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Midterm Review Math 2 Topics Equations Inequalities Lines Systems of Linear Equations in Two Variables Factoring Laws of Exponents Functions Equations 1. You may add or subtract the same expression on both sides of an equation. 2. You may multiply or divide both sides of an equation by the same non-zero expression. 3. You may replace any expression by an equivalent expression. Linear Absolute Value Polynomial Inequalities 1. 2. 3. 4. You may add or subtract the same expression on both sides of an equation. You may multiply or divide both sides of an equation by the same positive expression. You may multiply or divide both sides of an equation by the same negative expression if you switch the sense of the inequality. You may replace any expression by an equivalent expression. Interval Notation and Graphing Solution Set Types of Inequalities Lines General linear equation: ax + by = c Slope-intercept equation: y = mx + b Point-slope equation: y-y1= m(x – x1) rise y2 y1 Slope: m run x 2 x1 Parallel lines have the same slope. Perpendicular lines have slopes that are opposite reciprocals. Vertical lines have no slope. Systems of Equations Two equations in two Variables • Represent two lines • Solution represents the intersection of the lines • May have 0, 1, or infinitely many ordered pairs in the solution set Three equations in Three Variables • Represents three planes • Solve by reducing to a system of two equations in two variables. Factoring Greatest Common Factor Special Binomial Forms Trinomial Forms Polynomials with Four Terms Laws of Exponents 1. 2. 3. 4. 5. am an = am+n am /an = am-n (am)n = amn (ab)m = am bm (a/b)m = am / bm Definitions: Let a 0 then a0 = 1 and a-n = 1/an Functions A rule that assigns to each element of the domain a unique element of the range. A set of ordered pairs (x,y) such that each x corresponds to one and only one y. The graph of a function intersects any vertical line in at most one point. f(x) notation. Types of Inequalities 1. Linear: Isolate the variable 2. Intersection or overlap when the inequalities are joined by “and” 3. Union, include both sets when the inequalities are joined by “or” 4. Absolute Value: Two inequalities using “and” or “or” Linear equation 3(x + 5) – 7 = 4x – (x+3) 3x + 15 – 7 = 4x – x – 3 3x + 8 = 3x – 3 8=–3 No Solution Absolute Value Equation 3 |2x – 5| +7 = 28 3 |2x – 5| = 21 |2x – 5| = 7 Interpret: 2x – 5 = 7 or 2x – 5 = – 7 2x = 12 or 2x = – 2 x=6 or x = – 1 {– 1, 6} Polynomial Equation 2x2 + x = 10 2x2 + x – 10 = 0 (2x + 5)(x – 2) = 0 2x + 5 = 0 or x – 2 = 0 2x = – 5 or x = 2 x = – 5/2 or x = 2 {– 5/2, 2} Linear Inequalities • • • • • 3(x + 2) < 2(4x – 7) distribute 3x + 6 < 8x –14 subtract 8x and 6 -5x < -20 divide by –5 and flip x>4 solution (4, ) interval notation and graph 0 4 Absolute Value Inequality “and” • • • • • • |3x – 5| 10 3x – 5 10 and 3x – 5 -10 Can be written –10 3x – 5 10 -5 3x 15 -5/3 x 5 [-5/3, 5] -5/3 0 5 Absolute Value Inequality “or” • • • • • |2x – 5| > 3 2x – 5 > 3 or 2x – 5 < -3 2x > 8 or 2x < 2 x>4 or x < 1 Interval notation: (,1) (8, ) 0 1 4 Interval Notation and Graphing • Interval Notation: (-3,5] • Graph: -3 5 • < or > don’t include the value, parenthesis use open circle on graph • or include the value, square bracket use closed circle on graph • Always use parentheses with infinity (, ) or (, 4]or (3, ) Solving Systems of Equations Substitution Elimination Cramer’s Rule Substitution 1. Solve one of the equations for one of the variables. Avoid introducing fractions if possible. 2. Substitute this expression into the other equation. 3. Solve this equation in a single variable. 4. Use the substitution equation to find the value of the other variable in the ordered pair. 5. Example Elimination 1. Multiply one or both equations by the appropriate numbers so that one of the variables has coefficients that are opposites. 2. Add the equations together to eliminate one variable. 3. Solve this equation in a single variable. 4. Use either equation to find the other value of the variable in the ordered pair. 5. Example Cramer’s Rule The value for each variable is the ratio of two determinants. x = Dx / D y = Dy / D If D = 0 and Dx = 0 the system is dependent and there are infinitely many solutions. If D = 0 and Dx 0 the system is inconsistent and there are no solutions. Example of Substitution The system: 1. 2. 3. 4. 5. 2x + y = 7 3x + 5y = 7 Solve for y in first equation: y = 7 – 2x Substitute: 3x + 5 =7 Solve this equation: 3x + 35 – 10x = 7 -7x + 35 = 7 -7x = -28 x=4 Use the substitution equation: y = 7 – 2 Solution: (4, -1) = -1 Example of Elimination The system: 2x + 3y = 13 3x + 2y = 12 1. -2(2x + 3y) = 13(-2) -4x – 6y = -26 3(3x + 2y) = 12(3) 9x + 6y = 36 5x = 10 2. x = 2 2(2) + 3y =13 4 + 3y = 13 3y = 9 so y = 3 3. Solution: (2,3) Determinants The determinant is a number that we get from four numbers arranged in two rows with two numbers in each row. Given the system: ax + by =c dx + ey =f a b c b a c D , Dx and Dy d e f e d f Example Example of Cramer’s Rule 3x +4y = 2 5x – 2y = 7 D = 3(-2) – 5(4) = -6 – 20 = -26 Dx = 2(-2) – 7(4) = -4 – 28 = -32 Dy = 3(7) – 5(2) = 21 – 10 = 11 x = -32/-26 = 16/13 y = 11/-26 Greatest Common Factor For each variable look at the value of its exponent in each term. Use the lowest value. Binomials a2 – b2 =(a + b)(a – b) a3 – b3 =(a – b)(a2 + ab + b2) a3 + b3 =(a + b)(a2 – ab + b2) CAUTION: THESE DO NOT FACTOR a2 + b 2 a2 + ab + b2 a2 – ab + b2 Trinomials x2 + 2xy + y2 = (x + y)2 x2 – 2xy + y2 = (x – y)2 x2 + bx + c = (x + r)(x + s) Where c = rs and b = r + s ax2 + bx + c = ax2 + rx + sx + c then group Where ac = rs and b = r + s Example Example of ac Method 18x2 – 9x – 20 Find factors of ac = -360 whose sum is – 9 They are 15 and –24 18x2 + 15x – 24x – 20 (18x2 + 15x) – (24x + 20) 3x(6x + 5) – 4(6x + 5) (3x – 4)(6x + 5) Tetranomial ab +2b – 3a – 6 (ab +2b) – (3a + 6) b(a + 2) – 3(a + 2) (b – 3)(a + 2) Domain The set of all inputs to the function. Since this variable is usually called x, the domain is the set of all values for x. Range The set of all outputs of the function. Since this variable is usually called y, the range is the set of all values for y. f(x) Notation f(x) = 3x – 7 x is the input and f(x) is the output. 3x – 7 is the rule. Input 2 f(2) = 3(2) – 7 = –1 is output. (2, –1) corresponds to a point on the graph of this linear function. More Graph of a Function The set of all points corresponding to the ordered pairs (x,f(x)) is the graph of the function, i.e. let y = f(x). If a vertical line intersects the graph in more than one point then it is not the graph of function. More Function Notation Let f(x) = x2 – 2x f( ) = ( )2 – 2( ) Find f(x – 3) f(x – 3) = (x – 3)2 – 2(x – 3) = x2 – 6x + 9 – 2x + 6 = x2 – 8x + 15 The End Relax. Remember your Picture ID. Sleep well the night before the exam. And most important of all… relax.