This page describes various common concepts presented in Algebra I. See the list of topics below. It is applicable to both Algebra I and Algebra II, but essential for Algebra I. To help you with you knowing the appropriate formulas, you are permitted to use this document on up-coming quizzes and exams or copy the important formulas and use that copy on your quizzes and exams.
- Order of Operation
- Distribution
- Rates
- Percentage Problems
- Solutions to Solution Problems
- Trains Approaching
- Representation of Inequalities
Order of Operation
Order of operation is the order in which you perform mathematical operations., usually called PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction). Some people say Grouping instead of Parentheses, either way will do, just do them in the proper order. Multiplication and division can be done left to right, while addition and subtraction can also be done together in the same order (left to right).
Distribution
When multiplying across one or more parenthetical expressions, the element on the outside of the parentheses must be multiplied by each element in the parentheses. If there are two sets, then each element of each set must be multiplied by each element of the second set. I will only use expressions with two elements in each of the parentheses.
| Problem | Hint |
|---|---|
| Distribution | When a minus/negative sign comes before a parenthetical expression, the negative sign needs to be distributed to all elements of the expression 2x + 4 -(x – 7) = 2x + 4 – x + 7 = x + 11 2x + 4 -(x + 7) = 2x + 4 – x – 7 = x – 3 |
| Squaring | Again, parentheses are important here. If the squaring is outside the parentheses, then the number including the sign must be included in the multiplication -12 = -(1)2 = -1 (-1)2 = (-1)(-1) = 1 |
| Two sets | Use FOIL – First, Outer, Inner, Last (2x-3)(x+2) = (2x)(x) + (2x)(2) + (-3)(x) +(-3)(2) = 2x2+4x-3x-6 |
Rates
Rates are just ratios of two quantities. For example, miles per hour, miles per gallon, tax per dollar spent, or amount of acid per unit of water or total units of liquid. We have used all these in this class. Throught this and the Percentage sections, any rate is represented by the variable r.
Note: Rate and percentage rate are the same calculations, the only difference is that rate is reduced to a number over 1 while percentage rate is reduced to a number over 100.
All rates have the general formula of: answer = p * rate, where p is either principal or price, depending on the problem. Since simple interest rates are per year, time in years has to be included:
General: answer = p * r Simple Interest: answer = p * r * t
Remember that the answer only gives you the change from the original value. For all cases except for discounts, the final total is p + change.
final total = p + p * r final balance for simple interest = p + p * r * t price with discount = p - p * r
| Rate or Ratio | Definition/Formula/Example |
|---|---|
| Miles per gallon | How many miles can you travel on one gallon of gasoline. r mpg = m miles/g gallons r mpg = 420 miles/21 gallons = 20 mpg m miles = r mpg * g gallons g gallons = m miles/r mpg |
| Miles per hour | How many miles can you travel in one hour r mph = m miles/h hours r mph = 434 miles/7 hours = 62 mph |
| Tax Rate | Amount of tax for every dollar spent t tax = p price * r tax rate t tax = $4.50 * 8.1/100 = $0.36 The total cost = p + p * tax rate (adding the cost of the item to the tax) |
| Commission Rate | Amount of money paid to sales person per $100 of sale. c commission = p price * r commission rate profit after commission = p – p * r |
| Discount Rate | Discount of price of item per $100 of sale. d discount = p price * r discount rate profit after discount = p – p * r |
| Mark-up Rate | Mark up of price of item per $100 of sale. m mark-up = p price * r mark-up Rate price after mark-up = p + p * r |
| Interest | Amount of money earned or paid per $100 of principal per year. simple interest is principal * interest rate * years total invested or owed = p + p * r * y |
Percentage Problems
Percentages are just like Rates, with the denominator always being 100.
Note: Rate and percentage rate are the same calculations, the only difference is that rate is reduced to a number over 1 while percentage rate is reduced to a number over 100.
All rates have the general formula of: answer = p * rate, where p is either principal or price, depending on the problem. Since simple interest rates are per year, time in years has to be included:
General: answer = p * r Simple Interest: answer = p * r * t
Remember that the answer only gives you the change from the original value. For all cases except for discounts, the final total is p + change.
final total = p + p * r final balance for simple interest = p + p * r * t price with discount = p - p * r
| Percentage | Definition/Formula/Example |
|---|---|
| Tax % | Amount of tax for every $100 spent t tax = p price * r tax rate t tax = $4.50 * 8.1/100 = $0.36 The total cost = p + p * tax percentage (adding the cost of the item to the tax) |
| Commission % | Amount of money paid to sales person per $100 of sale. c commission = p price * r commission percentage profit after commission = p – p * r |
| Discount % | Discount of price of item per $100 of sale. d discount = p price * r discount percentage profit after discount = p – p * r |
| Mark-up % | Mark up of price of item per $100 of sale. m mark-up = p price * r mark-up percentage price after mark-up = p + p * r |
| Interest | Amount of money earned or paid per $100 of principal per year. simple interest is principal * interest % * years total invested or owed = p + p * r * y |
| Solution % | Amount of solute (like acid) per 100 parts of liquid. s solute = l liquid * r solution percentage |
Solutions to Solution Problems
The formulas for solving solution problems are:
where v1 = volume of solution 1 v2 = volume of solution 2 vt = volume of total solution v1 + v2 = vt r1 = percentage of the solute of solution 1 r2 = percentage of the solute of solution 2 rt = percentage of the solute of total solution v1 * r1 + v2 * r2 = vt * rt
Trains Approaching
Distance traveled for one train is m miles = r (miles / hour) * h hours
r1 = speed of train 1 r2 = speed of train 2 h1 = hours travelled of train 1 h2 = hours travelled of train 2 mt = combined distance of both trains r1 * h1 + r2 * h2 = mt
In most cases, the time is the same for both trains, h1 = h2 and the speed of one train is expressed in terms of the other train. For example, if the distance between the starting points of the two trains is 500 miles, the speed of the second train is half again (1.5) times as fast as the first train, and they take 4 hours to meet, then the formula becomes:
r * 4 hours + 1.5 r * 4 hours = 500 miles 2.5 r * 4 hours = 500 miles 2,5 r = 500 miles/4 hours = 125 miles/hour r = 125 miles/hour/2.5 = 50 miles/hour for the first train 1.5 * 50 miles/hour = 75 miles/hour for the second train.
Representation of Inequalities
So far, the equations we use are equalities, two expressions separated by an equal sign (=). When there is a bounded range of possible answers, inequalities are used. The separator can be one of the following:
| Symbol | Meaning |
|---|---|
| > | greater than |
| < | less than |
| ≥ | greater than or equal to |
| ≤ | less than or equal to |
If you consider the symbol as a fish with its mouth open, the mouth is always facing the larger number. There are several ways to represent this inequality.
Inequality Formula
An inequality formula is two expressions separated by an inequality sign. Examples are:
| # | Inequality line |
|---|---|
| 1. | 3x – 5 > 7 3x > 7 + 5 3x > 12 x > 12/3 x > 4 |
| 2. | 4x + 5 ≤ 9 4x ≤ 9 – 5 4x/4 ≤ 4/4 x ≤ 1 |
| 3. | -5x -2 ≥ 8 -5x ≥ 8 + 2 -5x ≥ 10 -5x/-5 ≤ 10/-5 # Notice the flip of the sign when divided by negative number. x ≤ -2 |
| 4. | -23 < 3x – 5 ≤ 16 -23 + 5 < 3x – 5 + 5 ≤ 16 + 5 -18 < 3x ≤ 21 -18/3 < 3x/3 ≤ 21/3 -6 < x ≤ 7 |
Inequality Graph
1. 4 ∞ (———————> 0———————>
| # | Inequality line |
|---|
Set Builder Notation
Sets are described within curly braces “{ }” with a description of the variable and its limits. The general form is:
| # | Set Builder Notation |
|---|---|
| 1. x > 4 | { x | x > 4 } |
| 2. x ≤ 1 | { x | x < 1 } |
| 3. x ≤ -2 | { x | x ≤ -2 } |
| 4. -6 < x ≤ 7 | { x | -6 < x ≤ 7 } |
Interval Notation
Interval notation uses square brackets and parentheses to delimitate the domain of x. Square brackes “[” include the number, while parentheses do not.
<td(4, &infit;)
| # | Interval Notation |
|---|---|
| 1. x > 4 | (4,∞) |
| 2. x ≤ 1 | (-∞, 1] |
| 3. x ≤ -2 | (-∞, -2] |
| 4. -6 < x ≤ 7 | (-6,7] |