Keystone Exam Concept Review Name: Properties and Order of Operations COMMUTATIVE Property of: Addition ASSOCIATIVE Property of: Addition ( ) ( ) IDENTITY Property of Addition ZERO PRODUCT PROPERTY Let a and b be real numbers. Multiplication Multiplication ( ) ( ) Multiplication If, then. ADDITION PROPERTY OF EQUALITY Definition: If a=b, then a+c = b + c. PROPERTY OF ZERO (INVERSE PROPERTY) The sum of a number and its opposite is 0. a + (-a) = 0 MULTIPLICATION PROPERTY OF EQUALITY Definition: If a=b, then. You Try: Identify the property used in each equation. a) ( ) ( ) b) c) d) ( ) e) ( ) ( ) f) ( ) ( ) g) PROPERTY OF OPPOSITES The property of a number and -1 is the opposite of the number. (-1) a = -a SUBTRACTION PROPERTY OF EQUALITY Definition: If a=b, then a c = b c. PROPERTY OF ZERO (MULTIPLICATION) The product of a number and 0 is 0. a(0) = 0 DIVISION PROPERTY OF EQUALITY Definition: If a=b and, then. DISTRIBUTIVE PROPERTY ( ) ( ) You Try: Simplify the expression (PEMDAS) 1) ( ) ( )( ) 2) [ ] 3) ( ) Linear Equations and Inequalities Solve the equations. 1) Solve the inequalities and graph the solution. 1) ( ) 2) ( ) ( ) 2) ( )
Compound Inequalities Solve and graph the solution. 1) ( ) Absolute Value Equations *Get the absolute value alone first! *Make two equations: one positive and one negative, then solve both. *You can have 2 solutions, 1 solution, or no solution. 1) 2) OR 2) 4 x 6 8 0 3) 5 x 7 2 2 Absolute Value Inequalities: 1) Get the absolute value bars alone first!!! 2) Decide if it s an and or an or. 3) Take the bars away and write 2 inequalities. First inequality Keep the inequality sign and right side number the same. Second inequality Flip the inequality sign and make the right side number negative! 4) Solve both. 5) Graph the solution. GreatOR Less thand Linear Equation Word Problem Marcus works as a salesman at a car dealership. He is paid a base salary of $1,307.74 each month, and he receives a commission of $163.70 for each vehicle he sells. If last month Marcus earned $5,727.64, how many cars did he sell last month? A. 27 B. 48 C. 70 D. 54 Linear Inequality Word Problem Ron plans on visiting an amusement park this weekend. Admission to the park costs $11.00, and each attraction costs $1.01 to use. If Ron has saved $59.00 for this weekend, what is the highest number of attractions he can visit at the park? A. 48 B. 5 C. 47 D. 58
Functions, Relations, Domain and Range Fill in the blanks: 1. A relation is a pairing of input values with output values, or a set of. In a relation, the set of all of the x-coordinates/input is called the, while the set of all of the y-coordinates/ output is called the. 2. A function is a in which every has exactly one. In other words, there can be no repeats in the. In order to determine whether a graph is a function, use the line test. 3. In function notation, the symbol ( ) replaces. The way that you say ( ) is. Two other ways you can write the answer ( ) are: and. You try: 1. Determine if the relation is a function. State the domain and range. A) B) 2. Evaluate ( ) when. Slope and Intercepts Fill in the blanks: 1. The slope measures the of a line, and is represented with the letter. 2. A line that rises to the right has a slope, while a line that falls to the right will have a slope. 3. A line has a slope of zero (0), while a line has an undefined ( ) slope. 4. The two formulas used to calculate the slope of a line are: A) (use this formula when given 2 ordered pairs) You try: Find and classify the slope for #1-4. 1. ( ) ( ) 2. ( ) ( ) 3. 4. ( ) ( ) B) (use this formula when given a graph) 5. The x-intercept is located where the graph crosses the - axis. To find the x-intercept of an equation, let = 0, then solve for. Write your answer as an ordered pair in the form ( ). 6. The y-intercept is located where the graph crosses the - axis. To find the y-intercept of an equation, let = 0, then solve for. Write your answer as an ordered pair in the form ( ). 5. Find the x & y intercepts for: Graphing a Linear Equation in Two Variables Graph each equation using the method stated. 1) (use x-y table) 2) (use m and b) 3) (use x and y-int.)
Graphing a Linear Inequality in Two Variables 1) 2) 3) 4) Writing Equations of Lines and Inequalities Fill in the blanks. 1) In order to write the equation of the line, you need to know the and. 2) The general equation for slope-intercept form is, while the general equation for standard form is. 3) The two special rules for standard form are: A) B) 4) Parallel lines have the slope, while lines that are perpendicular have slopes. You Try: Write the equation of the line in standard form form. 4) Parallel to and through (10, 7). You try: Write the equation of the line in slopeintercept form. 1) m = 4 ; through (6, -2) 3 2) m = 0; through (-1, 7) 3) Through (-6, 2) and (-4, 11) You try: 8) Write the equation of the line in standard form. 5) Perpendicular to and through (-3, 1). 9) Write the inequality for each graph in slopeintercept form. A) B) 6) Parallel to and through (-5, -2). 7) Perpendicular to and through (-5, -2). 8) Vertical line through (5,4). 9) Horizontal line through (-3, 2).
Writing Equations of Lines Word Problems 1) A hot tub that holds 300 gallons of 2) You are stacking chairs after a water drains at a rate of 8 gallons per school dance. The height of one chair minute. is 5 feet. The height of 4 stacked a. Write an equation that represents chairs is 12 feet. how many gallons of water are left a. Write an equation using x and y to in the tub after it has drained for x find the height of a stack based minutes. on the number of chairs. 3) You rent a moving truck. The total cost includes a flat fee plus a mileage fee of $2 per mile. After 60 miles, the total cost is $160. a. Write an equation that represents the total cost y after x miles. b. After how many minutes will the hot tub be half full? b. What is the height in feet of 12 stacked chairs? b. If you only have $300 to spend, for how many miles can you rent the moving truck? c. When will the hot tub be empty? Systems of Linear Equations and Inequalities 1) Solve by graphing. 2) Solve by substitution. A) 3) Solve by elimination. A) B) B) Systems of Linear Equations Word Problems 1) A movie theater charges $7.50 for adults and $4.50 for children. The receipts for one showing of a movie were $540. If 80 tickets were sold, find how many tickets of each type were sold. 2) A group of students go out to lunch. If two have hamburgers and five have hot dogs, the bill will be $8. If five have hamburgers and two have hot dogs, the bill will be $9.50. What is the price of a hamburger?
Exponents Rules Multiplying Like Raising to a Power Dividing Like Bases Zero Exponents Negative Bases m n 0 Exponents a m a a a n 1 m a a b You Try Simplify the exponential expression. No negative exponents in your answer! 1) 2 3 2 22 2) ( ) 3) ( ) 4) 6 2 4 (5 xy z)( 2 x y z) 5 3 4 3 7 3x y 5) (3 xy ) (2 x ) 6) 7) 8) 3 7 21x y ) 4x 3x 2 1 2 10) ( ) ( ) 11) 3xy 6x y y 4 3 2 5 2x y 4 Polynomials Write the polynomial in standard form, then classify by degree and number of terms. 1) 2) 3) 4) Adding and Subtracting Polynomials be sure to put the result in standard form. 5) 6) Find the product be sure to put the result in standard form (except for #9). 7) 8) 9) ( ) 10) ( )( ) Other Topics on the Keystone Exam Estimation Factoring Polynomials Completely GCF and LCM of Monomials Rational Expressions Simplifying, Multiplying, Dividing, Adding, Subtracting, Excluded Values Radicals Simplifying, Multiplying, Dividing, Adding, Subtracting Probability and Odds of an Event Probability of Compound Events (And, Or) Measures of Central Tendency (Mean, Median, Mode) and Dispersion (Range, Quartiles) Data Displays and Analysis Interpreting graphs, scatterplots, line of best fit