ALGEBRA II SUMMER SKILLS PACK WELCOME TO ALGEBRA II!

Transcription

1 ALGEBRA II SUMMER SKILLS PACK WELCOME TO ALGEBRA II! Let's fine-tune those algebra skills! This packet contains procedures, sample problems, and solutions on fundamental concepts that should be mastered prior to taking Algebra II. Please work through the problems in the packet. Similar to "Summer Reading", students in Algebra II will be tested on this material shortly after school begins in September. STUDENTS PLEASE KEEP THIS PACKET AS IT WILL BE PART OF YOUR NOTES AND HOMEWORK FOR THE ALGEBRA II CLASS. To be successful while working through this packet please do the following. Work through the packet in order. Try to work on the example problems by following the procedures outlined. Check your work in the solutions after each section. Complete the practice problems at the end of the packet and use the example problems and procedure as a guideline. The practice problems will be collected and graded in September.

2 Topic 1: Real Number System & Vocabulary Numerical Expression: a symbol or group of symbols used to represent a number. Ex. 3x4, , 17 Value: the number represented by the expression. Ex. Twelve is the value of 3 x 4. Equation: a sentence formed by placing an equals sign " = " between two numerical expressions. Ex. 3x4 = 12, = 1x9 Inequality symbol: <, >, *, <, > Inequality: a sentence formed by placing an inequality symbol between two expressions, called the sides of the inequality. Sum: the result of adding numbers, called the terms of the sum. Difference: the result of subtracting one number from another. Product: the result of multiplying numbers, called the factors of the product. ^3 ^ _ Quotient Quotient: the result of dividing one number by another. Power: product of equal factors. Base: the repeated factor. Power or Divisor - * 4^692 «- Dividend Exponent: the number of times the base occurs as a factor. Grouping symbols: parentheses ( ), brackets [], or a bar _ used to enclose part of an expression. Variable: a symbol, usually a letter, used to represent any member of a given set, called the domain, of the variable. Algebraic Expression: a numerical expression, variable, sum, different, product, or quotient that contains one of more variables. Ex. A + 2 = B The Real Number System: The Real Number System consists of these sets of numbers: Natural Numbers: (1, 2, 3,...} Whole Numbers: (0, 1, 2, 3,...} Integers: (-3, -2, -1, 0, 1, 2, 3,...} Rational numbers: any number that can be written in the form, where b cannot equal zero b the set of rational numbers includes: fractions, decimals, repeating decimals, integers, whole numbers, and natural numbers Irrational Numbers: the set of numbers that are not rational, numbers such as -v/2, \/3, it

3 Topic 2: Simplifying and Evaluating Algebraic Expressions To simplify an algebraic expression use The Order of Operations. The Order of Operations: PEMDAS!!! 1. Simplify what is inside of grouping symbols 2. Simplify exponents 3. Simplify multiplication or division in order from left to right 4. Simplify addition or subtraction in order from left to right Left to Right Left to Right Parentheses Exponents /-Multiplication Division Addition *- Subtraction Examples: Simplify: 2. _i2-7-ll -[3-(-9)] = * -\Z-4-VZ. -(7-9)(7

4 To evaluate an algebraic expression: 1. Substitute the given value of each variable into the expression 2. Simplify using The Order of Operations Examples: Evaluate: 1. Evaluate ify =-4: 3y 3 +2y 2 -y + l 2. Evaluate if s = -8: 3. Evaluate if a = 5, b = 3, and c= 15 : + a-b a

5 Topic 3: Solving Equations & Inequalities in one variable, and corresponding applications, Solving Literal Equations To solve an equation in one variable: 1. Simplify each side a. Distribute (eliminate grouping symbols) b. Combine like terms 2. If the equation has variables on both sides, move the variables to one side 3. Undo any addition or subtraction to the variable term 4. Undo any multiplication or division to the variable itself * Equations in one variable may have one solution, no solution, infinitely many solutions (called the Identity) Examples: Solve for the variable. 1. 5(3 Z -7)-4(2 Z + 7) 2. -a + 3 = 2a-ll / jr i = -70

6 Solving Application Problems that involve simple equations: Example 1: A theater has 600 tickets to sell for a show. Of these, 225 tickets sell for $2 a piece more than the others. If all tickets are sold and $2250 is taken for the show, what price is each type of ticket? Solution and work: * f.5 * /?/i'<^ -o^v-uwv- - -<4rjW 3"75~X^'" ''' "- i ^ ^ 6 *C~ Example 2: At 1:30 two planes leave Chicago, one flying east at 540 km/h and the other flying west at 620 km/h. At what time will they be 1450 km apart? Solution and work: T f ~^ x x = ^ ^s'voy -azox /- ZS~^ou^> s / Uoi/r (5~*s\fs^> X- /- ' 2:^5r~ -^ Example 3: At noon a cargo plane leaves McHare Airport and heads east at 180 mi/h. At 1:00 p.m., a jet takes off from McHare airport and flies east after the cargo plane at 450 mi/h. At what time will the jet overtake the cargo plane? Solution and work: X») / %Voor ' llvo^r 4 O

7 Solving Literal Equations for a specified variable: A Literal Equation is a formula. To solve a Literal Equation for a specified variable, use the procedure for solving equations, to isolate the specified variable. Examples: Solve for the indicated variable. 1. 2x-5^ = 10; x 2. 2-

8 Topic 4: Solving & Graphing Inequalities in One Variable To Solve an inequality in one variable, follow the same procedure as solving equations in one variable EXCEPT: WHEN YOU MULTIPLY OR DIVIDE BOTH SIDES OF THE INEQUALITY BY A NEGAITVE NUMBER, YOU MUST CHANGE THE ORDER OF THE INEQUALITY (FLIP THE INEQUALITY SYMBOL) Graph the solution on a number line. If the inequality contains a < or > symbol, do not color in the solution "dot". If the inequality contains a < or > color in the solution "dot". Shade the appropriate region (left or right). Examples: Solve & graph the solution: x *<2(l-x) ^y. - ^\2 x^ 2 2C,

9 Topic 5: Solving and Graphing Conjunctions and Disjunctions: Conjunction: is an inequality resulting in an "And" statement Solutions to a conjunction must satisfy both inequalities These Problems Look Like: 4 2x+3>l and 5x-9^6 To solve a conjunction: Break the problem into 2 inequalities (if not done already). Solve each inequality and graph each inequality on the same number line. Remember "open dot" or "closed dot", and remember to shade the appropriate region (left or right). 90% of the time the graph of a conjunction will have overlap. The overlap in the shading is the solution. Looks like: Disjunction: is an inequality resulting in an "Or" statement Solutions to a disjunction satisfy at least one of the inequalities The word "Or" will always be in the problem These Problems Look Like: 2.x+ 3 >1 or 5x-9<6 To solve a disjunction: Solve each inequality and graph each inequality on the same number line. Remember "open dot" or "closed dot", and remember to shade the appropriate region. 90% of the time the graph of a disjunction looks like 2 arrows pointing in opposite directions. The shading in opposite directions is the solution. Looks like:

10 Examples: Solve and graph each conjunction or disjunction <-;c + 3<4 2. c + l<4 or l-c<l or

11 Topic 6: Solving Application problems that involve inequalities Translations to know: x > a: x is at least a x is no less than a x < b : x is at most b x is no greater than b a<x<b : x is between a and b a<x<b : x is between a and b inclusive Example 1: Bryce has bowling scores of 240,180, 220, and 200. Find the score that she must get to maintain an average of at least 220. Solution and work: l-t^r X r g^~ "Z'M.O -^ V.t?c3 Ir'ZSf-s-^ t^h2!l ' ^ --) J - ~~ ~~~~^. > ^- <--O J /Wr >- \\OO 2(00 Example 2: The two congruent sides of an isosceles triangle are each 7 cm less than three times the base. If the perimeter of the triangle is to be at most 63 cm, what is the maximum length of the base? Solution and work:

12 Topic 7: Solving Equations that contain Absolute Value To solve an equation that contains Absolute Value: 1. Isolate the absolute value Get the equation in this format: \x\ = a 2. Write 2 equations NOTICE ABSOLUTE VALUE IS BY ITSELF x = a x = -a The 1 st equation just removes the absolute value The 2 nd equation, the absolute value is removed and the sign of "a" is switched 3. Solve both equations *This produces 2 answers Examples: Solve. 1. 2/-3 +2 = -2 ' y T'T

13 Topic 8: Solving and Graphing Inequalities that contain Absolute Value: To solve an inequality that contains Absolute Value: 1. Isolate the absolute value Get the inequality in this format: <a <a >a > a NOTICE ABSOLUTE VALUE IS BY ITSELF 2. Write 2 inequalities, example: x<a x>-a The 1 st inequality just removes the absolute value The 2 nd inequality, the absolute value is removed, the sign of "a" is switchedalso, the inequality symbol is flipped 3. Solve both inequalities This produces either a conjunction (AND problem), or a Disjunction (OR Problem) 4. Graph the solution on a number line. Remember "open dot" or "closed dot", and remember to shade the appropriate region. Examples: Solve and graph the solution set. "T -i>t> >> 2. 3-k\<2 3. "*> -5 JO" -3

14 Topic 9: Fundamental Concepts of Linear Equations Open Sentence in 2 variables: an equation or inequality that has one or more variables x+2y=5 Look Like: y = -3x - 7 x+2y>4 The Solution Set of an Open Sentence in 2 variables is an ordered pair (x, y). The x values are called the domain. The y values are called the range. Basic Problems: ALL OF THESE PROBLEMS INVOLVE SIMPLE SUBSTITUTION Solve an equation for a given domain: Example 1: Solve the equation if the domain of x is {0, 1,2} 3x-2y=7 Procedure: 1. Substitute value for x into equation ' 7 " ^U"" ' 2. Solve equation for y. Repeat for all values in given domain. c 3 3' ~ -2- i " > ~' 2 ~ /I - "^ 0 " ^- (j 0 Complete each ordered pair to form a solution: x.. _ \. ^ * VcT^-i?. Example 2: 2x + 3y=12 given: ( 0, _ ), (_, 0) Procedure: 1. Substitute given x or y value into equation Solve for missing variable "Find the value of k" problems: Example 3: Find the value of k so that the ordered pair satisfies the equation. 3x-y = k; (-1,3)? l 3 r/v Procedure: 1. Substitute ordered pair into equation 3v~-\ -, / \ 3t~0 ~ 2. Solve fork '3>'

15 Equations of Lines: There are several forms of equations of lines (linear equations). Standard Form Slope-Intercept Form NOTICE BOTH FORMS HAVE 2 VARIABLES X and Y! The Graph of a Linear Equation is a straight line! All Linear Equations are Linear Functions-EXCEPT the equations of vertical lines. There are infinitely many points on a line, and to graph a line you need points. There are several ways to graph an equation of a line: Graph the line by making a table of values to obtain points, then plot the points, connect points: You pick values for x, substitute x values into equation and find y-values, repeat (Gets you points), plot points, connect Graph the line by finding the x and y-intercepts: x-intercept: is the point on the x-axis where the line crosses the x axis since it is a point, it has the format (x, y ). They part of the point will always be 0. To find the x-intercept: set y = to 0 in your equation and solve for x. This allows you to create the point: (x, 0) y-intercept: is the point on the y-axis where the line crosses the y-axis since it is a point, it has the format (x, y). The x part of the point will always be 0. To find the y-intercept: set x= to 0 in your equation and solve for y. This allows you to create the point: (0, y) Graph the line by plotting the y-intercept, then count off the slope to obtain another point: Slope of a Line: the slope of a line is the measure of the steepness of the line, or the measure of how slanted the line is. Slope is represented by the variable m. Recall: Slope-Intercept Form of an Equation of a Line: y = mx+b b is the y-intercept so you immediately have 1 point on the line: (0, b)

16 Plot that point, and count using the slope: up/down, then right Formula to calculate slope of a line given 2 points on the line: More about slopes of lines: Lines that travel uphill from left to right always have a positive slope. Lines that travel downhill from left to right always have a negative slope. Graphing Linear Equations: Special Cases The line might be a special case-these require some memorization on your part to graph. The slopes of these lines should also be memorized. Vertical Line: Has the form: x = a, where a is any real number. Example: x = -2 The slope of any vertical line is: undefined OR Does Not Exist *Vertical Lines ARE NOT LINEAR FUNCTIONS Horizontal Line: Has the form y = b, where b is any real number. Example: y = 4 The slope of any horizontal line is 0. Horizontal Lines are Linear Functions

17 Examples: Graphing linear equations: 1. Graph the line by making a table of values: 2x - 3y = -9? o \ 1 3 7/ Graph the line by finding the x and y-intercepts: 3x-4y=12 X - 3. Graph the line y = -2x - 4

18 4. Graph the line x = 4 and identify the slope. 5. Graph the line y = -1 and identify the slope. Is the line a function? Is the line a function? /K VVA a V Calculate the slope of each line that contains the given points. 6. (-2,3 V )1["4,-1^ 7. (-5,2) (8,2) ^f\z ^^-'^r^7 >^-*v M" 2 - l/v\~ jv\ ~Ji~ ~~&. z a /3 8. (5,-4) (-4,5) 9. (2,4) (2,7) M-

19 Finding the slope of a line, given an equation: It is easy to identify the slope of a line when the equation is written in slope-intercept form. You can also find the slope of a line that is written in Standard Form. Example: Find the slope of the line 5x-2y=20. Procedure: 1. Solve the equation for y, specifically re-writing the equation in slope-intercept form 2. Identify the slope. * The slope of the equation is the coefficient of the x-term, when the equation is in slope-intercept form. Examples: Find the slope of the line. 1. x + y = 7 6. y = 3 -x a^-x o 2. 3x-3y = x-2y=l 3. = x = -2

20 Writing Equations of Lines You will be asked to write an equation of a line in one of these formats: That means your final answer must be written in one of these ways: Slope-intercept form Standard Form Now, to get your final answer (equation of a line), you may be need to use this formula: The Point-Slope Formula: y ~ J\ = m ( x ~ *i Examples: 1. Find the equation of a line in both slope-intercept form and Standard Form of the line having, and y-interce^t 1. -T- K x, y, 2. Find the equation of a line in Standard Form of the line that contains the point (4, -3) and has

21 Key Facts about Parallel & Perpendicular Lines: Two Lines are Parallel if they have the same slope. What can you do with this fact? YOU CAN SOLVE A PROBLEM LIKE THIS: Example 1: Find the equation of a line in Standard form that passes through the point (0, -2) and is parallel to the line y = x - 3. Example 2: Find the equation of a line in Standard form that passes through the point (0,1) and is parallel to the line 3x+2y=l.

22 The slopes of Perpendicular lines are opposite reciprocals. This means that if the slope of a given line =4, the slope of a line perpendicular to the given line would have slope of: -V-* ' ' What can you do with this fact? YOU CAN SOLVE A PROBLEM LIKE THIS: *,«$, Example 3: Find the equation of a line in Standard form that passes through the point (1, 2) and is perpendicular to the line x + y ' = 4. d K \-i-~ x -/ ) -'Z f^u- X^/ x^f -x ->> x-n"-'^

23 Topic 10: Systems of Linear Equations and Applications Systems of linear equations are used to find the point of intersection of two lines. Systems of linear equations can be solved by 1) graphing, 2) the substitution method, 3) the elimination method. Example 1) The Graphing Method x + 2y = 7 3x- 2y = 5 By graphing the two lines you can find where their point of intersection is. (3,2). Example 2) The Substitution Method a) x = y + 3 2x-y = 5 b) 5x + 2y = 9 x + y = -3 Example 3) The Elimination Method a) -3x - 5/= -7-3C b) 8x-5y=ll - 3y ="V 0 Example 4] Applications X,_* a) A library is having a book sale to raise money. Hardcover books cost $4 each and paperback books cost $2 each. A person spends $26 for 8 books. How many hardcover books did she purchase? x b) You pay $24.50 for 10 gallons of gasoline and 1 quart of oil at a gas station. Your friend pays $22 for 8 gallons of gas and 2 quarts of the same oil. Find the cost of 1 quart of oil and 1 gallon of gas. *- c~\ vj y "if ^ ^in = 2- V2x---

24 Topic 11: Graphing Linear Inequalities A linear inequality in two variables, such as x - 3y < 6, is the result of replacing the = sign in a linear equation with <, <, >, or >. A solution of an inequality in two variables x and y is an ordered pair (x,y) that produces a true statement when the values of x and y are substituted into the inequality. The graph of an inequality in two variables would be the shaded region above or below the line that represents where all of the solutions lie. Example 1) Tell whether the ordered pair is a solution of -x + 2y< 8. a) [0,0) y b) MX_ c) (3,5)/ ci J/N X 4-H \/e^\v<? 1 s Example 2) Graphing a Linear Inequality in Two Variables in Slope-Intercept Form Procedure: 1. Graph the boundary line using a dashed line for < or > or a solid line for < or >. 2. Shade above the line for > or > and below the line for < or <. Graph the inequality y > 4x - 3 : * Notice how the graph goes through the point (0,-3) which is the y-intercept in y = mx + b form. Also notice that the slope of this line is 4. 4 Example 3) Graphing a Linear Inequality in Two Variables in Standard Form 3x + 5y < 8 ' ' '" x >^ ' *, 1 2 '\ 4 ;. - j, ;TS S^."" -, -4

25 Topic 12: Graphing Systems of Linear Inequalities The solution for a linear inequality is a shaded half-plane like the examples above. The solution to a system of linear inequalities is also a shaded region but it is where both shaded regions overlap each other. The only way to do this is graphing by hand or using a graphing calculator. Example 1) Graph the system of inequalities. y<5x+ 1 y >x- 2 *Notice that the first line in red is solid and shaded below the red line. Also notice that the blue line is dashed and shaded above. The solution region would be darker area primarily in the first quadrant. This would be where both shaded regions overlap each other. Example 2) Graph the system of inequalities. y<x-3 2x + >-l