A Quick Algebra Review
1. Simplifying Expressions
2. Solving Equations
3. Problem Solving
4. Inequalities
5. Absolute Values
6. Linear Equations
7. Systems of Equations
8. Laws of Exponents
9. Quadratics
10. Rationals
11. Radicals
Simplifying Expressions
An expression is a mathematical “phrase.” Expressions contain numbers
and variables, but not an equal sign. An equation has an “equal” sign. For
example:
Expression: Equation:
5+3 5+3=8
x+3 x+3=8
(x + 4)(x – 2) (x + 4)(x – 2) = 10
x² + 5x + 6 x² + 5x + 6 = 0
x–8 x–8>3
When we simplify an expression, we work until there are as few terms as
possible. This process makes the expression easier to use, (that’s why it’s
called “simplify”). The first thing we want to do when simplifying an
expression is to combine like terms.
,For example:
There are many terms to look
at! Let’s start with x². There Simplify:
are no other terms with x² in
them, so we move on. 10x x² + 10x – 6 – 5x + 4
and 5x are like terms, so we
add their coefficients = x² + 5x – 6 + 4
together. 10 + (-5) = 5, so we
write 5x. -6 and 4 are also = x² + 5x – 2
like terms, so we can combine
them to get -2. Isn’t the
simplified expression much
nicer?
Now you try: x² + 5x + 3x² + x³ - 5 + 3
[You should get x³ + 4x² + 5x – 2]
Order of Operations
PEMDAS – Please Excuse My Dear Aunt Sally, remember that from
Algebra class? It tells the order in which we can complete operations when
solving an equation. First, complete any work inside PARENTHESIS, then
evaluate EXPONENTS if there are any. Next MULTIPLY or DIVIDE
numbers before ADDING or SUBTRACTING. For example:
Inside the parenthesis,
look for more order of
Simplify:
operation rules -
PEMDAS.
-2[3 - (-2)(6)]
We don’t have any
exponents, but we do
= -2[3-(-12)]
need to multiply
before we subtract,
= -2[3+12]
then add inside the
parentheses before we
= -2[15]
multiply by negative 2
on the outside.
= -30
, Let’s try another one…
Inside the parenthesis, look for
order of operation rules - Simplify:
PEMDAS.
We need to subtract 5 from 3 then (-4)2 + 2[12 + (3-5)]
add 12 inside the parentheses. This
takes care of the P in PEMDAS, = (-4)2 + 2[12 + (-2)]
now for the E, Exponents. We
square -4. Make sure to use (-4)2 if = (-4)2 + 2[10]
you are relying on your calculator.
If you input -42 the calculator will = 16 + 2[10]
evaluate the expression using
PEMDAS. It will do the exponent = 16 + 20
first, then multiply by -1, giving
you -16, though we know the = 36
answer is 16. Now we can multiply
and then add to finish up.
Practice makes perfect…
Since there are no like terms
inside the parenthesis, we
need to distribute the negative Simplify:
sign and then see what we
have. There is really a -1 (5a2 – 3a +1) – (2a2 – 4a + 6)
there but we’re basically lazy
when it comes to the number = (5a2 – 3a +1) – 1(2a2 – 4a + 6)
one and don’t always write it
(since 1 times anything is = (5a2 – 3a +1) – 1(2a2)–(-1)(- 4a )+(-1)( 6)
itself). So we need to take -1
times EVERYTHING in the = (5a2 – 3a +1) –2a2 + 4a – 6
parenthesis, not just the first
term. Once we have done = 5a2 – 3a +1 –2a2 + 4a – 6
that, we can combine like
terms and rewrite the = 3a2 + a - 5
expression.
Now you try: 2x + 4 [2 –(5x – 3)]
[you should get -18x +20]
, Solving Equations
An equation has an equal sign. The goal of solving equations is to get the
variable by itself, to SOLVE for x =. In order to do this, we must “undo”
what was done to the problem initially. Follow reverse order of operations –
look for addition/subtraction first, then multiplication/division, then
exponents, and parenthesis. The important rule when solving an equation is
to always do to one side of the equal sign what we do to the other.
For example
Solve: To solve an equation we need to get our
x + 9 = -6 variable by itself. To “move” the 9 to the
other side, we need to subtract 9 from
-9 -9 both sides of the equal sign, since 9 was
added to x in the original problem. Then
x = -15 we have x + 9 – 9 = -6 – 9 so x + 0 = -15
or just x = -15.
Solve:
5x – 7 = 2 When the equations get more
complicated, just remember to “undo”
+7 +7 what was done to the problem initially
using PEMDAS rules BACKWARDS
5x = 9 and move one thing at a time to leave the
term with the variable until the end.
5x = 9 They subtract 7; so we add 7 (to both
5 5 sides). They multiply by 5; we divide by
five.
x = 9/5
Solve:
When there are variables on both sides of 7(x + 4) = 6x + 24
distribute
the equation, add or subtract to move
them to the same side, then get the term 7x + 28 = 6x + 24
with the variable by itself. Remember, -28 - 28
we can add together terms that are alike!
7x = 6x - 4
-6x -6x
Your Turn: 2(x -1) = -3 x = -4
(you should get x = -1/2)
1. Simplifying Expressions
2. Solving Equations
3. Problem Solving
4. Inequalities
5. Absolute Values
6. Linear Equations
7. Systems of Equations
8. Laws of Exponents
9. Quadratics
10. Rationals
11. Radicals
Simplifying Expressions
An expression is a mathematical “phrase.” Expressions contain numbers
and variables, but not an equal sign. An equation has an “equal” sign. For
example:
Expression: Equation:
5+3 5+3=8
x+3 x+3=8
(x + 4)(x – 2) (x + 4)(x – 2) = 10
x² + 5x + 6 x² + 5x + 6 = 0
x–8 x–8>3
When we simplify an expression, we work until there are as few terms as
possible. This process makes the expression easier to use, (that’s why it’s
called “simplify”). The first thing we want to do when simplifying an
expression is to combine like terms.
,For example:
There are many terms to look
at! Let’s start with x². There Simplify:
are no other terms with x² in
them, so we move on. 10x x² + 10x – 6 – 5x + 4
and 5x are like terms, so we
add their coefficients = x² + 5x – 6 + 4
together. 10 + (-5) = 5, so we
write 5x. -6 and 4 are also = x² + 5x – 2
like terms, so we can combine
them to get -2. Isn’t the
simplified expression much
nicer?
Now you try: x² + 5x + 3x² + x³ - 5 + 3
[You should get x³ + 4x² + 5x – 2]
Order of Operations
PEMDAS – Please Excuse My Dear Aunt Sally, remember that from
Algebra class? It tells the order in which we can complete operations when
solving an equation. First, complete any work inside PARENTHESIS, then
evaluate EXPONENTS if there are any. Next MULTIPLY or DIVIDE
numbers before ADDING or SUBTRACTING. For example:
Inside the parenthesis,
look for more order of
Simplify:
operation rules -
PEMDAS.
-2[3 - (-2)(6)]
We don’t have any
exponents, but we do
= -2[3-(-12)]
need to multiply
before we subtract,
= -2[3+12]
then add inside the
parentheses before we
= -2[15]
multiply by negative 2
on the outside.
= -30
, Let’s try another one…
Inside the parenthesis, look for
order of operation rules - Simplify:
PEMDAS.
We need to subtract 5 from 3 then (-4)2 + 2[12 + (3-5)]
add 12 inside the parentheses. This
takes care of the P in PEMDAS, = (-4)2 + 2[12 + (-2)]
now for the E, Exponents. We
square -4. Make sure to use (-4)2 if = (-4)2 + 2[10]
you are relying on your calculator.
If you input -42 the calculator will = 16 + 2[10]
evaluate the expression using
PEMDAS. It will do the exponent = 16 + 20
first, then multiply by -1, giving
you -16, though we know the = 36
answer is 16. Now we can multiply
and then add to finish up.
Practice makes perfect…
Since there are no like terms
inside the parenthesis, we
need to distribute the negative Simplify:
sign and then see what we
have. There is really a -1 (5a2 – 3a +1) – (2a2 – 4a + 6)
there but we’re basically lazy
when it comes to the number = (5a2 – 3a +1) – 1(2a2 – 4a + 6)
one and don’t always write it
(since 1 times anything is = (5a2 – 3a +1) – 1(2a2)–(-1)(- 4a )+(-1)( 6)
itself). So we need to take -1
times EVERYTHING in the = (5a2 – 3a +1) –2a2 + 4a – 6
parenthesis, not just the first
term. Once we have done = 5a2 – 3a +1 –2a2 + 4a – 6
that, we can combine like
terms and rewrite the = 3a2 + a - 5
expression.
Now you try: 2x + 4 [2 –(5x – 3)]
[you should get -18x +20]
, Solving Equations
An equation has an equal sign. The goal of solving equations is to get the
variable by itself, to SOLVE for x =. In order to do this, we must “undo”
what was done to the problem initially. Follow reverse order of operations –
look for addition/subtraction first, then multiplication/division, then
exponents, and parenthesis. The important rule when solving an equation is
to always do to one side of the equal sign what we do to the other.
For example
Solve: To solve an equation we need to get our
x + 9 = -6 variable by itself. To “move” the 9 to the
other side, we need to subtract 9 from
-9 -9 both sides of the equal sign, since 9 was
added to x in the original problem. Then
x = -15 we have x + 9 – 9 = -6 – 9 so x + 0 = -15
or just x = -15.
Solve:
5x – 7 = 2 When the equations get more
complicated, just remember to “undo”
+7 +7 what was done to the problem initially
using PEMDAS rules BACKWARDS
5x = 9 and move one thing at a time to leave the
term with the variable until the end.
5x = 9 They subtract 7; so we add 7 (to both
5 5 sides). They multiply by 5; we divide by
five.
x = 9/5
Solve:
When there are variables on both sides of 7(x + 4) = 6x + 24
distribute
the equation, add or subtract to move
them to the same side, then get the term 7x + 28 = 6x + 24
with the variable by itself. Remember, -28 - 28
we can add together terms that are alike!
7x = 6x - 4
-6x -6x
Your Turn: 2(x -1) = -3 x = -4
(you should get x = -1/2)
