Algebra (Period 5)

Solving Inequalities 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 Honors ( Period 4)

Solving Inequalities 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 ( Period 5)

Solving Inequalities by Addition and Subtraction 5-1

An open sentence that contains < , >,   ≤, or  ≥  is an inequality.

Graphing an inequality
    Open dot  (○) is used for  <  or >
    Closed dot  (●) is used  for   ≤, or  ≥
(represents that it is either “ less than or EQUAL to”   or it is  “greater than or EQUAL to” )

Think of the = sign as a crayon that you can use to COLOR INT THE DOT
It is different from equations—Inequalities have many answers—in fact most of the time it is an infinite number of possibilities

n > 3 means that every real number greater than 3 is part of the solutions set  ( but NOT 3)
n ≥  3 still means that every real number greater than 3 is part of the solutions set, but now 3 is also a solution

Remember…graphing equation’s solution is easy! Let’s say you found that y = 5, you would just put a dot on 5 on the number line

Now with y ≥  5 you still put the dot on 5 but you also darken a line with an arrow going to the right to indicate all the numbers greater than 5 that are also solutions

With y > 5 you still have that line with an arrow going to the right of 5 but you use the OPEN DOT on 5 to indicate that 5 is NOT A SOLUTION

Translating words:
Some KEY words to know:

AT LEAST: means greater than or equal
NO LESS THAN : also means greater than or equal

I need at least $100 to go to the mall means I must have $100 but I would LOVE to have even more

AT MOST: means less than or equal
NO MORE THAN : also means less than or equal

I want at most 15 minutes of homework means that I will be okay with 15 minutes but I am hoping for even less!

Solving Inequalities with adding and subtracting

Use the additive Inverse Property as you would if you were balancing an equation. The only difference is that now you have more than one possible solution

5y + 4 > 29
You would subtract 4 from BOTH sides , then divide by 5 on each side

y > 5
Your answer is infinite because any real number bigger than 5 will work!

ALWAYS finish with the VARIABLE on the LEFT.

If you don’t, you may misunderstand the answer and graph it in the opposite position.
5 > y is NOT the same as y > 5

5> y means that y < 5

Check with whatever solution is easier in the solutions set—but NEVER USE the boundary number! It really does not help you see if your solutions are true!

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 4)

Solving Inequalities by Addition and Subtraction 5-1

An open sentence that contains < , >,   ≤, or  ≥  is an inequality.

Graphing an inequality
    Open dot  (○) is used for  <  or >
    Closed dot  (●) is used  for   ≤, or  ≥ 
(represents that it is either “ less than or EQUAL to”   or it is  “greater than or EQUAL to” )

Think of the = sign as a crayon that you can use to COLOR INT THE DOT
It is different from equations—Inequalities have many answers—in fact most of the time it is an infinite number of possibilities

n > 3 means that every real number greater than 3 is part of the solutions set  ( but NOT 3)
n ≥  3 still means that every real number greater than 3 is part of the solutions set, but now 3 is also a solution

Remember…graphing equation’s solution is easy! Let’s say you found that y = 5, you would just put a dot on 5 on the number line

Now with y ≥  5 you still put the dot on 5 but you also darken a line with an arrow going to the right to indicate all the numbers greater than 5 that are also solutions

With y > 5 you still have that line with an arrow going to the right of 5 but you use the OPEN DOT on 5 to indicate that 5 is NOT A SOLUTION

Translating words:
Some KEY words to know:

AT LEAST: means greater than or equal
NO LESS THAN : also means greater than or equal

I need at least $100 to go to the mall means I must have $100 but I would LOVE to have even more

AT MOST: means less than or equal
NO MORE THAN : also means less than or equal

I want at most 15 minutes of homework means that I will be okay with 15 minutes but I am hoping for even less!

Solving Inequalities with adding and subtracting

Use the additive Inverse Property as you would if you were balancing an equation. The only difference is that now you have more than one possible solution

5y + 4 > 29
You would subtract 4 from BOTH sides , then divide by 5 on each side

y > 5
Your answer is infinite because any real number bigger than 5 will work!

ALWAYS finish with the VARIABLE on the LEFT.

If you don’t, you may misunderstand the answer and graph it in the opposite position.
5 > y is NOT the same as y > 5

5> y means that y < 5

Check with whatever solution is easier in the solutions set—but NEVER USE the boundary number! It really does not help you see if your solutions are true!

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.