ࡱ> ` bjbj .BDDDDdDlJ*vEvEvEvEvEvEvEvEIIIIIII$Kh3NIivEvEvEvEvEIvEvE[JLILILIvEvEvEILIvEILILILIvEjE Wkb6DpFLII$qJ0JLINI(NLINLI\vEvELIvEvEvEvEvEIIBI vEvEvEJvEvEvEvE(V-DV- Do all sections in NB: All work must be shown for sections I-IV!!! Section I: Solving Equations 1) 20 - (5/8)x = 40 2) 6(7x - 2) = 8(4x + 1) 3) 2(5x - 4) - 3(4x + 3) = -43 4) x2+ 44 = 15x 5) 3x2+ 18x = 81 6) 3x2 = 5x + 5 Section II: Factoring 7) x2 676 8) 4x2 12x 112 9) 6x3 + 72x2 + 216x 10) 18x2 9x 5 11) 18x2 + 15x 12 Section III: Absolute Value 12) 3|9x 6| - 11 = 88 13) |4x + 8| + 13 = 37 14) 3|7x + 14| - 4 _ = 16 15) 4|12x 18| - 11 = 29 5 3 16) 5|6x - 3| + 7 < 82 17) - | 5x + 10 | - 41 > -56 18) 2 | 3x - 12 | + 4> 2 19) 2|7x + 14 | - 14 > -7 11 10 Section IV: Systems (must be checked in both original equations) 20) x = 4- 3y 21) 5x + 3y = -7 22) y = x2 13x + 22 7x + 10y = -5 7x + 2y = -34 y = x - 18 Simplifying and Combining Like Terms Exponent Coefficient 4x2 Variable (or Base) * Write the coefficients, variables, and exponents of: a) 8c2 b) 9x c) y8 d) 12a2b3 Like Terms: Terms that have identical variable parts {same variable(s) and same exponent(s)} When simplifying using addition and subtraction, combine like terms by keeping the "like term" and adding or subtracting the numerical coefficients. Examples: 3x + 4x = 7x 13xy 9xy = 4xy 12x3y2 - 5x3y2 = 7x3y2 Why cant you simplify? 4x3 + 4y3 11x2 7x 6x3y + 5xy3 Simplify: 1) 7x + 5 3x 2) 6w2 + 11w + 8w2 15w 3) (6x + 4) + (15 7x) 4) (12x 5) (7x 11) 5) (2x2 - 3x + 7) (-3x2 + 4x 7) 6) 11a2b 12ab2 WORKING WITH THE DISTRIBUTIVE PROPERTY Example: 3(2x 5) + 5(3x +6) = Since in the order of operations, multiplication comes before addition and subtraction, we must get rid of the multiplication before you can combine like terms. We do this by using the distributive property: 3(2x 5) + 5(3x +6) = 3(2x) 3(5) + 5(3x) + 5(6) = 6x - 15 + 15x + 30 = Now you can combine the like terms: 6x + 15x = 21x -15 + 30 = 15 Final answer: 21x + 15 Solving Equations Golden Rule of Algebra: Do unto one side of the equal sign as you will do to the other Whatever you do on one side of the equal sign, you MUST do the same exact thing on the other side. If you multiply by -2 on the left side, you have to multiply by -2 on the other. If you subtract 15 from one side, you must subtract 15 from the other. You can do whatever you want (to get the x by itself) as long as you do it on both sides of the equal sign. Solving Single Step Equations: To solve single step equations, you do the opposite of whatever the operation is. The opposite of addition is subtraction and the opposite of multiplication is division. Solve for x: 1) x + 5 = 12 2) x 11 = 19 3) 22 x = 17 4) 5x = -30 5) (x/-5) = 3 6) T! x = - 8 Solving Multi-Step Equations: 3x  5 = 22 To get the x by itself, you will need to get rid of the 5 and the 3. +5 +5 We do this by going in opposite order of PEMDAS. We get rid of addition and subtraction first. 3x = 27Then, we get rid of multiplication and division. 33 x =9 We check the answer by putting it back in the original equation: 3x - 5 = 22, x = 9 3(9) - 5 = 22 27 - 5 = 22 22 = 22 (It checks) Simple Equations: 1) 9x - 11 = -38 2) 160 = 7x + 6 3) 32 - 6x = 53 4) -4 = 42 - 4x 5) x - 11 = 16 6) 37 = 25 - (2/3)x 7) 4x 7 = -23 8) 12x + 9 = - 15 9) 21 4x = 45 10) (x/7) 4 = 4 11) (-x/5) + 3 = 7 12) 26 = 60 2x Equations with more than 1 x on the same side of the equal sign: You need to simplify (combine like terms) and then use the same steps as a multi-step equation. Example: 9x + 11 5x + 10 = -15 9x 5x = 4x and 4x + 21 = -15 Now it looks like a multistep eq. that we did in the 1st 11 + 10 = 21 -21 -21 Use subtraction to get rid of the addition. 4x = -36 4 4 Now divide to get rid of the multiplication x = -9 13) 15x - 24 - 4x = -79 14) 102 = 69 - 7x + 3x15) 3(2x - 5) - 4x = 33 16) 3(4x - 5) + 2(11 - 2x) = 43 17) 9(3x + 6) - 6(7x - 3) = 12 18) 7(4x - 5) - 4(6x + 5) = -91 19) 8(4x + 2) + 5(3x - 7) = 122 Equations with x's on BOTH sides of the equal sign: You need to "Get the X's on one side and the numbers on the other." Then you can solve. Example: 12x 11 = 7x + 9 -7x -7x Move the xs to one side. 5x 11 = 9 Now it looks like a multistep equation that we did in the 1st section. +11 +11 Add to get rid of the subtraction. 5x = 20 5 5 Now divide to get rid of the multiplication x = 4 20) 11x - 3 = 7x + 25 21) 22 - 4x = 12x + 12623) x - 12 = x -6 24) 5(2x + 4) = 4(3x + 7) 25) 12(3x + 4) = 6(7x + 2)26) 3x- 25 = 11x - 5 + 2x Solving Quadratic Equations Solving quadratic equations (equations with x2 can be done in different ways. We will use two different methods. What both methods have in common is that the equation has to be set to = 0. For instance, if the equation was x2 22 = 9x, you would have to subtract 9x from both sides of the equal sign so the equation would be x2 9x 22 = 0. Solve by factoring: After the equation is set equal to 0, you factor the trinomial. x2 9x 22 = 0 (x-11) (x+2) = 0 Now you would set each factor equal to zero and solve. Think about it, if the product of the two binomials equals zero, well then one of the factors has to be zero. x2 9x 22 = 0 (x-11) (x+2) = 0 x 11 = 0 x + 2 = 0 +11 +11 -2 -2 x = 11 or x = -2 * Check in the ORIGINAL equation! Solving Quadratics by Factoring: 20) x2 - 5x - 14 = 0 21) x2 + 11x = -30 22) x2 - 45 = 4x 23) x2 = 15x - 56 24) 3x2 + 9x = 54 25) x3 = x2 + 12x 26) 25x2 = 5x3 + 30x 27) 108x = 12x2 + 216 28) 3x2 - 2x - 8 = 2x2 29) 10x2 - 5x + 11 = 9x2 + x + 83 30) 4x2 + 3x - 12 = 6x2 - 7x - 60 Solve using the quadratic formula: When ax2 + bx + c = 0 x = -b "b2  4ac . 2a a is the coefficient of x2 b is the coefficient of x c is the number (third term) Notice the is what will give your two answers (just like you had when solving by factoring) x2  9x  22 = 0 x = -b "b2  4ac . a = 1 2a b= - 9 c = -22 x = -(-9) " (-9)2  4(1)(-22) -4(1)(-22) = 88 2(1) x = 9 "81 + 88 2 x=9"169 . 2 Split and do the+ side and - side 9 + 13 9  13 2 2 x = 11 or x = -2 * Check in the ORIGINAL equation! Solving Quadratics Using the Quadratic Formula: 31) 2x2 - 6x + 1 = 032)3x2 + 2x = 3 33) 4x2 + 2 =-7x 34) 7x2 = 3x + 235) 3x2 + 6 = 5x 36) 9x - 3 = 4x2 HOW TO FACTOR TRINOMIALS Remember your hints: A. When the last sign is addition B. When the last sign is subtraction x2 - 5x + 6 1)Both signs the same x2 + 5x 36 1) signs are different 2) Both minus (1st sign) (x - )(x - ) (x - )(x + ) 2) Factors of 36 w/ a 3) Factors of 6 w/ a sum differenceof 5 (9 of 5. (3 and 2) and 4) 3) Bigger # goes1st sign, + (x - 3)(x - 2) (x - 4)(x + 9) FOIL Check!!!!! Case II Factoring Factoring a trinomial with a coefficient for x2 other than 1 Factor: 6x2 + 5x 4 1) Look for a GCF: a. There is no GCF for this trinomial b. The only way this method works is if you take out the GCF (if there is one.) 2) Take the coefficient for x2 (6) and multiply it with the last term (4): 6x2 + 5x 4 6 * 4 = 24 x2 + 5x 24 3) Factor the new trinomial: x2 + 5x 24 (x + 8)(x 3) 4) Take the coefficient that you multiplied in the beginning (6) and put it back in the parenthesis (only with the x): (x + 8)(x 3) (6x + 8)(6x 3) 5) Find the GCF on each factor (on each set of parenthesis): (6x + 8) ( 2(3x+ 4) (6x 3) ( 3(2x 1) 6) Keep the factors left in the parenthesis: (3x + 4)(2x 1) 7) FOIL CHECK Solving Absolute Value Equations Solving absolute value equations is almost the exact same as solving regular equations with one major difference. In most cases you have 2 solutions. Example: | x | = 5 We know that when x = 5, | 5 | will also equal 5, but it is also true that | -5 | will equal 5. So, for |x | = 5, x = {-5, 5}. They both work. How to solve absolute value equations 1) Isolate the absolute value. 2) Split into two separate equations, setting one to the negative and one to the positive. Example: | 2x + 6 | - 3 = 13 1) Isolate the absolute value: ** The steps are the same as if you were getting the x by itself. You move away all other numbers by doing the opposite operation:** | 2x + 6 | - 3 = 13 +3 +3 | 2x + 6 | = 16 2) Now split into two separate equations and 2x + 6 = -16 2x + 6 = 16 solve each. - 6 - 6 -6 -6 2x = -22 2x = 10 2 2 2 2 x = -11 x = 5 3) Check by substituting in the original equation. Absolute Value Inequalities Solving absolute value inequalities combine the strategies you used in: 1) Solving and Graphing Compound Inequalities 2) Solving Absolute Value Equations Every absolute value inequality is a compound inequality. The 2 separate inequalities come from when you split the inequality once the absolute value is isolated. So first, you isolate the absolute value following all the same steps as you did when isolating the absolute value when solving an absolute value equation. Then you split and set to the negative and the positive. The new stuff: 1) Once the absolute value is isolated, you choose whether it will be an AND problem or an OR problem. - Greater than (> , > ) will be an OR problem. (Graph both and keep) - Less than (<, < ) will be an AND problem. (Graph both and keep the intersection) 2) When you split the absolute value YOU MUST TURN THE INEQUALITY AROUND WHEN SETTING TO THE NEGATIVE. Example: | 2x 3 | - 10 > -5 Isolate the abs. val.: +10 +10 | 2x 3 | > 5 Split and turn the ineq. around (to less than) when setting to -5 and 2x 3 < -5 2x 3 > 5 solve each ineq: +3 +3 +3 +3 2x < -2 2x > 8 2 2 2 2 Since its a Greater than, x < - 1 OR x > 4 it is an OR problem and will be graphed accordingly. Solving Systems of Equations In order to solve for two variables, you need to have two equations. If you only have one equation there are an infinite amount of ordered pairs (x,y) that will work. For example: 4x 2y = 16 you can have x = 4 and y = 0 (4,0) and (2, -2) and (0, -4) and an infinite amount of others. To be able to solve for a single ordered pair, you need a second equation. When we introduce the second equation, we will be able to solve for a single ordered pair that will work in both equations. There are two ways to solve a system of equations (algebraically and graphically). We will focus on solving algebraically. There are two methods of solving algebraically (substitution and elimination). The key to both of them is changing one (or both) equations so there is only one variable to solve for. Then you follow all the rules of solving for the one variable. Then plug the value back into one of the original equations to find the value of the second variable. Always state your answer as an ordered pair. SUBSTITUTION Example: x = 3y + 8 5x + 2y = 6 5(3y + 8) + 2y = 6 Substitute 3y +8 for the x in the2nd equation 15y + 40 + 2y = 6 Distribute and solve. 17y + 40 = 6 17y = -34 y = -2 x = 3(-2) + 8 substitute the value for y back in to find x. x= -6 + 8 x = 2 (2, -2) Check in BOTH ORIGINAL EQUATIONS! Solve each system and check (in both equations): a) x = 2y + 1 b) y = 3x + 4 c) 5x y = 7 5x 6y = 13 9x + 2y = -37 4x + 2y = 28 d) x + 3y = 11 e) 7x + 9y =-74 f) 10x y = 1 6x 5y = 20 4x + y = -5 8x + 3y = -8 Solving Systems with Linear Combinations (Elimination): Sometimes solving a system of equations using substitution can be very difficult. For these problems we solve using Linear Combinations (or Elimination). With elimination you solve by eliminating one of the variables. This is accomplished by adding the 2 equations together. Before you can add the equations together, you need one of the two variables to have two things: 1) Same Coefficient 2) Different Signs (one positive and one negative) When you add terms with the same coefficient and different signs, the term drops out. You then solve for the variable that is left. After you have solved for one variable, you plug the value into one of the original equations and solve for the 2nd variable (just like Substitution). Then, you check the solution in both original equations. The only difference between Substitution and Elimination is how you solve for the 1st variable. After that they are the same. Examples: A) Sometimes it works out that the 2 equations already have a variable with the same coefficient and different signs. You can then just add the equations: 3x + 4y = 10 (The +4y and -4y cancel out Plug x = -6 in: 5x 4y = -58 leaving you with just 8x.) 3(-6) + 4y = 10 8x = -48 -18 + 4y = 10 8 8 +18 +18 4y = 28 x = -6 4 4 y = 7 Final Solution: (-6, 7) CHECK IN BOTH!!!! B) Sometimes (usually) the equations do not have same coefficient and different signs, so we have a little bit of manipulating to do. 3x + 8y = 25 With this system, nothing will drop out if we just add the 5x + 4y = 23 equations. So we will multiply the bottom one by (-2). 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C. Sometimes we need to manipulate both equations. We can do this by criss crossing the coefficients. 6x + 7y = 11 This is different than Example B, because no coeffcient 5x 6y = -50 goes into another evenly. -5(6x + 7y = 11) You need the negative sign to change the 6x to negative 6(5x 6y = -50) so the signs will be different. You can also use 5 and -6. You can also criss cross the y coefficients. -30x 35y = -55 30x 36y = -300 Plug in y = 5 - 71y = -355 5x 6(5) = -50 -71 -71 5x 30 = -50 +30 +30 5x = -20 y = 5 5 5 x = -4 Final Solution: (-4, 5) CHECK IN BOTH!!!! K||||||||| }"}'}(}[}}|\ hY5>* hY>*hYCJaJhYU}:~d~~~Hegy HXYgdY,1h/ =!"#$% @`@ YNormalCJ_HaJmH sH tH B`B Y Heading 1$$@&a$>*CJ DA@D Default Paragraph FontRi@R  Table Normal4 l4a (k@(No ListB^`B Y Normal (Web)dd[$\$0>`0 YTitle$a$CJ,4 `4 YFooter  !*W`!* YStrong5\B DSz{| WX%]5XYOPZ"#$stv )*BCUVno  < ( W X v ^  % 3 ? 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