Algebra Honors Summer Packet 018 Solving Linear Equations with Fractional Coefficients For these problems, ou should be able to: A) determine the LCD when given two or more fractions B) solve a linear equation with fractional coefficients Solve each equation with fractional coefficients. Check our solutions with the video. 5 1 V. 4 3 3 3 6 3 5 5 1V. 3 Independent Practice: 3. What is the LCD of the three fractions????,, 4 5 3 4. Solve: 3 4 + 5 = 1 3 ( ) 5. Solve: 5 3 1 4 = 5 6 3 4 Solving Literal Equations For these problems, ou should be able to: A) solve a literal equation for a specified variable B) identif an restrictions in the original problem and final answer If ou need assistance, watch the corresponding video on MathCamp31. For questions 1-3, solve the literal equation for the indicated variable. Include restrictions. 1. Solve for b1: A 1 1 h b b. Solve for r: 4 V r 3 3 3. Solve for w: z = w 3+w 1
Solving Absolute Value Equations For these problems, ou should be able to: A) solve an absolute value equations using two cases B) identif etraneous solutions b checking our answer If ou need assistance, watch the corresponding video on MathCamp31. Solve each absolute value equation showing work as per the video. 1V. 3 1 15 V. 5 1 10 3V. 4 8 Independent practice: 4. Solve: 3 7 5. Solve: 5 10 6. What is an etraneous solution? 7. The equation, 1, ields the solutions 1 and 1. Show the check for each of these solutions and decide which, if an, is the solution to the equation. For questions 8 and 9, solve, clearl showing the two cases. 8. 8 6 9. 3 3 10. Solve and identif an etraneous solutions: 3 8 6 1
Flip WS 1.5: Interval Notation For these problems, ou should be able to: A) epress an inequalit using a number line, using an algebraic sentence and most importantl, b using interval notation. B) identif included boundaries and non-included boundaries 1V. Practice Fill out this worksheet as ou watch the corresponding video on MathCamp31. Algebraic inequalit Number line Interval notation Set-builder notation 5V. 5 6V. 3 7V. 4 4 8V. or 3 9V. 0 10 6, 30V. 31V.,4 5, 3V. 7 Independent Practice:. Multiple choice: Give the interval over which the inequalit is true: 10 A., 10 B. 10, C.,10 D. 10, 3. Application: Use interval notation to describe the solution set for the range in temperature in a certain cit on Jul 4th: 71 t 94 4. Use interval notation to describe the solution set: 4 or 0 5. Eplain wh the interval shown is incorrect based on the solution set shown on the number line: is the same as,3. 3 student sees student concludes 6. Multiple choice: Which grouping smbol will alwas be adjacent to the infinit smbol? 3
A. brace {,} B. bracket [,] C. parenthesis (,) Solving Compound Inequalities For these problems, ou should be able to: A) determine the solution set using interval notation Fill out this worksheet as ou watch the corresponding video on MathCamp31. Using interval notation, find the solution set to each compound inequalit. 1V. < 3 and ³ 0 V. -1 or > 1 3V. 4 6 4V. 5 > 5V. < 0 or > 0 6V. < 0 and > 0 Independent Practice: 7. Classif the following compound inequalit as an and or or problem: -10 10 8. Does the shading on the number line and the solution suggest an and or or problem? a. solution set:, 5 b. solution set: 5 onl Match the solution set shown on the number line with the corresponding solution set in interval notation: 9. 10. -3-3 A. [ 3,) ( ) È é, B. -,-3 C. ( 3,] ë ) D. (-,-3ù û È (, ) For questions 11-13, find the interval over which the compound inequalit is true. Show the word and or or in addition to our shaded number line before ou state the solution interval. 11. 3 and 7 10 1. 5 1 8 13. 6 0 or 1 4
Flip WS: Solving Absolute Value Inequalities For these problems, ou should be able to: A) determine the solution set to an absolute value inequalit B) use greator or less thand to distinguish between and and or Fill out this worksheet as ou watch the corresponding video on MathCamp31. Start with this: write down the three ke points identified as 1,, and 3. 1.. 3. Solve each absolute value inequalit as per the video. 1V. 1 5 V. 1 3 9 3V. 1 3 Independent Practice: For questions 4-7, match the solution set shown on the number line with the corresponding compound inequalit: 4. 5. A., B.,, 6. 0 7. 0 C. no solution D. all real numbers E. 0 onl For questions 8-10, find the interval over which the absolute value inequalit is true. Show the word and or or in addition to our shaded number line before ou state the solution interval. 8. 1 9. 3 3 9 10. 1 5
Flip WS: Point-slope Form For these problems, ou should be able to: A) write a linear equation in point-slope form B) understand that a single line has infinitel man point-slope forms C) transform a linear equation in point-slope form into slope-intercept form Point-slope form: 1V. Show the calculation to find the slope of. V. Write the point-slope form using point A. D(5, 3) C(, 1) B(-1, -1) 3V. Write the point-slope form using point B. A(-4, -3) 4V. Write the point-slope form using point D. Independent Practice: 5. Using the traditional formula to calculate slope, cross-multipl and see what ou get! m 1 1 6. True or false: the following point-slope form is simplified: 5 1 3 4 7. Write the point-slope form using point C in the line shown on this page. 6
Practice: Slope and Equations of Lines Slope-Intercept Form: m b Standard Form: a b c, where a, b, and c are integers and a 0. General Form: a b c 0, where a, b, and c are integers and a 0. Point-Slope Form: m 1 1 Equation of a Vertical Line: Equation of a Horizontal Line: c c, where c is an real number., where c is an real number. Neatl show all work. Use the indicated form. Bo in our final answers. 1. Write the following in slope-intercept form: 5 8. Transform the following into standard form: 5 3 4 3. Manipulate the following into general form: 3 4 1 4. Write the line with the following conditions into point-slope form: passes through 1 7, and has a slope of 9. 7
5. Write the equation of the line shown in standard form: -7 6. In general form, write the equation of the line that is parallel to the line shown and passes through the point (-5, 8). 4 7. In general form, write the equation of the line that passes through the points (-5, -1) and (4, 5). 8. Write the equation of the line whose -intercept is -6 and whose -intercept is in standard form. 9. If line a b and line a has the equation 5 6 = 10, determine the equation of line b, in pointslope form, if b passes though (1, -). 8
Practice: Domain and Range from a Graph For these problems, ou should be able to: determine the domain and range given a sketch. 1. When determining the domain b analzing a sketch, ou must scan the sketch from to to see where the -values eist.. When determining the range b analzing a sketch, ou must scan the sketch from to to see where the -values eist. Using interval notation, give the domain and range b analzing the sketch. 3. 4. 5. (, 7) (-1,-) (-6, -5) (4,-4) (-6,-7) D: D: D: R: R: R: Practice: Finding Domain of an Epression Ke concept: For what value(s) of will the given epression ield a real number output? Two Rules: 1. denominator 0. if even R, then R 0 But there s no graph?! Find the domain of each epression. Epress solution using interval notation. 1. 5 1. 4 3. 3 5 5 4. 10 5. + 4 6. 5 9
Practice: Graphing Inequalities Solid or dotted? Where to shade? FLIP THE SWITCH? A) Write the related equation and sketch. B) Select test points in each resulting region and shade accordingl. 1. Sketch: < 4. Sketch: 4 3. Sketch: 3 < 6 4. Sketch: + 1 1 Practice: Sstems I. Solve each sstem b elimination. Epress our solution as an ordered pair [i.e. (-3, 6)] 1. 4 5 = 17. 5 3 = 46 4 3 + 4 = 5 3 7 = 10 8 10
II. Solve each sstem b substitution. Epress our solution as an ordered pair. 11 3. 6 6 3 1 4. 8 III. Applications: For the following problems, set up a sstem of equations and solve using an method. 5. 35 lights are needed to light up the stage in The Lion King on Broadwa. Onl 100 watt and 150 watt fitures are available. The total allowable wattage is 4000 watts. How man of each tpe of fiture will be used? 6. On Januar 1 st, Heather has $00 in her bank account, and earns $60 per week babsitting. On Januar 1 st of the same ear, Sall has $350 in her bank account, and earns $45 per week working at Dair Queen. Using slope-intercept form, write a linear function describing the amount of mone in each girl s bank account as a function of time. a. Linear model for Heather: Amount in Savings Amount in Savings b. Linear model for Sall: Time (weeks) c. In which week will the girls have the same amount of mone in their accounts? d. What is the amount of mone in each girl s account at the week when the amounts are the same? e. Which girl has more mone after 15 weeks? How much more mone does the girl with the 11
larger account have over the girl with the lesser amount at this time? Flip WS: Monomials Rules of Eponents A monomial is an epression that is a number, a variable, or the product of a number and a variable. Monomials can not contain variables in denominators, variables with negative eponents, or variables under radicals. A constant is a monomial that contains no variables. A coefficient in the number that proceeds a variable. The degree of a monomial is the sum of its eponents. RULES OF EXPONENTS 0 1 Anthing raised to the power of zero equals 1. ( 0) a b a b and add the eponents. a ab When dividing powers of the same base, retain the base b subtract the eponents. a b a b When raising a power to a new power, multipl the eponents. When multipling powers of the same base, retain the base a a a a a a a 1 a ; a 1 a a a When several factors are raised to a power, each factor will feel the effect of the eponent. When a fraction is raised to a power, both the numerator and denominator will feel the effect of the eponent. When a term is raised to a negative eponent, the term is sent to the denominator, and the eponent changes sign (from negative to positive) and vice-versa. When a fraction is raised to a negative eponent, the fraction flips and the eponent changes sign. For #s 1-10, simplif on our own. Check our answers b watching the video link on MATHCamp31. 1V. [video 1] 3 4 a b 5ab V. [video 1] s 10 s 3V. [video 1] 3 3 10 6 10 4V. [video 1] b 3 5V. [video ] 5 3 3c d 6V. [video ] 4 3 1
7V. [video ] 5 5 3a 6 4 b 8V. [video ] a 4a 7c 3 6 b 3 4 a 9V. [video ] 0 36 0 0 0 0 6 4 10V. 4 6 3 5a 4a a a a 11. Epress in scientific notation: 1. Evaluate using scientific notation: 510 3 7 10 8 a. 4,560,000 a. b. 0.00009 b. 4 7 1.8 10 4 10 Practice: Polnomials Operations A polnomial is a monomial or a sum of monomials. A binomial is a polnomial with terms. A trinomial is a polnomial with 3 terms. The terms are the monomials that make up a polnomial. The degree of a polnomial is the degree of monomial of highest degree. Simplif on our own. Check our answers b watching the video link on MATHCamp31. 1V. 3 4 5 3V. 3 5 3 4 Video 1: 1:41 (#s 1-15 ODD) Video : 7:14 (#s 17-3 ODD) 5V. 3 7V. 3 5 13
9V. 3 4 1 11V. 3 3 13V. 3 5 17V. u v 3 Flip WS: FACTORING POLYNOMIALS Video 1 [1:14] Video [14:35] GCF greatest common factor DOTS difference of two squares SOC/DOC sum of cubes / difference of cubes FAST factoring a simple trinomial - trinomial factoring w/ leading coefficient = 1 Nobes trinomial factoring w/ leading coefficient 1 PST perfect square trinomial grouping Factor completel. Check our answers b watching the video link on MATHCamp31. 6 5 7 4 5 3 5 6 4 1. 5a b c 35a b c 15a b c V. a ( ) b( ) ( ) 3V. 4a 81 b 4. 16 5 5V. 4 9 3 9V. 9 36 10. 3 8 11V. z 5z 100 1. a 5ab 6b 14
13V. 6 14. 10 1 10 15V. 1 13 3 16. 1 3 10 17V. 8 16 18. 10 5 19V. a 18ab 81b 0. 36 60 5 1V.. a 3a b ab 6b 3 Practice: Rational Square Roots and Radicals For these problems, ou should be able to: simplif rational square roots. Find each square root. 1. 13. 5 3. 17 8 4. 5 1 5. 1 6. 100 49 7. 9 3 50 8. 15 5 15
Practice: Irrational Square Roots For these problems, ou should be able to: simplif irrational square roots. Find each square root. Use simplest radical form for non-perfect squares. 9. 75 10. 3 11. 48 1. 7 13. 6 45 14. 5 8 Practice: Multipling and Dividing Radicals Rule: When multipling radicals, multipl the numbers on the OUTSIDE together, then multipl the numbers on the INSIDE of the radical together and simplif if possible. Multipl. 15. 3 5 16. 3 3 17. 50 18. 5 6 8 19. 3 The process b which we eliminate the radical in the denominator is called rationalizing. Simplif. 0. 4 6 1. 1 3. 10 3. 5 5 4. 5 1 5. 3 8 16
Combine. Practice: Combining Radicals 6. 0 5 3 7. 16 5 8. 8 63 9. 45 80 Practice: Solving Basic Quadratic Equations Rule: For an real numbers a and b: if a b,then a b Samples that require ou to solve (be sure to isolate the squared term first): 30. 16 31. 3a 75 3. c 1 33. d 11 0 17
Defn.: A quadratic function has the form Flip WS: Graphing Quadratic Functions f( ) a b c. a is called the quadratic term, b is called the linear term and c is the constant. The graph of a quadratic function is called a parabola. Some of the parabolas we will stud will open upwards while others will open downwards. Parabolas have an ais of smmetr which runs through the verte and splits the parabola in half. The -coordinate of the verte can be found using the formula: Tr This Out!: 1. Determine a, b, and c for f( ) 4. Then determine the verte and create a table of 5 values. Find the -intercept and identif the ais of smmetr. Sketch. a = b = c = -coordinate of verte b a. *Complete this eample using the MathCamp31 video. 1V. Determine a, b, and c for f( ) 3 1. Then determine the verte and create a table of 5 values. Find the -intercept and identif the ais of smmetr. Sketch. a = b = c = -coordinate of verte For f( ) a b c, if a > 0, then the parabola will open upward and the verte will be a minimum. if a < 0, then the parabola will open downward and the verte will be a maimum.. Consider f( ) 3. Determine whether the function has a minimum or maimum value. Find the value of the minimum or maimum. 18
Practice: Solving Quadratic Equations b Factoring Zero Product Propert: for an real numbers a and b, if ab = 0, then either a = 0, b = 0, or both a and b equal 0. Use factoring to solve to solve each quadratic equation. 6 0. 10 0 3. 1. 4. 9 5. 5 0 6. 4 9 0 7. 6 0 8. 8 15 0 9. 14 3 10. 3 5 0 11. 6 7 0 1. 3 Practice: The Quadratic Formula If a b c 0, then a b b 4ac Use the quadratic formula to solve to solve each quadratic equation. 1. 8 0. 9 10 0 3. 19
Practice: Solving Quadratic Equations b Completing the Square Procedure for completing the square: Manipulate the equation so that the leading coefficient is positive one. Isolate the constant. Take ½ of the coefficient of the linear term. Square the result of step 3 and add to both sides of equation. Factor the left side as a PST and square root both sides. Use completing the square to solve each quadratic equation. 1. 6 1 0. 7 3. 1 0 4. 1 6 0 Practice: Quadratic Graph Analsis Use the graphs shown to answer each question. Use interval notation where appropriate. 5. The graph shown below is f(). 4 3 1-4 -3 - -1-1 1 3 4 - -3-4 -5-6 -7-8 For which -values does f()=0? Find the -intercept: Give the interval for which f() 0: Give the interval for which f()<0: 0
Calculator Review The calculator required for an level of algebra is the Teas Instrument, TI-84 (either plus, silver edition, or CE is fine). Practice Graphing Functions Activit I: Go to Y screen Let Y1 7 3 Let Y 5 4 8 (either ^3 or MATH, option 3: 3 ) Find a viewing window that shows where the two graphs intersect. (press WINDOW and adjust parameters as needed) 1. Find the point of intersection, rounding to the nearest hundredth. ( nd, TRACE, option 5:intersect). Find the -intercept of Y1 7 ( nd, TRACE, option :zero) 3. Find the -intercept of 3 Y 5 4 8 ( nd, TRACE, option :zero) Go to Y screen Let Y3 1 (MATH, NUM, option 1:abs ) 4. Find the points (ordered pair) where the absolute value graph intersects Y 1 and Y. Activit II: Go to Y screen Let Y1 0.5 1 Let Y 0.1 0 Find a viewing window that shows where the two graphs intersect. (press WINDOW and adjust parameters as needed) 5. Find the point of intersection, rounding to the nearest hundredth. ( nd, TRACE, option 5:intersect) Activit III: Go to calculating screen ( nd, MODE) Press ALPHA, Y, option 1:n/d Practice with the Fraction Template 1
6. Tpe in and evaluate: 3 10 4 rounding to the nearest hundredth.