Algebra 1 Review. Properties of Real Numbers. Algebraic Expressions
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1 Algebra 1 Review Properties of Real Numbers Algebraic Expressions
2 Real Numbers Natural Numbers: 1, 2, 3, 4,.. Numbers used for counting Whole Numbers: 0, 1, 2, 3, 4,.. Natural Numbers and 0 Integers:, -3, -2, -1, 0, 1, 2, 3, Positive, Negative, and 0
3 Real Numbers n Rational Numbers: 7/5, -3/2, 0, 0.3, -1.2 q Fractions q Terminating or repeating decimals q Integers n Irrational Numbers: 7/5, -3/2, 0, 0.3, -1.2, π q Decimals that go on and on without repeating
4 Real Numbers Rational Numbers: ½, 0.3, 1, 2 2/3, -5/4, Integers:, -2, -1, 0, 1, 2, 3, Whole Numbers: 0, 1, 2, 3, 4,. Natural Numbers 1, 2, 3, Irrational Numbers - 3, π,
5 Properties of Real Numbers n The opposite or additive inverse of any number a is a; You just change the sign n The reciprocal or multiplicative inverse of any nonzero number a is 1/a
6 Properties of Real Numbers Property Addition Multiplication Closure a + b is a real number ab is a real number. Commutative a + b = b + a ab = ba Associative (a + b) + c = a + (b + c) (ab)c = a(bc) Identity a + 0 = a a * 1 = a Inverse a + (-a) = 0 a * 1/a = 1 Distributive a(b + c) = ab + ac
7 Properties of Real Numbers n The absolute value of a real number is its distance from zero on the number line. Absolute value is always positive because distance is always positive.
8 Evaluating Expressions n Vocabulary: q Variable A symbol, usually a letter of the alphabet, such as the letter n, that is used to represent a number. q Variable expression (A.K.A. - Algebraic Expression) An expression, such as n 5, that consists of one or more numbers and variables along with one or more arithmetic operations. (Note: No equal sign) q Evaluate a Variable Expression write the expression, substitute a number for each variable, and simplify the result.
9 How do you describe a variable expression? Variable Expression 5x, 5 x, (5)(x) (same 5 as x 5),5 x x 5 + x (same as x + 5) Meaning Operation 5 times x Multiplication 5 divided by x Division 5 plus x Addition 5 x 5 minus x subtraction
10 Evaluate a Variable Expression Solution: Solution: n Example 1: Evaluate each expression when n = 4. Simplify (means to solve the problem or perform as many of the indicated operations as possible.) a. n + 3 n + 3 = = 7 b. n 3 n 3 = 4 3 = 1 Substitute 4 for n. Simplify Substitute 4 for n. Simplify
11 Evaluate an Algebraic Expression n Example 2: Evaluate each expression if x = 8. Solution: Solution: a. 5x 5x = 5(8) = 40 b. x 4 x 4 = 8 4 = 2 Substitute 8 for x. Simplify Using parenthesis is the preferred method to show multiplication. Additional ways to show multiplication are 5 8 and 5 x 8. Substitute 8 for x. Simplify Recall that division problems are also fractions this problem could be written as: x 4 = = 8 4 2; because x 4 = x 4
12 Evaluating More Expressions Solution: Solution: n Example 3: Evaluate each expression if x = 4, y = 6, and z = 24. a. 5xy Substitute 4 for x; 6 for y. simplify 5xy = 5(4)(6) = 120 b. z y z y = = Substitute 24 for z; 6 for y. Simplify.
13 Now You Try Evaluate each expression given that a = 6, b = 12, and c = ac 2. a c 3. a + b + c 4. ba 5. b c 6. c b A A A A A A
14 You Try #1 Evaluate each expression given that a = 6, b = 12, and c = ac 4ac = 4(6)(3) = (24)(3) = 72 Substitute the value for a = 6 and c = 3 into the problem and multiply Click to return to You try it slide
15 You Try #2 Evaluate each expression given that a = 6, b = 12, and c = a c a c = 6 3 = 2 Substitute the value for a = 6 and c = 3 into the problem and divide Click to return to You try it slide
16 You Try #3 Evaluate each expression given that a = 6, b = 12, and c = a + b + c a + b + c = = = 21 Substitute the value for a = 6, b=12, and c = 3 into the problem, then add. Click to return to You try it slide
17 You Try #4 Evaluate each expression given that a = 6, b = 12, and c = ba ba = (12)(6) = 72 Substitute the value for b=12 and a = 6 into the problem, then multiply. Click to return to You try it slide
18 You Try #5 Evaluate each expression given that a = 6, b = 12, and c = b - c b c = 12 3 = 9 Substitute the value for b=12 and a = 3 into the problem, then subtract. Click to return to You try it slide
19 You Try #6 Evaluate each expression given that a = 6, b = 12, and c = c b Divide both numerator and denominator by the GCF = (3) to reduce this fraction. c b 3 12 = 3 3 c b = 3 = Substitute the value for c=3 and b = 12 into the problem, then Divide Note: It is better to rewrite this division problem as a fraction. This fraction can now be reduced to its simplest form. It is OK to have a fraction as an answer. Click to return to You try it slide
20 Combining Like Terms n Now that we have seen some algebraic expressions, we need to know how to simplify them. n Vocabulary q Like terms: In an expression, like terms are the terms that have the same variables, raised to the same powers (same exponents). n i.e. 4x and -3x or 2y 2 and y 2 q Coefficient: A constant that multiplies a variable. n i.e. the 3 in 3a or the -1 in b
21 Combining Like Terms n In algebra we often get very long expressions, which we need to make simpler. Simpler expressions are easier to solve! n To simplify an expression we collect like terms. Like terms include letters that are the same and numbers.
22 Let s try one n n n n n Step One: Write the expression. 4x + 5x -2-2x + 7 Collect all the terms together which are alike. Remember that each term comes with an operation (+,-) which goes before it. 4x, 5x, and -2x -2 and 7 Simplify the variable terms. 4x+5x-2x = 9x-2x = 7x Simplify the constant (number) terms = 5 You have a simplified expression by writing all of the results from simplifying. 7x + 5
23 Another example n 10x 4y + 3x 2 + 2x 2y 3x 2 10x, 2x -4y 2y n 3x x 6y Remember you cannot combine terms with the same variable but different exponents.
24 Now you try Simplify the following: n 5x + 3y - 6x + 4y + 3z n 3b - 3a - 5c + 4b n 4ab 2a 2 b + 5 ab + ab 2 + 2a 2 b + 4 n 5xy 2yx + 7y + 3x 4xy + 2x A A A A
25 You Try #1 n Simplify the following: 1. 5x + 3y - 6x + 4y + 3z 5x, -6x 3y, 4y 3z -x + 7y + 3z
26 You Try #2 n Simplify the following: 2. 3b - 3a - 5c + 4b 3b, 4b -3a -5c -3a + 7b 5c
27 You Try #3 n Simplify the following: 3. 4ab 2a 2 b + 5 ab + ab 2 + 2a 2 b + 4 4ab, -ab -2a 2 b, 2a 2 b 5, 4 ab 2 3ab + ab 2 + 9
28 You Try #4 n Simplify the following: 4. 5xy 2yx + 7y + 3x 4xy + 2x 5xy, -2yx, -4xy 7y 3x, 2x -xy + 7y + 5x
29 Conclusion n A variable or algebraic expression is an expression that consists of one or more numbers and variables along with one or more arithmetic. operations (Note: No equal sign) n To evaluate an expression write the, expression substitute a number for each variable, and simplify the result.
30 Conclusion Continued n In an expression, like terms are the terms that have the same, variables raised to the same power (same exponents). n A coefficient is a number that multiplies a variable.
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