Algebra 1 Review

Properties of Real Numbers Algebraic Expressions

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

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

Real Numbers

Irrational Numbers

-√3, π,

1.234897037… Natural Numbers 1, 2, 3, …

Rational Numbers: ½, 0.3, 1, 2 2/3, -5/4, -1.07

Integers: …, -2, -1, 0, 1, 2, 3, …

Whole Numbers: 0, 1, 2, 3, 4, ….

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

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

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.

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.

How do you describe a variable expression?

Variable Expression

Meaning Operation

5x, 5·x, (5)(x) (same as x·5)

5 times x Multiplication

5 divided by x

Division

5 + x (same as x + 5)

5 plus x Addition

5 – x 5 minus x subtraction

xx

÷5,5

Substitute 4 for n. Simplify

Simplify (means to solve the problem or perform as many of the indicated operations as possible.)

Solution:

Substitute 4 for n. Simplify

Solution:

Evaluate a Variable Expression

n  Example 1: Evaluate each expression when n = 4. a. n + 3 n + 3 = 4 + 3 = 7 b. n – 3 n – 3 = 4 – 3 = 1

Substitute 8 for x. Simplify

Solution:

Solution:

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:

4

4

2; 48

4

xx

because

x

=÷

=

=

Evaluate an Algebraic Expression

n  Example 2: Evaluate each expression if x = 8. a. 5x 5x = 5(8) = 40 b. x ÷ 4 x ÷ 4 = 8 ÷ 4 = 2

Substitute 4 for x; 6 for y. simplify Solution:

Evaluating More Expressions

n  Example 3: Evaluate each expression if x = 4, y = 6, and z = 24. a. 5xy 5xy = 5(4)(6) = 120 b.

= 4

Solution:

yz

624

=yz

Substitute 24 for z; 6 for y. Simplify.

A

A

A

A

A

A

Now You Try… Evaluate each expression given that a = 6, b = 12, and c = 3. 1.  4ac 2.  a ÷ c 3.  a + b + c 4.  ba 5.  b – c 6.  c ÷ b

Substitute the value for a = 6 and c = 3 into the problem and multiply

Click to return to “You try it” slide

You Try #1

Evaluate each expression given that a = 6, b = 12, and c = 3.

1.  4ac 4ac = 4(6)(3) = (24)(3) = 72

Substitute the value for a = 6 and c = 3 into the problem and divide

Click to return to “You try it” slide

You Try #2

Evaluate each expression given that a = 6, b = 12, and c = 3.

2. a ÷ c a ÷ c = 6 ÷ 3 = 2

Substitute the value for a = 6, b=12, and c = 3 into the problem, then add.

Click to return to “You try it” slide

You Try #3

Evaluate each expression given that a = 6, b = 12, and c = 3.

3.  a + b + c a + b + c = 6 + 12 + 3 = 18 + 3 = 21

Substitute the value for b=12 and a = 6 into the problem, then multiply.

Click to return to “You try it” slide

You Try #4

Evaluate each expression given that a = 6, b = 12, and c = 3.

4. ba ba = (12)(6) = 72

Substitute the value for b=12 and a = 3 into the problem, then subtract.

Click to return to “You try it” slide

You Try #5

Evaluate each expression given that a = 6, b = 12, and c = 3.

5. b - c b – c = 12 – 3 = 9

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.

Divide both numerator and denominator by the GCF = (3) to reduce this fraction. It is OK to have a fraction

as an answer. Click to return to “You try it” slide

You Try #6

Evaluate each expression given that a = 6, b = 12, and c = 3.

6. c ÷ b

123

==÷bcbc

41

33

123

=÷

÷

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 2y2 and –y2

q  Coefficient: A constant that multiplies a variable. n  i.e. the 3 in 3a or the -1 in –b

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.

Let’s try one…

n  Step One: Write the expression. 4x + 5x -2 - 2x + 7

n  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

n  Simplify the variable terms.

4x+5x-2x = 9x-2x = 7x n  Simplify the constant (number) terms.

-2+7 = 5 n  You have a simplified expression by writing all of the results from

simplifying. 7x + 5

Another example…

n  10x – 4y + 3x2 + 2x – 2y

3x2 10x, 2x -4y – 2y

n  3x2 + 12x – 6y

Remember you cannot combine terms with the

same variable but different exponents.

Now you try…

Simplify the following: n  5x + 3y - 6x + 4y + 3z n  3b - 3a - 5c + 4b n  4ab – 2a2b + 5 – ab + ab2 + 2a2b + 4 n  5xy – 2yx + 7y + 3x – 4xy + 2x

A

A

A

A

You Try #1

n  Simplify the following: 1.  5x + 3y - 6x + 4y + 3z

5x, -6x

3y, 4y 3z -x + 7y + 3z

You Try #2

n  Simplify the following: 2.  3b - 3a - 5c + 4b

3b, 4b

-3a -5c -3a + 7b – 5c

You Try #3

n  Simplify the following: 3.  4ab – 2a2b + 5 – ab + ab2 + 2a2b + 4

4ab, -ab

-2a2b, 2a2b 5, 4 ab2 3ab + ab2 + 9

You Try #4

n  Simplify the following: 4.  5xy – 2yx + 7y + 3x – 4xy + 2x

5xy, -2yx, -4xy

7y 3x, 2x -xy + 7y + 5x

Conclusion

n  A variable or algebraic expression is an expression that consists of one or more ________ and _________ along with one or more ________ _________. (Note: No _______ sign)

n  To evaluate an expression write the _________, substitute a _______ for each variable, and _________ the result.

numbers variables

arithmetic operations equal

expression number

simplify

Conclusion Continued…

n  In an expression, __________ are the terms that have the same ________, raised to the same ________ (same exponents).

n  A coefficient is a number that ________ a variable.

like terms

variables

power

multiplies