Lesson 6
III. Basic Algebra
A. Simplifying Algebraic Expressions
Combining like terms
Ex1) 6x 2,15x 2
Multiplying Binomials
Ex1) (x – 3)(2x + 4)
5y, 6y 2
Ex2) 3x – 2(x + 3y) – 4y
Ex2) (2x + 5)2
B. Solving Equations and Inequalities
1. Equations
Steps for Solving
1) Remove all fractions
2) Distribute
3) Combine like terms
4) Get variable on one
side
5) Add or subtract
6) Multiply or divide
Solve:
3(x - 2)
(x + 2)
+5=
4
2
2. Inequalities
Same steps as equations EXCEPT: when you
multiply or divide both sides by a negative you
must FLIP THE SIGN!!!
EX) Solve and graph on a number line
6x – 3 > 8x + 7
2. Inequalities: Special Cases
No Solutions
Ex) 3x + 4 < 2(x + 2) + x
All real number solutions
Ex) 2x + 3 < 2x – 2
2. Inequalities
C. Solving Rational Equations
Ex)
(x - 3) (x - 2)
=
(x + 4) (x + 2)
Ex)
1
1
6
+
=
(x + 4) (x + 2) (x - 4)(x + 2)
D. Evaluating and Solving Formulas
Ex) Given a = -1, b = 2, c =
-2, what is the value of
a b-b c
2
2
Ex) Solve for a:
1 2
d = × at
2
E. Factoring
Types of Factoring:
1) GCF (any number of terms)
2) Difference of 2 Squares & Sum of Difference
of cubes (2 Terms)
3) Trinomial into 2 Binomials (3 terms)
4) Grouping (4 terms)
E. Factoring: GCF
Factor out the largest number and the largest
variable. Once you have your answer, you should
be able to distribute it and result in the original
problem.
Ex)
Ex) a7bc3 + a 4b2c2 - a2bc 4
E. Factoring: Difference of 2 squares or
Sum/Difference of Cubes
Diff of 2 Squares Pattern
a - b = (a + b)(a - b)
2
Ex)
2
4x 2 - 9
Sum/Diff of 2 cubes
a3 - b3 = (a - b)(a2 + ab + b2 )
a3 + b3 = (a + b)(a2 - ab + b2 )
Ex)
125x 3 +8
Ex)
27x 3 -1
E. Factoring: Trinomial  2 Binomials
• x should go first in each binomial
• The numbers should multiply to give you c,
and add or subtract to give you b
Ex) 2
Ex)
2
x - 3x -10
2x +11x - 6
E. Factoring: Grouping
Step 1) Group the first 2 terms, factor out GCF
Step 2) Group last two terms, factor out GCF
Step 3) Write new binomials
Ex) ax + bx + ay + by
Ex) 2y + 4z + xy + 2xz
E. Factoring: Extra Step
Note: You may have to factor out a GCF then
apply one of the factoring methods!!!
Ex) 6x 3 - 6x
Ex)
x -1
6
E. Factoring: BOTTOM LINE
Distributing leads back to original expression in
standard form!!!
(x - 5)(x + 2) = x - 3x -10
2
E. Factoring
F. Solving Quadratic Equations
Ways of Solving
1) Square Root Method
2) Factoring
3) Quadratic Formula
1. Square Root Method
• Algebraic, using square rooting
• Don’t forget +/Ex) (x - 3)2 = 25
Ex) 3x 2 - 5 = 70
2. Factoring
• Get equation = to zero
• Factor
• Set each factor = to zero and solve
Ex) x 2 + 9x = -14
Ex) x 2 - 6x = 0
3. Quadratic Formula
• Set equation = to zero
• Plug a, b, and c into the
formula and solve
-b ± b - 4ac
x=
2a
2
Ex)
x + 6x + 4 = 0
2
Solve using any of the methods we
have covered
G. Exponents
Operation
Multiplying same
bases
Power to a Power
Example
Answer
x2 × x3
3x 3 y × 4xy 5
(x 2 )4
(3x 3 )4
Dividing same bases
Negative Exponents
Zero Exponents
x4
x3
10x 2 y 5
5xy
3x 2
x0
x -2 y -3
z -4
4x 0
(4x)0
Rule
Use rules of exponents to solve
H. Radicals
Approximating Square Roots – being able to
guess between 2 whole numbers
Ex) Approximate 50
Simplifying
Ex) 200
Ex)
96
H. Radicals
Adding and Subtracting – simplify first! Then add
or subtract like radicands
Ex) 3 20 + 45
H. Radicals
Multiplying and Dividing – Combine the
numbers on the outside (mult/div), and
combine the numbers on the inside (mult/div)
Ex)
Ex) 5 27
6 20 ×3 30
10 3
H. Radicals
Rationalizing the Denominator – The radical
symbol is NOT allowed to stay in the
denominator
Ex) 2
Ex. 5
3
3- 2
Use rules of radicals to complete the
problems
I. Systems of Equations
Methods of Solving:
1) Graphing – finding where the 2 lines intersect
2) Substitution – plugging one equation into the
other
3) Elimination – line up equations and eliminate
a variable
4) Plugging in the answers
I. Systems of Equations
Possible Solutions:
1) One coordinate point: (x, y)
2) No solution: (lines are parallel)
3) Infinitely many solutions: (line are the same)
Elimination
Solve the following systems using the
elimination method:
1) 3x + 2y = 10
2) 3x + 5y = 13
x – 5y = -8
2x – 2y = -2
Substitution
Solve the systems using the substitution method
1) 3x + 2y = 10
2) 2x + 4y = 6
x = 5y – 8
x + 2y = -4