Simplify. 0 or 00 Undo multiplication by dividing both sides by 875. Reverse the inequality. Simplify. McDougal Littell, Pre-Algebra 8, Unit 05: Rational Numbers; Solving Equations & Inequalities Page of