Math 6A ( Periods 2 & 5)

The Coordinate Plane 6-5

The Coordinate Plane is formed by the intersection of a horizontal number line ( x-axis) and a vertical number line ( y-axis)
They intersect at the origin (0,0) and the ORIGIN separates the Coordinate Plane into FOUR regions called QUADRANTS


A PAIR of numbers whose ORDER is important is called an   ordered pair
(ordered, pair) (x, y)

(2,3) is not the same as (3,2)

The two perpendicular lines are called axes.
The x-axis deals with the 1st number of the ordered pair and the y-axis deals with the 2nd number of the ordered pair.
The AXES meet at a point called the Origin (0,0)
The plane is called the coordinate plane

There are 4 quadrants, You MUST use Roman Numerals to name them!!
Quadrant I ---> both the x and y coordinates are positive
(x,y) (+,+)

Quadrant II --> the x coordinate is negative but the y is positive
(-x,y) (-,+)

Quadrant III -->. both the x and y coordinates are negative
(-x,-y) (-,-)

Quadrant IV --> the x coordinate is positive but the y coordinate is negative
(x,-y) (+,-)

Reflect a point in the x-axis
Use the same x-coordinate and take then opposite of the y- coordinate

Reflect a point in the y-axis
Use the same y-coordinate and take then opposite of the x- coordinate

Reflect a point in both axes:
1) x followed by y
First follow the steps above so for (2,1) reflect in x followed by y means
(2,1)--> (2, -1), then --> (-2,-1)  What do you notice? Both coordinates are now the opposites of what they started as!

Algebra Honors ( Period 6)

Solving Multi- Step Inequalities 5-3

Again you will use your equation skills

1.      Do Distributive Property first (if necessary) do it carefully

2.      Combine like terms on each side of the “WALL”

3.      “JUMP” the variables to one side of the wall—that is get all the variables on one side of the inequality by using the Additive Inverse Property (add or subtract using the opposite sign of the variable term, constants second by adding or subtracting

4.      Do it just like a 2 step or 1 step equation but remember;  FLIP THE SYMBOL if you multiply or divide by a NEGATIVE to balance

5.      If you end up with the variable on the right side, SWITCH the SIDES and the SYMBOL

REMINDERS:

Set builder notation

Get familiar with the following notation

{x│ x≥ 5} is read “ x SUCH THAT c is greater than or equal to 5”

Checking your solutions is an important set. Many students skip this step! Checking the solutions is especially important with inequalities because the direction of the inequality sign is often changed when writing solutions in set builder notation.

Algebra (Periods 1 & 4)

Solving Multi- Step Inequalities 5-3

Again you will use your equation skills

1.      Do Distributive Property first (if necessary) do it carefully

2.      Combine like terms on each side of the “WALL”

3.      “JUMP” the variables to one side of the wall—that is get all the variables on one side of the inequality by using the Additive Inverse Property (add or subtract using the opposite sign of the variable term, constants second by adding or subtracting

4.      Do it just like a 2 step or 1 step equation but remember;  FLIP THE SYMBOL if you multiply or divide by a NEGATIVE to balance

5.      If you end up with the variable on the right side, SWITCH the SIDES and the SYMBOL

REMINDERS:

Set builder notation

Get familiar with the following notation

{x│ x≥ 5} is read “ x SUCH THAT c is greater than or equal to 5”

Checking your solutions is an important set. Many students skip this step! Checking the solutions is especially important with inequalities because the direction of the inequality sign is often changed when writing solutions in set builder notation.

Algebra Honors ( period 6)

Solving Inequalities  ( All 4 Op's) 5-1 &5-2

Solving Inequalities 5-2

Again you will use your equation skills but this time use the Multiplicative Inverse Property as you would if you were balancing an equation.

ONE MAJOR DIFFERENCE FROM EQUATIONS:

When you multiply or divide by a NEGATIVE coefficient (to balance) you must SWITCH the inequality SYMBOL (this does NOT apply to adding or subtracting negatives) You must rewrite the problem as you divide by a negative as shown below:

If you want to understand why:    3 < 10      you know that is true

Now multiply both sides by -1 ( multiplication property of equality lets you do that)
but you get -3 < -10 but THAT IS NOT TRUE

You have to SWITCH THE SYMBOL to make it true  -3 > -10

REMEMBER: When you MULTIPLY or DIVIDE by a NEGATIVE, the symbol SWITCHES!

Doing 2 Steps with Inequality Signs:  same as equations except make sure you switch the symbols if you multiply or divide by a negative. (Rewrite that portion of the problem—as you multiply or divide)  Always finish with the variable on the left!

Check with whatever solution is the easiest in the solution set. If 0 fits—use it! 

NEVER USE THE BOUNDARY NUMBER   for instance if your solution was x ≤  4  You could check with any number less than 4—BUT NEVER USE 4!

With two steps—before you start, you may want to clear fractions or decimals but if you don’t mind using them—just get started with the checklist below: If you want to clear them- you should do that right after you distribute ( which is between Steps 1 and 2 below)

1     1, Do Distributive Property first (if necessary) do it carefully

2      2.  Combine like terms on each side of the “WALL”

3   3.  “JUMP” the variables to one side of the wall—that is get all the variables on one side of the inequality by using the Additive Inverse Property (add or subtract using the opposite sign of the variable term

4     4.   Add or subtract

5      5.  Multiple or divide ( only FLIP THE SYMBOL if you multiply or divide by a NEGATIVE to balance)

     6.   Make sure the variable is on the LEFT side when finished.

Set builder notation

Get familiar with the following notation

{x│ x≥ 5} is read “ x SUCH THAT c is greater than or equal to 5”

Checking your solutions is an important set. Many students skip this step! Checking the solutions is especially important with inequalities because the direction of the inequality sign is often changed when writing solutions in set builder notation.

Algebra ( Periods 1 & 4)

Solving Inequalities ( All 4 Op's) 5-1& 5-2

Solving Inequalities 5-2

Again you will use your equation skills but this time use the Multiplicative Inverse Property as you would if you were balancing an equation.

ONE MAJOR DIFFERENCE FROM EQUATIONS:

When you multiply or divide by a NEGATIVE coefficient (to balance) you must SWITCH the inequality SYMBOL (this does NOT apply to adding or subtracting negatives) You must rewrite the problem as you divide by a negative as shown below:

If you want to understand why:    3 < 10      you know that is true

Now multiply both sides by -1 ( multiplication property of equality lets you do that)
but you get -3 < -10 but THAT IS NOT TRUE

You have to SWITCH THE SYMBOL to make it true  -3 > -10

REMEMBER: When you MULTIPLY or DIVIDE by a NEGATIVE, the symbol SWITCHES!

Doing 2 Steps with Inequality Signs:  same as equations except make sure you switch the symbols if you multiply or divide by a negative. (Rewrite that portion of the problem—as you multiply or divide)  Always finish with the variable on the left!

Check with whatever solution is the easiest in the solution set. If 0 fits—use it! 

NEVER USE THE BOUNDARY NUMBER   for instance if your solution was x ≤  4  You could check with any number less than 4—BUT NEVER USE 4!

With two steps—before you start, you may want to clear fractions or decimals but if you don’t mind using them—just get started with the checklist below: If you want to clear them- you should do that right after you distribute ( which is between Steps 1 and 2 below)

1     1, Do Distributive Property first (if necessary) do it carefully

2      2.  Combine like terms on each side of the “WALL”

3   3.  “JUMP” the variables to one side of the wall—that is get all the variables on one side of the inequality by using the Additive Inverse Property (add or subtract using the opposite sign of the variable term

4     4.   Add or subtract

5      5.  Multiple or divide ( only FLIP THE SYMBOL if you multiply or divide by a NEGATIVE to balance)

     6.   Make sure the variable is on the LEFT side when finished.

Set builder notation

Get familiar with the following notation

{x│ x≥ 5} is read “ x SUCH THAT c is greater than or equal to 5”

Checking your solutions is an important set. Many students skip this step! Checking the solutions is especially important with inequalities because the direction of the inequality sign is often changed when writing solutions in set builder notation.