1. Graph Real Numbers on a Number Line

Transcription

1 (Chapters and ) A. Real Numbers and Expressions The set of real numbers consists of all rational and irrational numbers. Rational numbers can be expressed as fractions, and irrational numbers can not be expressed as fractions. All real numbers can be graphed on a number line, as seen below. The operations that combine real numbers have specific properties, which can be extended directl to algebraic expressions.. Graph Real Numbers on a Number Line Number Line Real Numbers can be graphed as points on a line called a real number line, on which numbers increase from left to right. A. Real Numbers and Expressions Graph the following real numbers on a number line. a. } b. } c. Ï } 7 Note that Ï } 7 is between Ï } and Ï } 9. The calculator approximation of Ï } 7.6 supports this insight. When graphing rational numbers, as in a. and b., consider the mixed number expression. That is, } } and } }. To graph the number on the number line, begin at zero and move to the integer part of the number. Then move the fractional part of the next unit (in the alread determined direction) to graph the number. 7 0 Graph the following real numbers on a number line.. }. }. Ï }. Use Properties and Definitions of Operations Opposite The opposite, or additive inverse, of an number b is b. Reciprocal The reciprocal, or multiplicative inverse, of an nonzero number b is } b. Subtraction: a b a (b) Division: a b a } b, b 0 Properties of addition and multiplication Let a, b, and c be real numbers:. Closure: a b and a b are real numbers. Commutative: a b b a and a b b a. Associative: (a b) c a (b c) and (a b) c a (b c). Identit: a 0 0 a a and a a a. Inverse: a (a) 0 and a } a, a 0 6. Distributive: a (b c) a b a c Algebra Benchmark Chapters and

2 A. Real Numbers and Expressions (Chapters and ) Be aware that subtraction and division do not possess the commutative and associative properties. But b rewriting them as addition and multiplication, we can appl these useful properties! Use properties and definitions of operations to show that 0 (x ) x. Justif each step. 0 (x ) 0 x } Definition of division. 0 } x Commutative propert of multiplication. 0 } x Associative propert of multiplication. () x x B multiplication. Using real numbers, give and verif an example of the following properties.. Closure propert of multiplication. Commutative propert of addition 6. Associative propert of addition 7. Associative propert of multiplication Use properties and definitions of operations to show the following. Justif each step. 8. ( x) 7 } x, x 0 9. ( c) c Switching the numerator and denominator would ield the unit minutes per word. In this rate, a smaller number would indicate a faster tpist. Check this with a person who can tpe 00 words in minutes.. Use Unit Analsis Unit Analsis Observing the units of measurement serves two main functions. First, it provides a quick check of the validit of a solution. Secondl, it can be used to help converting quantities from one unit of measurement to another. You can tpe 00 words in minutes. What is our tping rate? 00 words } 60 words per minute minutes Using unit analsis, our answer is in words per minute, which is an acceptable unit for measuring a tping rate. The speed of light is approximatel 00,000 kilometers per second. What is the approximate speed of light in miles per hour? 00,000 kilometers }} 60 seconds 60 minutes second } minute } hour 67,000,000 miles per hour 0.6 mile } kilometer Algebra Benchmark Chapters and

3 (Chapters and ) Note that unit analsis helped us determine how to write each unit multiplier we used. 60 seconds For instance, we multiplied our original rate b }, which equals. Note that minute minute } also equal, but this would be an unfortunate choice. Our units seconds 60 seconds would fail to cancel. 0. You travel 0 miles while driving at a rate of 60 mile per hour. How long have ou been driving?. Convert ard to centimeters, given inch. centimeters. A. Real Numbers and Expressions. Evaluate an Algebraic Expression Variable A variable is a letter that is used to represent one or more numbers. Algebraic Expression An expression involving variables is called an algebraic expression. Evaluate When ou substitute a number for each variable in an algebraic expression and simplif, ou are evaluating the algebraic expression. Conscientiousl appl the order of operations. Don t take too man steps at once, as this can result in an incorrect answer. Evaluate k k 0 when k. k k 0 () () 0 Substitute for k. () () 0 Evaluate the powers first. 8 (6) 0 Perform the multiplications Rewrite the subtraction as an addition. Add. Evaluate the expression for the given value of the variable.. x x when x 0. w 0 (w 6) when w. Simplif an Algebraic Expression Terms In an expression that can be written as a sum, the parts added together. Variable Term A term that has a variable part. Constant Term A term that has no variable part. Coefficient The number when a term is a product of a number and a power of a variable. Like Terms Terms that have the same variable parts. Algebra Benchmark Chapters and

4 A. Real Numbers and Expressions (Chapters and ) Combine like terms b simpl combining their coefficients. Be sure to include the sign of the coefficient when combining them. Once the coefficients are combined, do not combine the variable parts of the like terms. Simplif b combining like terms. a. 7x x x 7x x x (7x x ) x ( ) b. (x ) (x ) (x ) (x ) Group like terms x x Combine like terms 6x 9 x 8 Appl distributive propert (6x x) ( 8) (9 ) Group like terms x Combine like terms Simplif the expression.. r s t r s t. x x x x 6. (w ) (w ) 7. ( ) ( ) Quiz Graph the following real numbers on a number line.. } 8. } 7. Ï } Using real numbers, give and verif an example of the following properties.. Inverse propert of multiplication. Distributive propert Use properties and definitions of operations to show the following. Justif each step. 6. a (b c) (b a) c 7. (6 b) 6 b 8. Given U.S. dollar 6 Japanese en, and Japanese en, Turkish liras, find the value of U.S. dollars in Turkish liras. Evaluate the expression for the given value of the variable. 9. d d when d 0. w 0 (w 6) when w Simplif the expression.. (x ) (x ). a(a a) (a a ) Algebra Benchmark Chapters and

5 (Chapters and ) B. Problem Solving The abilit to solve word problems should be one the highest priorities for ever student in a mathematics class. The skills necessar to successfull solve a word problem are accessible to everone, but often neglected. Often times, solving these problems is attempted half-heartedl, or not at all! The three skills presented here can help make word problems a pleasure, instead of something to dread. The onl wa to achieve this is to invest the necessar practice time, with a positive attitude along the wa. B. Problem Solving. Write a Verbal Model Verbal Model A verbal model is a mathematical expression with words, instead of numbers and variables. You are planning on baking cookies for holida gifts. You alread have 6 cookies baked, and ou can bake cookies ever hour. Write an expression that shows how man cookies ou will have read after h hours. How man cookies will ou have after hours? STEP : Write a verbal model. Then write an algebraic expression. Cookies baked per hour Hours Number of cookies alread baked A word problem should have a word answer. Alwas write our final answer as a sentence. h 6 An expression that shows how man cookies ou will have after h hours is h 6. STEP : Evaluate the expression from step when h. h 6 () 6 Substitute for h. 0 6 Multipl. 6 Add. You will have 6 cookies after hours. Write an expression and answer the question.. A boat traveling miles per hour can cover a distance of miles before needing to refuel. At this speed, how man hours can the boat travel before needing to refuel?. A supermarket has 8 pounds of a bulk trail mix that is worth $. per pound. How much mone does the supermarket have invested in this trail mix? Algebra Benchmark Chapters and

6 (Chapters and ) B. Problem Solving. Look for a Pattern A coffee shop has fixed dail expenses which include rent, insurance, utilities and wages. The coffee shop also has variable expenses per cup of coffee served. The table below shows the total dail expense e when c cups of coffee are served. Find the total dail expenses for the coffee shop if 0 cups of coffee were served. Cups Served, c 0 Dail Expenses ($), e Alwas investigate and interpret what happens when the independent variable is zero. In this case, ou find the fixed expenses. Most often, ou will gain some insight to the problem, if not ke information. The expenses increase b $0.0 for each cup of coffee served You can use this pattern to write a verbal model for the expenses. Total Expenses Fixed Expenses Expense per cup cups served ($) ($) ($/cup) (cups) e c An equation for the total dail expenses is e c. So the total dail expenses for the coffee shop when 0 cups of coffee are served is e (0) $0.00. Look for a pattern in the following tables. Then write an equation that represents the table... x x 0 } 7 } } }. Draw a Diagram You decide to build a picture frame for an 8 0 portrait, using wide molding. How man linear inches of molding will ou need? That is, what is the outside perimeter of the frame? Algebra Benchmark Chapters and

7 (Chapters and ) When possible, include our variable(s) in the drawing. It ma help ou see a phsical relationship between what ou know, and what ou are looking for. Begin b drawing and labeling a diagram. Tr to include all the information given in the problem. L From the diagram, we can write and solve an equation to find the outside perimeter of the frame. Perimeter L W L W L W Definition of perimeter. W W (8 ) (0 ) Because L 8 and W 0 () () Add. 8 Multipl, then add. You will need linear inches of molding to make this frame. 80 L B. Problem Solving. A compan prints targets for archer ranges, and is designing a new target. The bull s ee will have a diameter of inches. Then there will be an additional concentric circles, each with inch width. What are the dimensions of the smallest size paper the compan can use to print this target? 6. A 0 0 ceramic wall tile has a 0 0 rectangle removed for an electrical outlet. What is the area of the tile that remains? Quiz. Kendra pas $ plus $ per hour to rent a scooter. Write an expression for how much she pas for renting a scooter h hours. How much does she pa for renting the scooter for 6 hours?. Look for a pattern in the table. Then write an equation that represents the table. x An equilateral triangle with side length 0 will have a height of Ï } 0. A ield sign is an equilateral triangle with a perimeter of 7. feet. To the nearest inch, what is the height of a ield sign? Algebra Benchmark Chapters and

8 (Chapters and ) C. Linear Equations The title Linear Equations actuall refers to two separate ideas. The first tpe of linear equations we work with are in one variable. We will be asked to solve these linear equations. That is, we will need to use fundamental transformations to determine what value(s) of the variable make the equation true. The second tpe of linear equations we will work with are in two variables. These tpes of equations have so man solutions that the onl wa to accuratel present them is graphicall. And naturall, the graph of these solutions form a line. C. Linear Equations. Solve a Linear Equation Equation A statement that two expressions are equal. Linear equation An equation that can be written in the form ax b 0 where a and b are constants and a Þ 0. Solution A number is a solution of an equation in one variable if substituting the number for the variable results in a true statement. Equivalent equations Two equations are equivalent equations if the have the same solution(s). Solve x (x ) ( x) Be sure to distribute the negative two to ever term in the parenthesis. When checking, alwas use the original printed problem. A mistake could have occurred when ou wrote down the original problem. x (x ) ( x) x () (x ) ( x) x (0x 8) 0 x 7x 8 0 x Write original equation. Definition of subtraction. Distributive propert Combine like terms. 8 0 x Add 7x to both sides. x Add 0 to both sides. } x or x } Divide each side b. CHECK } } 0 } Substitute } for x. 6 } } } 8 Simplif. 8 } } } } 8 Simplif. Solve the equation. Check our solution. 7 } } 7 Solution checks.. x.. b 6 b 0. n (n ) ( n) Algebra Benchmark Chapters and

9 (Chapters and ) C. Linear Equations Before ou begin, find all the places where the desired variable occurs. Using fundamental properties, tr to combine multiple occurrences of the variable into one.. Rewrite a Formula Formula An equation that relates two or more quantities, usuall represented b variables. Solve for a variable Rewrite an equation as an equivalent equation in which the variable is on one side and does not appear on the other side. Solve the formula F } 9 C for C. Then find the Celsius temperature of 77 Fahrenheit. STEP : Solve the formula for C. F 9 } C F 9 } C Write temperature conversion formula. Subtract from both sides. } 9 (F ) C Multipl both sides b } 9. STEP : Substitute the given value into the rewritten formula. C } 9 (F ) } 9 (77 ) 77 Fahrenheit is equivalent to Celsius.. Solve the formula x 80 for. Find when x 00. Substitute 77 for F and simplif. When asked to write the equation of a graphed line, tr to use points that have integer coordinates. 6. Solve the girth formula G l w h for h. Find h when G 0, l 00, and w 0.. Write an Equation of a Line Slope-Intercept Form of a Line mx b given slope m and -intercept b. Point-Slope Form of a Line m(x x ) given slope m and point (x, ). Write an equation of the following lines. a. That passes through (0, ) and has a slope of. b. That passes through (, ) and (, ). a. Since (0, ) is actuall the -intercept, we can use the slope-intercept form of a line. mx b Use slope-intercept form of a line. x () Substitute for m, and for b. x Simplif. b. The first thing we must do is find the slope of this line. m } x x } () } 6 () Now use either of the two given points and the point-slope form of a line. Algebra Benchmark Chapters and

10 (Chapters and ) We could have used an point on the line for the point-slope form. Expressing our final equation in slope-intercept form ields the same answer, regardless of which point was chosen. m(x x ) Use point slope form of a line. 6 } (x ) Substitute 6 } for m, and (, ) for (x, ). 6 } x 8 } Distributive propert 6 } x 7 } Write in slope-intercept form. Write an equation of the line that has the given slope and -intercept. 7. m, b 8. m 0, b C. Linear Equations Write an equation of the line that passes through the given point and has the given slope. 9. (, ), m 0. (, ), m Write an equation of the line that passes through the given points.. (, ), (, ). (0.,.), (.,.) We could have looked at the ratio of area to weight and still discovered direct variation between a and w. In that case our equation would have been a w.. Use Ratios to Identif Direct Variation Direct Variation The equation ax represents direct variation between x and, and is said to var directl with x. Constant of variation The nonzero constant a is called the constant of variation. A batch of cookies is made, and certain data about each cookie is recorded. The table below gives the area of the bottom of the circular cookies, and their respective weight. Tell whether cookie area and weight show direct variation. If so, write an equation that relates the quantities. Area of cookie, a (cm ) Weight of cookie, w (grams) Find the ratio of weight w to area a for each cookie. } 6 0. } } 9 6 < 0. } < } 0. } 9 < 0.9 Since the ratios are approximatel equal, the data show direct variation. An equation relating the area of a cookie to its weight is } w a 0., or w 0.a. Algebra Benchmark Chapters and

11 (Chapters and ) C. Linear Equations Tell whether the data in the table show direct variation. If so, write an equation relating x and.. x x 7 Quiz Solve the equation. Check our solution.. } z. a a. (c ) (c ). (m ) m m (m 7). Solve the formula s t 00 for t. Find t when s Write an equation of the line with slope } and -intercept }. 7. Write an equation of the line that passes through, } and has slope }. 8. Write an equation of the line that passes through }, and 7 },. 9. Tell whether the data in the table show direct variation. If so, write an equation relating x and. x Algebra Benchmark Chapters and

12 (Chapters and ) D. Functions and Graphs The idea of the function is one of the most fundamental ideas in the field of mathematics. It actuall transcends mathematics and is extremel important in man other fields, including the sciences, engineering, and business. B studing the graphs of functions, we can often see characteristics that are obscure algebraicall.. Identif Functions Relation A mapping, or pairing, of input values with output values. Domain The set of input values for a relation. Range The set of output values for a relation. Function A relation for which each input has exactl one output. If an input of a relation has more than one output, the relation is not a function. D. Functions and Graphs Tell whether the relation is a function. a. Input Output b. Input Output Functions are so useful because for a given input, there is exactl one output. Functions are definitel not ambiguous! 7 7 a. This relation is a function because each input is mapped onto exactl one output. b. This relation is not a function because the input is mapped onto both 7 and. Tell whether the relation is a function. Explain.. Input Output. Input Output. Input Output Use the Vertical Line Test Vertical line test A relation is a function if and onl if no vertical line intersects the graph of the relation at more than one point Algebra Benchmark Chapters and

13 D. Functions and Graphs (Chapters and ) A vertical line intersecting the graph more than once indicates one input of the relation has more than one output. The first graph below presents 7 students scores on a 0 point quiz versus time spent on the quiz. The second graph presents an individual student s scores on such quizzes. Are the relations represented b the graphs functions? Explain. 0 x Minutes Spent on Quiz Score on Quiz Score on Quiz 0 x Quiz Number The graph for the group of seven students does not represent a function. The vertical lines at x,, and each intersect the graph at more than one point. The graph for the individual student s scores does represent a function, because there is no vertical line that intersects the graph at more than one point. Use the vertical line test to tell whether the relation is a function or not. When calculating the slope of a line, it is not important which point is first and second. However, the order of subtraction in the numerator and denominator must be preserved.. x. Find a Slope. x Slope The slope m of a non-vertical line is the ratio of vertical change (the rise) to horizontal change (the run). m } x x where (x, ) and (x, ) are an two distinct points on the line. Undefined Slope A vertical line has an undefined slope. In the above formula, an two points on a vertical line will lead to division b zero, which is undefined. What is the slope of the line passing through (, ) and (, )? Let (x, ) (, ) and (x, ) (, ). m ( ) } x x } } 7 7 The slope of this line is m. Find the slope of the line passing through the given points. 6. (, ), (, ) 7. (, ), (, ) 8. (, ), (, ) 9. (0, ), (0, 7) Algebra Benchmark Chapters and

14 (Chapters and ). Graph an Equation Slope-Intercept Form A line with equation mx b has slope m and -intercept b. The equation mx b is said to be in slope-intercept form. Standard Form The standard form of a linear equation is Ax B C where A and B are not both zero. Intercepts A x- and -intercept is the x- and -coordinate of the points where a graph intersects the x- and -axis, respectivel. a. Graph } x b. Graph x D. Functions and Graphs Slope-intercept form is characterized b having isolated on one side of the equation. STEP : The equation is in slope-intercept form. STEP : The equation is in standard form. STEP : Identif the -intercept. STEP : Identif the x-intercept. The -intercept is, so plot the x (0) Let 0. point (0, ) where the line crosses x Solve for x. the -axis. The x-intercept is. So, plot the point (, 0). STEP : Identif the slope. The slope STEP : Identif the -intercept. is }, so plot a second point on the (0) Let x 0. Both of these equations are linear. Because of their different forms, we use different methods to graph them. Be sure to recognize the different forms, and know how to convert between them. line b starting at (0, ) and then Solve for. moving up units and right units. The second point is (, ). The -intercept is. So, plot the point (0, ). STEP : Draw a line through the two STEP : Draw a line through the two points. points. x Graph the linear equation. x 0. x.. x 0. x 0. x 8. x 6. Estimate Correlation Coefficients Scatter Plot A scatter plot is a graph of a set of data pairs (x, ). Positive Correlation If tends to increase as x increases, then the data have a positive correlation. Algebra Benchmark Chapters and

15 D. Functions and Graphs (Chapters and ) Negative Correlation If tends to decrease as x increases, then the data have a negative correlation. Correlation Coefficient A correlation coefficient, denoted b r, is a number from to that measures how well a line fits a set of data pairs (x, ). If r is near, the points lie close to a line with positive slope. If r is near, the points lie close to a line with negative slope. If r is near 0, the points do not lie close to an line. Tell whether the correlation coefficient for the data is closest to, 0., 0, 0., or. If the data lie exactl on a horizontal line, r will equal 0. This is because there is no correlation between the x and values. An x produces the same! a x b x a. The scatterplot shows a clear but fairl weak positive correlation. So, r is between 0 and, but not too close to either one. The best estimate given is r 0.. (The actual value is r 0..) b. The scatter plot shows approximatel no correlation. So, the best estimate given is r = 0. (The actual value is r 0.0.) Tell whether the correlation coefficient for the data is closest to, 0., 0, 0., or x x Algebra Benchmark Chapters and

16 (Chapters and ) 6. Approximate a Best-Fitting Line Best-Fitting Line The best-fitting line is the line that lies as close as possible to all the data points. Twelve cooking students are required to prepare a main course entrée and a dessert plate. Their dishes are then tasted and scored b a panel of judges. Their average scores on a scale from to 0 are presented in the table below. Approximate the best-fitting line for this data. D. Functions and Graphs Entrée Dessert From statistics, there is an exact formula for the one best-fitting line. It is best calculated with the use of a calculator or computer. When ou are approximating the best-fitting line, tr to use points that are convenient. Also, be aware that the points ma or ma not be original data points. STEP : Draw a scatter plot of the data. STEP : Sketch the line that appears to best fit the data. One possibilit is shown. STEP : Choose two points on the line. For the line shown, ou might choose (0, 6) and (0, 8). Note that neither is an original data point. STEP : Write an equation of the line. First find the slope using (0, 6) and (0, 8). m } } x Since one of the points is the -intercept, use the slope-intercept form to write the equation. mx b Slope-intercept form. 0.x 6 Substitute for m and b. An approximation of the best-fitting line is 0.x 6. In the following, (a) draw a scatter plot of the data, and (b) approximate the best-fitting line x x x Algebra Benchmark Chapters and

17 D. Functions and Graphs (Chapters and ) Quiz. Input Output. Input Output. Input Output x..x. x 6 6. x 6 Algebra Benchmark Chapters and

18 (Chapters and ) E. Inequalities Here two basic tpes of inequalities are presented. First we look at inequalities in one variable, and then inequalities in two variables. The are closel related, but the methods we use to solve them are dramaticall different. Also, both tpes of inequalities generall have an infinite number of solutions. Therefore, the onl wa we can accuratel represent the solution set is graphicall. E. Inequalities. Graph Inequalities Linear inequalit in one variable An inequalit that can be written in one of the following forms, where a and b are real numbers and a Þ 0: ax b 0 ax b 0 ax b 0 ax b 0 Solution of an inequalit in one variable A value that when substituted for the variable, results in a true statement. Graph of an inequalit in one variable Consists of all points on a number line that represent solutions. Graph the following: a. x b. x c. x d. x or x In a compound inequalit like problem c., make sure the inequalit makes sense. For instance, there are no solutions to x. Without the x, this is an alwas false statement,. a. The solutions are all real numbers less than or equal to. A closed dot is used in the graph to indicate is a solution. b. The solutions are all real numbers greater than. An open dot is used in the graph to indicate is not a solution. c. The solutions are all real numbers that are greater than or equal to and less than. d. The solutions are all real numbers that are less than or greater than or equal to. Graph the inequalit x. x. x. x or x Algebra Benchmark Chapters and

19 (Chapters and ) E. Inequalities We could have added x to both sides and arrived at the same solution. It would be written as x.. Solve an Inequalit Equivalent inequalities Inequalities that have the same solutions. Solve x x. Then graph the solution. x x Write original inequalit. x Subtract x from each side. x 0 Subtract from each side. x Divide each side b and reverse the inequalit. The solutions are all real numbers less than or equal to. The graph is below. 0 Solve the inequalit and then graph the solution.. x x x 7. } x. Solve Compound Inequalities Solve the inequalit and then graph the solution. a. x 7 b. x 0 or x This tpe of compound inequalit can be written as two inequalities. In problem a. the would be x and x 7. This tpe of compound inequalit ma have the entire number line as its solution set. For instance, x or x. a. x 7 Write original inequalit. x 7 Subtract from each expression. 8 x Simplif. x Divide all expressions b and reverse inequalities. x Rewrite the inequalit, preserving the relationships. The solutions are all real numbers greater than or equal to and less than. The graph is shown below. 0 b. A solution of this compound inequalit is a solution of either of its parts. First Inequalit Second Inequalit x 0 Write original inequalit. x Write original inequalit. x 9 Subtract from each side. x 8 Subtract from each side. x Divide each side b. x Divide each side b and reverse inequalit. The solutions are all real numbers greater than or less than or equal to. The graph is shown below. 0 Algebra Benchmark Chapters and

20 (Chapters and ) Pa attention to the inequalit sign. If it is strict, equalit is not permitted. If our substitution ields a statement like, our ordered pair is not a solution. Solve the inequalit and then graph the solution x x or x. Solve a Linear Inequalit in Two Variables Linear inequalit in two variables An inequalit that can be written in one of these forms: Ax B C Ax B C Ax B C Ax B C Solution of an inequalit in two variables An ordered pair (x, ) is a solution if the inequalit is true when the values of x and are substituted into the inequalit. Tell whether the ordered pair is a solution of 7x. a. (, ) b. (, ) c. (, 0) d. (, ) Ordered pair Substitute Conclusion a. (, ) 7() () 7 9 Y (, ) is not a solution b. (, ) 7() () 0 (, ) is a solution c. (, 0) 7() (0) 0 (, 0) is a solution d. (, ) 7() () 8 Y (, ) is not a solution E. Inequalities Tell whether the given ordered pair is a solution of x (, ). (, ). },. (, ). (, 0). Graph a Linear Inequalit in Two Variables Graph a linear inequalit in two variables The set of all points in a coordinate plane that represent solutions of the inequalit. Boundar line Line that divides the plane into two halves one half is made up entirel of solutions of the inequalit, and the other has no solutions.the boundar line is the graph of the equation formed b replacing the inequalit smbol with the equal sign. Graph a. x 6 and b. x in a coordinate plane. a. Graph the boundar line x 6. Use a dashed line because the inequalit smbol is. Ordered pairs that ield equalit are not solutions. Algebra Benchmark Chapters and

21 (Chapters and ) E. Inequalities In both of these examples, we use the point (0, 0) to determine which side to shade. If the boundar line passes through (0, 0), select a different test point. Notice that the inequalit x could be considered a linear inequalit in one variable. The directions of graphing in a coordinate plane clarif the ambiguit. Test the point (0, 0). Because (0, 0) is not a solution of the inequalit, shade the half-plane that does not contain (0, 0) x 6 8 b. Graph the boundar line x. Use a solid line because the inequalit smbol is. Ordered pairs that ield equalit are solutions. Test the point (0, 0). Because (0, 0) is a solution of the inequalit, shade the half-plane that contains (0, 0). x Graph the linear inequalit in a coordinate plane.. 6. x 6 7. x 7 Quiz Graph the inequalit.. x 0. x. 0 x. x or x Solve the inequalit and then graph the solution.. x 6. (x ) x 6 7. } x 7 8. x x or (x ) 6 Tell whether the given ordered pair is a solution of the inequalit. 0. ; (0, ). x ; (, 7). x ; (, ) Graph the linear inequalit in a coordinate plane.. x. } x. x Algebra Benchmark Chapters and

22 (Chapters and ) F. Absolute Value and Transformations The set of real numbers consists of all rational and irrational numbers. Rational numbers can be expressed as fractions, and irrational numbers can not be expressed as fractions. All real numbers can be graphed on a number line, as seen below. The operations that combine real numbers have specific properties, which can be extended directl to algebraic expressions.. Solve an Absolute Value Equation Absolute value For a number x, written x, the distance the number is from 0 on a number line. It is alwas positive. Extraneous solution An apparent solution that must be rejected because it does not satisf the original equation. F. Absolute Value and Transformations When solving an absolute value equation, alwas isolate the absolute value expression on one side. If the other side is negative, there can be no solution. Solve the following: a. x and b. x 0 x a. Zx Z Write original equation. x orx Expression can equal or. x 6or x Add to each side. x or x } Divide each side b. Check these solutions in the original equation. Zx Z Zx Z Z() Z 0 Z } Z 0 ZZ 0 ZZ 0 The solutions are and }. b. Zx 0Z x Write original equation. x 0 x or x 0 x Expression can equal x or x. 0 x or 0 x Add x to both sides. x or x 0 Solve for x. Check these solutions in the original equation. Zx 0Z x Z() 0Z 0 () Z(0) 0Z 0 (0) Z6Z 0 6 Z0Z The onl solution is 0. is an extraneous solution. Algebra Benchmark Chapters and

23 F. Absolute Value and Transformations (Chapters and ) Solve.. ZxZ. Zx Z. Zx Z x. Write an Absolute Value Inequalit Tolerance The maximum acceptable deviation of a product from some ideal or mean measurement. A marina can hold boats between 8 feet and 0 feet, inclusive. Write an absolute value inequalit describing the acceptable boat size range. STEP : Calculate the mean of the extreme boat sizes. Mean of extremes } Subtracting the lower extreme from the mean will also give ou the tolerance. STEP : Find the tolerance b subtracting the mean from the upper extreme. Tolerance 0 9 STEP : Write a verbal model. Then write an inequalit. ZActual length (feet) Mean of extremes (feet)z Tolerance (feet) Z 9Z A boat can dock at the marina if its length satisfies Z 9Z. Consider writing x as x ( ). Compare this expression with the general x h. This is wh the x-coordinate of the vertex is.. A German Shepherd is considered to be of normal size if it weighs between 6 and 80 pounds. Write an absolute value inequalit describing the normal size range.. From Exercise, write an absolute value inequalit describing the abnormal size range of German Shepherds.. Graph an Absolute Value Equation Vertex of an absolute value graph The highest or lowest point on the graph of an absolute value function. The graph of a Zx hz k has the vertex at (h, k). Graph x. Compare it with the graph of x. STEP : Identif and plot the vertex, (h, k) (, ). STEP : Plot another point on the graph, such as (, ). Use smmetr to plot a third point, (, ). STEP : Connect the points with a V-shaped graph. STEP : Compare with ZxZ. The graph of Zx Z is the graph of ZxZ first stretched verticall b a factor of, then translated left units and down unit. (, ) x Algebra Benchmark Chapters and

24 (Chapters and ) Graph the following and compare them with the graph of x. 6. Zx Z 7. ƒ(x) 0.Zx Z. Appl Transformations to a Graph Transformations of General Graphs The graph of a ƒ(x h) k can be obtained from the graph of ƒ(x) b performing these steps:. Stretch or shrink the graph of ƒ(x) verticall b a factor of ZaZ if ZaZ. If ZaZ, stretch the graph. If ZaZ, shrink the graph.. Reflect the resulting graph from step in the x-axis if a 0. F. Absolute Value and Transformations. Translate the resulting graph from step horizontall h units and verticall k units. The graph of the function f(x) is shown. Sketch the graph of the given function. a. } ƒ(x) b. ƒ(x ) x The order of the transformations can sometimes be ver important. For instance, shifting up, then reflecting across the x-axis will not ield the same graph as reflecting, then shifting up. Follow the steps outlined here. a. The graph of } ƒ(x) is the graph of ƒ(x) shrunk verticall b a factor of }, and then reflected across the x-axis. To draw the graph, multipl the -coordinate of each labeled point b } and connect their images. Then reflect this across the x-axis to form the final image. b. The graph of ƒ(x ) is the graph of ƒ(x) translated right units and down units. To draw the graph, first translate the graph units to the right. Do so b adding to the x-coordinates of each labeled point and connecting their images. Then translate this graph down units b subtracting from the -coordinate of each point, and connect their images. (0, ) (0, ) (, ) (, 0) (, ) x (, ) (0, ) (0, ) (, ) (, ) (, ) (, ) (, 0) (0, 0) (, 0) x (0, ) Algebra Benchmark Chapters and

25 F. Absolute Value and Transformations (Chapters and ) Using the same graph of the function f(x) from the above example, sketch the graph of the given functions. 8. ƒ(x ) 9. ƒ(x) 0. ƒ(x ) Quiz Solve.. ZxZ. Zx Z. Zx Z x. The width of a wire on a certain transistor can var. The least acceptable diameter is 0. mm while the maximum acceptable diameter is 0.7 mm. Write an absolute value inequalit describing the acceptable size range. Graph the following and compare them with the graph of x.. Zx Z 6. ƒ(x) ZxZ Using the same graph of the function f(x) from the above example, sketch the graph of the given functions. 7. ƒ(x) 8. ƒ(x) 9. ƒ(x ) Algebra Benchmark Chapters and