Download Solving Mania KCAS: 7.EE.4, 8.EE.7, A.SSE.1, A.CED.4, A.REI.1, A

Solving Mania
KCAS: 7.EE.4, 8.EE.7, A.SSE.1, A.CED.4, A.REI.1, A.REI.3
Operations with Integers
7.NS.1
The mathematical operations are addition, subtraction, multiplication, and division.
Absolute Value is the distance from the ORIGIN on a number line. Absolute Value is NEVER NEGATIVE.
Integers are the whole numbers and their additive inverses (opposites). 0, ±1, ±2, ±3, …
ADDITION
pos + pos = pos
neg + neg = neg
Pos + neg = ???
To add numbers with
different
signs,
THINK of it like subt
of abs values & take
the sign of number
with greater abs value.
SUBTRACTION
MULTIPLICATION
pos * pos = pos
neg * neg = pos
pos * neg = neg
neg * pos = neg
DIVISION
pos ÷ pos = pos
neg ÷ neg = pos
pos ÷ neg = neg
neg ÷ pos = neg
Use the definition of
subtraction, and
THEN follow addition
rules
Factor * Factor = Product
Do integer quizzes using rules above, pictures, or common sense
An equation is a mathematical sentence that contains an = (equal sign). It can be solved or graphed.
There are absolute value equations like |x| = 7.
There are literal equations that contain 2 or more variables (like formulas, y=2x+3, etc.).
There are linear equations that graph as lines and are first degree (degree is exponent on variable). y=x
There are quadratic equations that graph as parabolas (U-like smooth curves) and are second degree.
y=x2
There are cubic equations that graph as “S-like” smooth curves and are third degree. y=x3
An inequality is a mathematical sentence that contains a < ≤ > ≥ ≠ < ≤ > ≥ (inequality symbol). It can
be solved or graphed.
< is read in words as “IS less than.”
≥ is read in words as “IS greater than or equal to.”
≠ is read in words as “IS NOT equal to.”
In math, “less than” means subtract and “greater than” means add.
2 less than 3 means 3-2. 2 is less than 3 means 2 < 3.
Terms: Separated by addition or subtraction signs NOT IN PARENTHESES.
Like Terms: Sep by + or – signs not in ( ) AND their variables and the exponents on those variables match
exactly. You can add and subtract like terms only. You can multiply anything.
3x2 + 4x2 = 7x2
3x2 * 4x2 = 12x4
3x2 + 4x3 cannot add because the two terms are NOT alike
3x2 * 4x3 = 12x5
Coefficient: a number that is multiplied by a variable. It is typically written in front of the variable.
1.
2.
3.
4.
5.
6.
7.
8.
8 steps to solve linear equations OR inequalities
*******GOAL: isolate the variable for which you are solving!!!*******
 * Means step is OPTIONAL.
* Use the definition of subtraction to change – to + - of the subtrahend.
* Clear fractions by multiplying EACH TERM on both sides of the eqn/ineq by the LCD of ALL
fractions. Clear decimals by multiplying EACH TERM on both sides of eqn/ineq by the proper power
of 10 that would eliminate ALL decimals.
Use the distributive property to clear parentheses.
Combine Like Terms on the LEFT of the = or ineq. Combine Like Terms on the RIGHT side of the = or
ineq.
Get all terms containing the variable on only one side of the = or ineq by using the additive inverse
property. Make sure to add the same thing to BOTH sides of the = or ineq.
Use the additive inverse property to isolate the variable term (get rid of the constant term). Make sure to
add the same thing to BOTH sides of the eqn or ineq.
Use the multiplicative inverse property to isolate the variable. Make sure to multiply BOTH sides of the
eqn or ineq by the same thing. ***REMEMBER that if you MULTIPLY or divide both sides of an
INEQUALITY by a NEGATIVE number that you must reverse the direction of the inequality
symbol!!!!!!!!!!!!!!!!!!!!!!
CHECK YOUR SOLUTION. Substitute your solution into the original eqn/ineq and see if it makes a
true eqn/ineq.
When solving linear equations, you can get the following “types” of results:
I.
X=4. This means you have one solution, and it is 4.
II.
-4 = 4 or 2=7 or 0=6, etc. If you get something untrue like this, there are NO SOLUTIONS to
your equation.
III.
4 = 4 or 2=2 or -6=-6, etc. If you get something always true like this, there are INFINITELY
MANY solutions to your equation, and they are ALL REAL NUMBERS. That means any real
number when substituted in for the variable will produce a true equation.
WHEN SOLVING ABSOLUTE VALUE EQUATIONS, YOUR GOAL IS TO ISOLATE THE ABS VALUE
AND THEN MAKE AN “OR” STATEMENT.
If |x|=7, then x=7 OR x=-7.
If |x|=0, then x=0
If |x|=-3, then there are NO SOLUTIONS because abs val can’t yield a neg.
When solving abs value inequalities, use the acronym GOLA.
When solving literal equations, follow the 8 steps above.