Class Notes

*HOW TO READ INEQUALITIES surprise

x < 3 : this means that x can be any number that is less than 3. For example 2 can be a solution 0 can be a solution -10 can be a solution.

> 20 : this means that x can be any number that is greater than 20 and 20 can also be a solution. 

 


*HOW TO GRAPH INEQUALITIES yes

STEP 1: Draw a circle around the solution

STEP 2: If it is a greater than or equal to sign or less than or equal to sign shade the circle in. If it is a less than sign or greater than sign leave the circle open. 

STEP 3: Draw an arrow in the direction of the solution set. (if the solution set is less than the variable draw the area to the left of the circle. If the solution set is greater than the variable draw the arrow to the right of the circle. 

       example of    x< 2       x < -4      x > -3

example of inequalities


*HOW TO EVALUATE INEQUALITIES smiley

VOCABULARY:

Greater than  >      Less than  <        Greater than or equal to >       Less than  or equal to <

STEP 1: Use inverse operations to isolate the variable. 

       Example 1: x + 4 < 15 ( the inverse of adding 4 is to subtract 4)

       x + 4 < 15       ⇒   x  < 15 – 4       ⇒       x  < 11

      Example 2:  a ÷ 10 > 30 (The inverse of dividing by 10 is to multiply by 10) 

      a ÷ 10 > 30      ⇒  a > 30(10)      ⇒       a > 300

      Example 3:  5x < 4  ( The inverse of multiplying by 5 is to divide by 5)

      5x < 40         x < 40 ÷ 5       ⇒       < 8

STEP 2: If the inverse operation is to multiply or divide by a negative number you need to switch the direction of the sign.

      Example 1:

     -3x > 15 ( the inverse of multiplying by -3 is dividing by -3)

     x > 15 ÷ -3   ⇒  x > -5   ⇒    x < -5 (since we divided by a negative number we need to switch the sign direction)

     Example 2: 

     x + -3 < 15 ( the inverse of adding a -3 would be subtracting a -3)

     x < 15 – (-3)     ⇒    x < 18  (Since we didn’t need to divide or multiply by a negative number we do not need to                                                          switch the direction of the sign.) 

** The only other time you would need to switch the direction of the sign is when you swap the left side and the right side of the inequalitiy 

                   EXAMPLE:   X – 4 < 10  WHEN YOU SWITCH SIDES THIS INEQUALITIY BECOMES:   10 > X – 4 


*HOW TO COMBINE LIKE TERMSenlightened

LIKE TERMS: Terms that have the same variable and are raised to the same power. For example 3x and -9x are like terms. 8z and -88z are like terms. 4h and 4h² are NOT LIKE TERMS because the second term is raised to the second power and the first term is not. 

COEFFICIENTS: A numerical quanitity that is placed before a variable in an algebraic expression. It signifies multiplication. 

STEP 1: Organize your like terms. You can color code your like terms or you can rearrange the expression so the like terms are next to each other. 

Example: 5h + 11g + 1h - 8g ⇒ 5h + 1h + 11g – 8g

STEP2: Combine the coefficents of the like terms. With negative numbers you might want to change subtraction into addition. Wtih fractions you might want to make common denominators. If you have fractions and decimals in the same expression you might want to change all the numbers into fractions or change all the numbers into decimals. 

Example: (5 + 1)h + (11 - 8)g ⇒ 6h + 3g.


*HOW TO DISTRIBUTE

STEP 1: Multiply each term by the numbers and/or variables outside of the parentheses.

Example: 2(4x - 3y) = 2(4x) - 2(3y) = 8x - 6y

Example: -(3x + 6) = -(3x) + -(6) = -3x + -6

Example: -3(2x + 6u) = -3(2x) + -3(6u) = -6x  + -18u


*HOW TO SOLVE EQUATIONS

Inverse operations: Addition and Subtraction are inverse operations. Multiplication and Division are inverse operations.  

STEP 1: Use inverse operations to get the variable on one side alone. 

Example: x – 5 = 34 ⇒ x = 34 + 5 ⇒ x = 39

Example: 4x = 20  ⇒ x = 20/4 ⇒x = 5

STEP 2: Replace your solution with the variable to be sure your answer is correct. 

Example: 39 – 5 = 34

Example: 4(5) = 20


*HOW TO FIND THE PERCENT OF A NUMBER

STEP 1: Locate the decimal point in the percent. If there is no decimal point add one to the end of the number

          Ex: 34% ⇒ 34.% or 345% ⇒ 345.%

STEP 2: Change the percent into a decimal by moving the decimal point two places to the left. If necessary use zeros as placeholders. 

         EX: 34.% ⇒.34 or 345.% ⇒ 3.45

STEP 3: Multiply the decimal by the number

*EXAMPLE OF HOW TO FIND THE PERCENT OF A NUMBER

What is the 5.9% of 45?

5.9% ⇒ .059 x 45 = 2.655


*HOW TO FIND MARKUPS/ SALES TAX/ TIP

STEP 1: Locate the decimal point in the percent. If there is no decimal point add one to the end of the number

          Ex: 34% ⇒ 34.% or 345% ⇒ 345.%

STEP 2: Change the percent into a decimal by moving the decimal point two places to the left. If necessary use zeros as placeholders. 

         EX: 34.% ⇒.34 or 345.% ⇒ 3.45

STEP 3: Multiply the decimal by the original price

STEP 4: If it is a markup, sales tax, or tip, ADD the original price and the product(answer) from step 3. If it is a markdown or discount, SUBTRACT the original price from the product(answer) from step 3.

*EXAMPLE OF HOW TO FIND THE MARKUP OF  NUMBER

A shoe store uses a 40% markup on the original price of items. Find the cost of a pair of shoes that originally cost $63.

Step 1: 40% = 40.% (I added a decimal point)

Step 2: 0.40 (I moved the decimal point over two places to the left)

Step 3: 0.40 x 63 = $25.20 (I multiplied to figure out how much the shoes will be marked up)

Step 4: $63 + $25.20 = $88.20 (I added the original price and the amount of the markup to find the retail price of the shoes or the selling price)

 

*EXAMPLE OF HOW TO FIND THE MARKDOWN OF  NUMBER

A shoe store is discounting its shoes 33% off for its holiday sale.  Find the cost of a pair of shoes that originally cost $63.

Step 1: 33% = 33.% (I added a decimal point)

Step 2: 0.33 (I moved the decimal point over two places to the left)

Step 3: 0.33 x 63 = $20.79 (I multiplied to figure out how much money the shoes will be discounted)

Step 4: $63 - $20.79 = 42.21  (I subtracted the original price and the amount of the discount to find the retail price of the shoes or the selling price)

 

 

 

Joyce Greene

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