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Math 9 Chapter 6: Solving EquationsLearning Outcomes:Solving algebraic equations requiring multiple stepsSolving algebraic equations involving rational numbersSolving inequalitiesGraphing inequalitiesName:__________________6.1 Solving Equations & Isolating the VariableTo solve an equation, the ______________ must be isolated.When solving, we can use the __________________________ to isolate the ______________.***Whatever you do to one side of the equation you ______________________________***Examples:A: Addition & Subtractionx + 4 = 10x + 17 = 235 = p + 11z – 3 = 10x – 4 = -1012 = m – 45B: Multiply & Divide4y = -8x8 = -913x = 21-6x = 546.1 Solving Simple EquationsWorksheetA. Solve for the variable by adding or subtracting:1.2.3.4.5.6.7.8.9.10. 11.12. 13.14.15.16.17.18.B. Solve for the variable by multiplying or dividing:1. 2.3. 4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.6.2 Solving Equations Requiring Several StepsMany equations require more than one step to isolate the variable.To solve these equations, isolate the term with _____________________ first.Examples:-5w + 9 = 245n – 8 = 120 = 7p – 359p – 2 = 6-1 = -7 – 0.3t4x + 34 = 744y + 35 = 33(x – 4) = -6***6.1 - 6.2 Quiz next classSolving Equations Requiring Several StepsWorksheetSolve for the variable.2w-5=118-2x=12-9p-81=0-11x-2=20-5r+6=8-4+10t=-1612=9-2t8-3z=-199=6-3w23=11-3r1.5x-3=-120.2x+4=7Collect like terms and solve for the variable.4x-2+2x=10-6+5m-4=-5-3y+3-2y=2816=5r+2+2r-9t+8-t=-12-18=6x+2-x17=3y-3+2y-4d+2-4d=18Solve for the variable. Expand first.4x-2=9-3=5z+76-x+2=58-20-2d=8-4t-3t+5=98-2-3+2y=-2-2=-3x-17.6=-2-3-y8.4=-6a+2.432+x3=0-43+t4=8-2x2+6=36.3 Solving Equations by Combining TermsTo solve an equation, it is necessary to _________________________ on one side of the equation. ________________________ are combined and any _____________________ ________________ must also be combined.Examples:6x – 5 = 2x + 76x + 7 = -2x – 5-4m + 2 = 6m + 124x – 3 = 2x – 35(y – 1) = 7(3 + y)5(2w – 3) = -2(3w + 2) + 53(y – 1) – 5y = 2y – (y – 2)6.3 Solving Equations by Combining TermsWorksheetSolve.7x-3=4x+35y+9=2y-3-4m+2=6m+12-8t-5=-9t-73r-2=-5r+146p-7=-6p-75-y=3-2y3-2t=5-5t3y-2=y+4y+7=3y-94-r=3-2rr-1=5r-76y-2=5y+4-11+6v=-6v+11Expand first, then solve.3x-1=129y-3=3y-46-t-3=-3t-1-4+2p=42-2p7x+3=5x-16-2-x=-52x+4-4-2y-1=-3y+15x-1=81-x-2x-1=-2x-x+4-3y+1=-2y-14x-2+5=3+2x-3-2m+4+12=35-m-83n-2+12=6n-34-n1+5x-1=4x-3+66.4 Solving Equations with Fractional CoefficientsWhen an equation contains fractions, ____________________________ ___ by a ___________ _________________ of the fractions. This allows us to obtain an equivalent equation without fractions.Examples:x2 = 14a3 – 3 = 34a + 12n3 - 3n2 = 16 – nx- 23 = 4 - 3(4-x)53x+22 - x+13 = x14(2 – x) + 12 = 12(13x + 7)***6.3 – 6.4 Quiz next class6.4 Solving Equations with Fractional CoefficientsWorksheetA. Solve for the variable.1.2.3.4.5.6.7.8.9.10.B. Solve.1.2.3.4. 5. 6.7.8.9. 10. 6.3 – 6.4 Review1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 3x10 - 2x5 = 3x2 + 12 6.5 Introduction to InequalitiesWe use an ________________ to model a situation that can be described by a _______________ instead of just one.Inequality Signs:Examples:a is less than 3d is greater than or equal to -5.4Contest entrants must be at least 18 years old (a = age)Scientists have identified over 400 species of dinosaurs ( )Inequality Solutions:Is each number a solution of the inequality b ≥ -4?-8-3.5-4-4.50Use a number line:1431232194060Graphing Inequalities:Graph each inequality on a number line and write three possible solutions.2235206794500t > -52247907574720-2 ≥ x22479086677500.5 ≤ a2260609131300p < 2536.5 Introduction to InequalitiesWorksheetGraph each inequality AND give 3 possible solutions for each.1674826118110001. 1674826112395002. 1693876154305003. 1674826208280004. 1674826119380005. 169387694615006. 168749951435007. 1672921121920008. 1679244125730009. 16729211485900010. 16729211238250011. 1672921895350012. 16729211790700013. 16976861568450014. 16976861428750015. 17230592038350016. 17. Use a symbol to write an inequality that corresponds to each statement:a) A number between 0 and 4999b) A number more than 5000c) x is less than -2c) p is greater than or equal to 6d) y is negativee) m is positivef) A coffee maker can hold no more than 12 cups of waterg) You must be at least 16 years old to obtain a learner’s permith) The maximum seating capacity of a school bus is 48 studentsi) Over 2500 people participate in the charity bike-a-thon each year18. Is each inequality true or false?a) 5 < 8b) 5 ≤ 5c) -5 < -8d) 0 ≥ -5e) 5 < -8f) 5.01 < 5.1g) 5 < 5h) 1/5 < 1/819. Is each number a solution of x < -2?a) 0b) -3c) -6.8d) -2e) -2.001e) -1/26.6 Solving Inequalities: Addition & SubtractionTo solve an inequality, we use the same strategy as for solving an equation. We isolate the variable by adding or subtracting from each side of the pare the following solutions:x + 3 = 5x + 3 < 5Examples: Solve the inequality and graph the solution6 ≤ x – 43x > 2x + 123.5 < x + 2.4x + 1 < 4-13 ≥ x – 116.7 Solving Inequalities: Multiplying & DividingWhen we multiply or divide by a _______________ we must _________________ the inequality sign.Proof: Consider the inequality-1 < 21987468148590019877161765300To make the inequality true, we must __________________________Examples: Solve and graph2704465908050-5s ≤ 252704465736600-4y > 1227044651651003x < 5x + 12-172085675336***6.6-6.7 Quiz next class00***6.6-6.7 Quiz next class26946367112003y + 8 ≥ 176.6 – 6.7 Solving Inequalities WorksheetSolve and graph.x-1≤3x+3>24>9-x9≤15-xSolve and graph.7x<149>-2x3y+8>1721-5z>1113≤1-34x18≥4.5-1.5aSolve and graph.4x-7≥2x+5-25+11z≤30-11z3x+2≤1242-x≤5x-22x+8≤43+x-3y+8<5-7y-35-8a>6a-718+10z≥-12-2z3-8+2x>4-23x+56-2c-11<-53c+86.6 – 6.7 Linear Solving InequalitiesExtra WorksheetSolve. Justify your work and show a graphical representation of your answer as well.1. 2. 3. 4. 5. 6. 7. 8.9.10.11.12.13. 14.15.16.B. For the following questions, write the inequality represented by each number line:17.45720000018.45720000019.45720000020.511810-88900021. A business must choose a company to print a promotional pany A charges $900 plus $0.50 per copyCompany B charges $1500 plus $0.38 per copyHow many brochures must be printed for company A to be less expensive?How many brochures must be printed fro company B to be less expensive?Chapter 6 Solving Equations ReviewSolve.5+x=-118-t=-215=x+2314-z=-145x=45-15=-3n15t=-3s5=138r=56x4=-72) Solve.6p-3=1513=4+3x-6-2r=88p-9=73y-11=14-2y5y-2=3y+4r-3=2r+4-2x+4=16+4x3) Solve. Collect like terms first.6x+4-2x=-3x+186x+3-9x=-2+12x+20-2z-3+5z=6-z-94) Solve.3x-15=4w-25=8514x-23=245x-13=12x32-4x=-54+x23w2+1=w+9214y+35=34y-1525+32x-15=12x4x-3=-23x+1=2x31-x=-2(2-x)4t+1=2t+(1-t)24+25x+3=3(x-4)4w-5-w=-9-w+7-9r+3-9r=-3r-3-r+856w-3-7=17+3w-5(2w+1)Solve and graph.3543300908055x-17<19-4x35433005842012y-4≤3y+14351472510287073-2z≤-27+2z351472511874534y+13≥12y+1435147251117602(3a-7)>4(-4+2a) ................
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