# Algebra 1 S1 Lesson Summaries. Lesson Goal: Mastery 70% or higher

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1 Algebra 1 S1 Lesson Summaries For every lesson, you need to: Read through the LESSON REVIEW which is located below or on the last page of the lesson and 3-hole punch into your MATH BINDER. Read and work through every page in the LESSON. Try each PRACTICE problem and write down the examples on the back of your lesson summary with the worked out solutions or on loose leaf paper to put in your MATH BINDER. Read the directions carefully for the assignment and submit the lesson ASSESSMENT. If you need to RESUBMIT (you see a BLUE ARROW) an assignment, make sure to view the feedback on the assignment by clicking on the assignment in your GRADES before resubmitting. Lesson Goal: Mastery 70% or higher Review 1. Types of Numbers Natural Numbers: All positive whole numbers, not including zero. Whole Numbers: All positive whole numbers including zero. This category does not include any negative whole numbers. Integers: Positive and negative whole numbers including zero. This category does not include any fractions or decimals. Rational: A number that can be expressed as a ratio (fraction) of two numbers. This includes decimals that terminate or repeat. Irrational: A number that cannot be expressed as a ratio (fraction) of two numbers. This includes decimals that never stop or repeat. Real Numbers: A number that is rational or irrational. The Real Number set includes all of the number groups listed above. 2. Addition and Subtraction of Integers Same signs, find the sum, keep the sign. Examples: 4+5=9, 4+ 5= 9 Different signs, find the difference, keep the sign of the larger number. Examples: 6+2= 4,6 2=4 Seeing Double: Circle the mess, clean it up. = = 7 = = 1 Opposite Signs: Circle the mess, clean it up. = 3 5 = 2 = 5 3 = 2

2 3. Multiplication and Division of Integers Same signs, positive answer. Examples: (3)(4)=12,( 3)( 4)=12, Different signs, negative answer. Examples: ( 3)(4)= 12, 4. The acronym PEMDAS was shown in the lesson. Each of the letters represents an operation in the correct order it must be performed. Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right). Example: (2+4) Add the in the parentheses first =(6) Simplify the term with the exponent next = Simplify the multiplication and division from left to right. = Addition and subtraction from left to right. Subtract first =24+4 Add = Review When combining like terms, it is important to remember the following: Variable parts must be identical. 2x and 3x are like terms. 2x and 3y are not like terms. 2x plus 3x is the same as saying two x s plus three more x s. 2x + 3x = 5x When using exponential expressions, it is important to remember the following: Coefficients are the numbers that come before the variable telling you how many times the variable has been added. 3x Factors are values you can multiply together to get a product. 1, 2, 3, and 6 are factors of 6. 1, 2, 2x, x, x 2, and 2x 2 are factors of 2x 2. The exponent is the small number to the right of the value and tells you how many times the value must be multiplied by itself.

3 7 2 = (7)(7) y 4 = (y)(y)(y)(y) If you do not see a variable or an exponent with a base number, the variable and the exponent are both understood to be 1. 3xy = 3(x) 1 (y) 1 EVALUATING EXPRESSIONS You can substitute values in for the variable and then follow the order of operations to simplify the expression. Find 2xy + 3y if x = 6 and y = 2. 2(6)( 2) + 3( 2) = 24 + ( 6) = 24 6 = Review Selecting Units When selecting units, be mindful of the situation you are modeling. Make sure the unit is not too large or too small to represent the measurement. Units in Formulas Use units that appropriately model the formula. Use units that match with the quantities measured by the formulas. Remember that units follow algebraic operations. (1 ft 1 ft = 1 ft 2 ) Division of units result in the use of per in the resulting unit rate. Miles divided by hours results in miles per hour. Dollars divided by gallons results in dollars per gallon. Converting Units Use the conversion as a ratio to assist in converting units. Any unit present in the numerator and the denominator can be canceled out. Conversion Table for Measurement Units Distance 1 inch = 2.54 centimeters 1 mile = 1,760 yards 1 mile = kilometers Mass / Weight 1 pound = 16 ounces 1 meter = inches 1 kilometer = 0.62 mile 1 pound = kilograms 1 mile = 5,280 feet 1 kilogram = 2.2 pounds

4 1 ton = 2,000 pounds 1 pint = 2 cups 1 gallon = liters Volume 1 cup = 8 fluid ounces 1 quart = 2 pints 1 gallon = 4 quarts 1 gallon = 128 fluid ounces 1 liter = gallons Scales of Graphs A graph is most useful when the units on the x-axis and the y-axis have the proper scales. The scale includes the low and high values and the increments (or steps) for each axis. The origin is the point (0, 0) on a graph. Use the scales to help interpret the axes and what the points represent Review Precision versus Accuracy High precision means that the data is grouped close together. High accuracy means that the data is near the expected value. Example: If 200 soda bottles are tested for their sugar contents, and they vary on all sides of the listed amount but are around to the listed amount, then they would have high accuracy but low precision. Rounding Check the significant figures in the measurements given to choose the appropriate accuracy in answers. Units of Measure When using descriptive modeling, make sure that the units of measure accommodate the situation. If a car is traveling across the United States, you would not want to measure distance in centimeters, but a larger distance measure such as miles would fit. Rates of Measure Rates of measure involves two units of measure and typically the word per. One unit will be the dependent variable and will be graphed on the y-axis. The other unit will be the independent variable and is graphed on the x-axis. This is typically the unit after the word per Review Addition Term Subtraction Terms sum of difference of plus minus more than less than increased by decreased by total subtracted from

5 Multiplication Terms product of times percent of per (5 ft per second means 5 feet times the number of seconds.) twice (indicates multiplying by 2) doubles (indicates multiplying by 2) triples (indicates multiplying by 3) Division Terms quotient of ratio of half of (indicates dividing by 2) third of (indicates dividing by 3) The terms sum and difference are special terms. In a situation where you need addition or subtraction to be done first in an expression, which is different from the order of operations, you use the special terms sum or difference. Step 1: Highlight Key Words Step 2: Talk It Out The word twice means to multiply by 2. The difference of n and 6 means to subtract 6 from n. Because the word difference indicates the answer to a subtraction problem, you must put the subtraction in parentheses so it is done first. Step 3: Translate 2(n 6) Review Algebraic Properties Commutative Property states that the order in which you perform an operation does not affect the outcome. This property works for Addition and Multiplication: = = 3 2 Associative Property states that grouping symbols does not affect the outcome. This property works for Addition and Multiplication: 2 + (3 + 4) = (2 + 3) (5 4) = (6 5) 4

6 Distributive Property states that any number multiplied to a sum or difference of two or more numbers is equal to the sum or difference of the products. 3(5 + 2) = 3(5) + 3(2) = = 21 2(x 8) = 2(x) 2(8) = 2x -16 Properties of Equality Reflexive Property says that anything is equal to itself. 2 = 2 Symmetric Property says that the order on either side of the equal sign does not matter. a = 6 is the same as 6 = a Transitive Property says for any real numbers a, b, and c if a = b, and b = c, then a = c = 7 and 7 = then = Addition Property says if a = b, then a + c = b + c. Subtraction Property says if a = b, then a c = b c. Multiplication Property says if a = b, then a c = b c. Division Property says if a = b, then a/c = b/c. Substitution Property of says if a = b, then b can replace a in any expression without changing the value of the expression. Steps to Solving an Equation 1. Use the distributive property to remove parenthesis if they exist in the equation. 2. Combine like terms on both sides of the equation 3. Read the equation and decide what operations (add, subtract, multiply, divide) are being applied to the variable. 4. Use the algebraic properties to undo each of these operations one at a time. 5. You are finished when the variable is isolated on one side of the equation by itself. 6. Check your solution by substituting it into the original equation and testing to see whether it gives you a true equation. Other Helpful Tips When dividing or multiplying by negatives, be careful with your signs. Remember, two negatives multiplied or divided make a positive answer. Multiplying or dividing a negative and a positive will make a negative answer. ( 2)( 3) = +6, (2)( 3) = 6 When dealing with a negative in front of a fraction, write the negative with the numerator and leave the denominator positive. If you were to write negatives in both the numerator and the denominator, you would create a positive value. The number ( ) is the same as 1/2. The number is the same as positive Work with the fraction to clear the denominator. Multiply all terms on both sides by the denominator.

7 02.01 Review The steps for solving an equation are below: Step 1: Simplify each side of the equation. Step 2: Get the variable on one side of the equation. Step 3: Get the variable by itself. Step 4: Check your solution. Special Cases If you solve the equation and the variables are eliminated, then: There is no solution if your final statement in the equation is false. Example: 2 = 5 The solution is all real numbers if your final statement in the equation is true. Example: 4 = 4 When working with fractions, multiply both sides of the equation by the Least Common Denominator (LCD) to eliminate the fractions first Review When solving application problems, it is important to fully understand the situation. Use the five-step problem solving process: 1. Read and understand the situation within. 2. Identify and pull out important information from the problem. 3. Assign variables to unknown values. 4. Set up and solve the equation. 5. Check that your answer makes sense within the context of the problem. Consecutive Integers First Integer x Second Integer x + 1 Third Integer x + 2 Consecutive Even/Odd Integers First Integer x Second Integer x + 2 Third Integer x + 4

8 02.03 Review When solving absolute value equations there are a few simple steps to remember. Isolate the absolute value first if it isn't already on one side by itself. Example: When the absolute value is set equal to a positive number, write the expression inside the absolute value bars equal to the positive and negative of the given number. x 2 = 6 x 2 = 6 x 2 = 6 Solve the two equations to find the two solutions to the absolute value equation. Check your solutions When an absolute value is equal to a negative number, the equation has no solutions. x 2 = 6 There is no solution because absolute value is the distance from zero on a number line and you cannot have a negative distance. When an absolute value is equal to zero, there is one solution. This equation cannot be split in two because zero is neither positive nor negative. Just drop the absolute value bars and solve for x. Solution: x = 5

9 02.05 Review Solve inequalities for the variable the same way you solve equations. When multiplying or dividing both sides of an inequality by a negative number, the inequality symbol must be flipped! Always read an inequality starting with the variable When graphing an inequality: Open circle for < and > Closed circle for and Shading to the right for greater than Shading to the left for less than Review Conjunctions are two inequalities connected by the word "and." Solutions to conjunctions have to fit the conditions of BOTH inequalities. Conjunctions may be written separately like x > 3 and x < 3. Conjunctions may be written together like 3 < x < 3. If the two inequalities have no intersection in their solutions, then there are "no solutions." Disjunctions are two inequalities connected by the word "or." Solutions to disjunctions can fit the conditions of EITHER of the inequalities. Disjunctions should be written separately like x < 3 or x > 3. If the solutions to a disjunction cover the entire number line, the solutions are "all real numbers." If the absolute value inequality contains a less than symbol (, <): 1. Write two separate inequalities to solve. A. Drop the absolute value bars. B. Create a second inequality with the flipped inequality symbol, opposite value, and the word "and" in between. 2. Solve the two inequalities. 3. Draw the graph between two values. Example: x + 1 < 2 x +1 > 2 and x +1 < x > 3 and x < 1

10 If the absolute value inequality contains a greater than symbol (, >): 1. Write two separate inequalities to solve. a. Drop the absolute value bars. b. Create a second inequality with the flipped inequality symbol, opposite value, and the word "or" in between. 2. Solve the two inequalities. 3. Draw the graph with two parts in opposite directions Example: x 5 1 x 5 1 or x x 4 or x Review Solving literal equations and inequalities is exactly the same as solving a standard linear equation or inequality. Determine which variable needs to be isolated. Highlight the variable and determine what operations are being applied to that variable. Reverse the order of operations to isolate the variable. When multiplying or dividing by a negative number, be sure to change the direction of the inequality symbol.

11 03.01 Review A relation describes a relationship and pairs input values with output values. Relations can be represented as ordered pairs, graphed on a coordinate plane, listed in an x/y table of values, or as a mapping. The domain of a relation is simply the input, or x-values of the relation. The range of a relation is simply the possible output, or y-values of the relation. Some relations have a special relationship in that each input value is paired with exactly one output value. When this happens the relation is called a function. You can test whether a relation is a function by comparing the x-values. The relation is not a function if any of the x-values repeat. You can test whether the graph of a relation is a function using the vertical line test. If the graph crosses any vertical line at more than one point, the relation is not a function Review Function notation simply sets an expression equal to f(x), read "f of x", which means "f is a function of x". You may use any letter to represent a function: f(x), g(x), h(x), r(x), etc. When given an equation in function notation (for example: f(x) = x + 1) you may be asked to do one of the following: Evaluate the function for a given x value substitute the value for x and simplify. Solve for x given a function value substitute the function value and solve for x. When given a table or a graph you may be asked to do one of the following: Given an output value, identify the input value that corresponds. Given an input value, identify the output value that corresponds. With any function, you can write inputs and outputs as ordered pairs: (input, output) Review Slope "rise over run calculated using two points and the formula =

12 Slope is the ratio of the vertical change to the horizontal change between two points. Given the line graphed on a coordinate plane, you find any two points on the line and count the rise (up or down) and the run (side to side) between the two points. Start with the point on the left. If you have to go up your rise is positive. If you have to go down your rise is negative. Since you always go to the right, your run will always be positive. To graph a line using its x- and y- intercepts: 1. Find the x-intercept: replace y with the number zero and solve for x. 2. Find the y-intercept: replace x with the number zero and solve for y. 3. Plot both intercepts on the coordinate plane, and then connect them to draw the graph. Remember that the x-intercept can be written as an ordered pair where y is zero. The y-intercept can be written as an ordered pair where x is zero. (2, 0) x-intercept (0, -3) y-intercept To graph a line using the slope-intercept form: 1. Manipulate the equation into slope-intercept form, y = mx + b by getting y alone. 2. Identify and plot the y-intercept of the line. Remember, the y-intercept is b. Don't forget: The sign goes with the number. 3. Identify and use the slope of the line to find a second point. Remember, the slope is m. Don't forget: Starting at the y-intercept, the numerator tells you the rise (count up if positive and down if negative), the denominator tells you the run (count right.) 4. Draw a straight line through the two points to complete the graph. Linear Functions: A linear function is a function that is defined by a linear equation. To write a linear function with function notation, first write it in slope-intercept form, and then replace y with f(x). The graph of a linear function is all points (x, f(x)), where x is in the domain of the function Review Real-world situations can be analyzed with linear models. Tables, equations, and graphs show the complete picture. Describe a linear function with its key features: What are the variables? What are the x- and y-intercepts? Is the function increasing, decreasing, or constant? What is the rate of change? What are the domain and range of the function?

13 The formula to find the average rate of change for any function, f(x), for any interval from a to b is: Average rate of change = The average rate of change in a linear function is the same as its slope. If the rate of change in a function is not constant, the function is not linear. Increasing the value of the y-intercept causes a vertical shift of a line up the y-axis; decreasing its value causes a vertical shift down. A function f(x) is shifted up or down the y-axis and becomes a new function g(x) depending on the value of k if g(x) = f(x) + k Review Point-slope form: y y1 = m(x x1) where (x1, y1) is any point on the line and m is the slope of the line. You can rearrange an equation from: point-slope form : y y1 = m(x x1) to slope-intercept form y = mx + b to standard form ax + by = c (where a is a positive integer) You can write any linear equation in function notation by replacing y with f(x). You can find the slope of a line if you know any two points on the line: = You can graph a line from point-slope form by first changing it to slope-intercept form and then graphing it. To solve problems that have defined variables, such as a real-world problem, you need to analyze which variable is independent and which is dependent. Then, choose point-slope or slope-intercept form based on the information that is given. To write the equation of a line parallel to a given line through a given point: 1. Find the slope of the given line. 2. Use that slope and the point on the new line to write the equation of the new line. y y 1= m(x x 1)

14 To write the equation of a line perpendicular to a given line through a given point: 1. Find the slope of the given line. 2. Use the opposite reciprocal of that slope and the point on the new line to write the equation of the new line. y y 1= m(x x 1) Review To write the equation of a horizontal or vertical line through a given point: Line Type Slope of the LIne Equation Form Horizontal Zero m = 0 y = number Vertical Undefined x = number Example: Given the point (4, 8): 1. The horizontal line through this point is y = 8 and has a slope of The vertical line through this point is x = 4 and has an undefined slope. Relationships of horizontal and vertical lines to other horizontal and vertical lines: Line Type Relationship to Horizontal Lines Relationship to Vertical Lines Horizontal Parallel Perpendicular Vertical Perpendicular Parallel Examples: 1. The lines y = 7 and y = 3 are parallel because they are both horizontal lines. 2. The lines y = 7 and x = 4 are perpendicular because the first line is horizontal and the second one is vertical.

15 04.01 Review Properties of Exponents Name of Property Example Explanation Zero Exponent Property X 0 = 1 (x 0) Any number (except 0) with an exponent of 0 equals 1. Negative Exponent Property x n = 1 x n Any number raised to a negative power is equivalent to the reciprocal of the positive exp the number. (x 0) Product of Powers Property Quotient of Powers Property Power of a Product Property x n x m = x n+m (x 0) x n x m = xn m (x 0) (xy) n =x n y n (x and y 0) To multiply two powers with the same base, add the exponents. To divide two powers with the same base, subtract the exponents. To find a power of a product, find the power of each factor, and then multiply. Power of a Quotient Property ( ) = To find a power of a quotient, find the power of each part of the quotient, and then divid canceling common factors. x and y 0 Power of a Power Property Rational Exponent Property (x a ) b = x a b (x 0) = To find a power of a power, multiply the exponents. Fractional powers, where a number is raised to a fraction, can be converted to a radical. numerator becomes the exponent, and the denominator becomes the index of the radica Unequal Bases = When working with unequal bases, it may be necessary to rewrite them with common bases. The Product of Powers Property and the Quotient of Powers Property will require the expressions to have the same base.

17 The domain for mathematical applications is usually all real numbers. However, limitations could exist depending on the scenario the question is in. The range will have a limit as standard exponential functions do not cross the x-axis. Similar to the domain, careful attention must be paid to the range in real-world scenarios. The average rate of change is the change in the output of the function over a section of the domain. Similar to SLOPE of a LINEAR EQUATION. Average Rate of Change = f(b) f(a) b a Review Exponential growth is when the graph is increasing from left to right. The base of the exponential expression must be greater than 1. Exponential decay is when the graph is decreasing from left to right. The base of the exponential expression must be less than 1. Comparing Functions In order to compare functions, focus on the key features available. Y-intercept Increasing versus decreasing, growth versus decay Rate of growth Average rate of change Effects on the Graph of a Function A graph will experience a vertical shift when f(x) + k = P(1 + r) x + k A positive value for k will shift the graph up k units. A negative value for k will shift the graph down k units. A graph will experience a horizontal shift when f(x + h) = P(1 + r) x + h A positive value for h will shift the graph to the left h units. A negative value for h will shift the graph to the right h units. A graph will be reflected over the y-axis when f( x) = P(1 + r) x

18 04.06 Review Arithmetic Sequences A list of numbers, called terms, which share a common difference. Example: -5, -7, -9, or 8, 23, 38, The arithmetic recursive formula finds each term based on the previous term. f(n) = f(n 1) + d, where n > 0 The arithmetic explicit formula finds each term based on the first term and number of terms. f(n) = f(1) + d(n - 1), where n > 0 The common difference of an arithmetic sequence is the same as the slope of the corresponding linear function. Geometric Sequences A list of numbers, called terms, which share a common ratio. Example: 1, 4, 16,... or 2, -6, The geometric recursive formula finds each term based on the previous term. f(n) = f(n 1) r, where n > 0 The geometric explicit formula finds each term based on the first term and number of terms. f(n) = f(1) r n-1, where n > Review Parameters of Linear Functions, f(x)=mx+b The x-variable will always represent the aspect that is changing in the situation. The f(x) will always represent the aspect that changes because of the change in the x-variable. The slope (m) will connect to the rate of change for the x-variable. The y-intercept (b) will represent a value that exists when x = 0. Parameters of Exponential Functions, f(x)=p(1+r) x The x-variable will always represent the aspect that is changing in the situation. The f(x) will always represent the aspect that changes because of the change in the x-variable. The P is the principal. This is the starting value, or y-intercept, when x = 0. The r is the rate of change. Remember that if the rate is a percentage, the r will be the decimal equivalent.

19 Exponential and Linear Growth An exponential growth function will always exceed a linear function eventually. As the x-values continue to get larger, the rate of change for the exponential function continues to increase, while the linear function's rate of change is constant. Creating Exponential Functions from Data Points Given two input-output pairs, it is possible to create the exponential function that contains those points. 1. Find the number of times the base is multiplied. 2. Use the properties of exponents to solve for the base. 3. Use either point and the base in the function f(x)=a(b) x to solve for the coefficient Review Solving Systems of Equations The solution to a system of equations (in two variables) is the pair of values that make both equations true. A system can be solved graphically or algebraically (elimination method or substitution method). To solve a system graphically, graph both equations on the same coordinate plane and find the point of intersection. This point is the solution because it is the point that lies on both lines. This ordered pair (x, y) will make both equations true. There are three types of systems of equations: Type of System Classification of System Number of Solutions Intersecting Lines Consistent-independent One the point of intersection The same line Consistent-dependent Infinitely many solutions Parallel lines Inconsistent-independent No solution the lines never intersect Solve inequalities for the variable the same way you solve equations Review A system of equations is a collection of two or more equations. The solution to a system of equations (in two variables) is the pair of values that make both equations true. This pair of values corresponds to the point where the two lines intersect when graphed on a coordinate plane. You have discovered three ways to solve a system of equations: 1. Graphically: graphing both equations and finding the point of intersection

20 2. Substitution Method: isolating one variable in one of the equations and substituting it into the other equation. Isolate one variable of one equation. - Choose an equation and solve for one of the variables. Substitute and solve for one variable. - Substitute the expression for the isolated variable into the other equation. Solve the new equation for the variable. Substitute and solve for the other variable. - Substitute the value from the first variable into one of the original equations and solve. Copy Example: 3. Elimination Method: eliminating one of the variables when combining the two equations. Identify/Create opposite coefficients. Add the equations vertically. Simplify and solve for the first variable.substitute and solve. Substitute the value of the first variable into one of the original equations and solve for the second variable. Copy Example: Equivalent systems are produced when any algebraic property is applied to either or both equations in a system. Equivalent systems have the same solutions as the original equations. There are two special cases of systems of equations: Parallel Lines Graphically these lines do not intersect Same Line Graphically these equations graph the same line When the variable disappears you are left with a... False Equation Final answer = No Solutions True Equation Final answer = Infinitely Many Solutions Review A graph of an equation in two variables or a function is a representation of an infinite number of solutions to the equation or function. A system of equations may not have an exact solution that meets the conditions of a real-world solution.

21 Using graphing technology is a very efficient way to find solutions to equations and systems of equations. The intersection point of two graphed functions is the solution for a system of equations. It is the point that makes both equations true. When two different functions f(x) and g(x) are graphed, the x-coordinates of the points of intersection is the solution to the equation form from f(x) = g(x) Systems of equations may be a combination of linear and non-linear functions. A table of values very rarely shows every possible solution to a system of equations. Finding the approximate solution that is between two values on the table can be a good answer in many situations Review Steps for Graphing Inequalities 1. Rearrange the inequality into slope-intercept form. 2. Plot the y-intercept and use the slope to find a second point. 3. Connect the points with a dashed or solid line. 4. Shade above or below the line. 5. Test the origin (0, 0) to make sure the correct side is shaded Review Steps for Solving Systems of Inequalities by Graphing 1. Graph both inequalities on the same coordinate plane. Remember to determine if the line should be solid or dashed and which side of the line to shade. 2. Emphasize the area where the shading is overlapped. 3. Test an ordered pair from the overlapping shaded region to be sure it makes both inequalities in the system true.

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