Midterm notes - MIDTERM NOTES Solve Linear Equations 0...

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Unformatted text preview: MIDTERM NOTES Solve Linear Equations 0. Cancel all denominators by multiplying every term by the LCD. 1. Simplify LHS and RHS. 2. Eliminate variable term on RHS. 3. Eliminate scalar term on LHS. 4. Eliminate the coefficient of the variable. Solve Absolute Value Equations 1. Isolate the absolute value. 2. Get rid of absolute value: IA]=B —> A=B or A=—B 3. Solve. Solve Linear Inequalities Cancel all denominators by multiplying every term by the LCD. Simplify LHS and RHS. Eliminate variable term on RHS. Eliminate scalar term on LHS. Eliminate the coefficient of the variable. PWNF‘Q Solve Compound Inequalities 1. Solve each inequality. 2. Graph the final solutions according to the appropriate set operation. Quay/«Ler Lxukumhaa la AA Solve Absolute Value Inequalities 9; 1. Isolate the absolute value. 2. Get rid of absolute value: [A] < B —> —B < A < B |A| g B —-+ —B 5 A g B IAI>B -—> A<—B or A>B IAIZB -—> AS—B 0r AZB 3. Solve the resulting compound inequalities. 2 Functions Function: one input, one output Domain: set of inputs Range: set of outputs Set representation: Function: no repeated r—coordinates Domain: set of atocoordinates Range: set of y—coordinates Evaluation: (input, output) Graph representation: Function: Vertical line test Domain: set of :c-coordinates Range: set of y—coordinates Evaluation: (input, output) Function as a formula: Function: f = an expression in 3: Evaluation: output 2 f(input) Lines Two points determine a line (to graph the line). Find two points: ac—intercept: Plug in y = 0 and solve for at. y—intercept: Plug in .r = 0 and solve for 3;. Given slope, m, and a point: Use the slope to find the second point. 112—311 $2*$1 Slope formula: m = Slope and a point determine a line (to find an equation of the line). Standard form: A$ + By = C SIOpe from standard form: m = _—I: Point-slope form: y — y1 = m(:c — $1) Slope—intercept form: y = ma: + b slope = m and y—intercept: (O,b) Parallel lines: same slope Perpendicular lines: opposite reciprocal slopes Graph Linear Inequalities 1. Graph the boundary line: < or >: dashed line S or 2: solid line 2. Pick a test point off the line and plug into the inequality: If True, shade the region containing the test point. If False, shade the other region. Systems of Equations Two variables: Solve system of linear equations by graphing: 1. Graph the lines on a single Cartesian plane. 2. Solution is the intersection point. Solve system of linear equations by substitution: 1. Pick an equation7 and solve for a variable (solve for a variable with coefficient = 21:1 if possible). 2. Substitute the result of (1) into the other equation. 3. Solve. 4. Back substitute the result of (3) into (1) to solve for the remaining variable. Solve system of linear equations by elimination: 0. Put both equations in standard form. 1. Pick a variable to eliminate, and multiply each equation by an appropriate factor to make the coefficients of the variable, +LCAI and ——LCM. 2. Add the two equations vertically. 3. Solve. 4. Back substitute the result of (3) to solve for the remaining variable. Three variables: 4 Solve system of linear equations by matrices (Gaussian elimination). Objective: Row-echelon form: 1 >k * 0 l * Row operations: 1. To make 1’s: Switch two rows. 2. To make 1’s: Divide/ multiply a row by a non—zero constant. 3. To make 0’s: Add a multiple of a row to another row (Only the row that you add onto changes). OOH Oi—Ax- h—‘** *** Graph System of Linear Inequalities 1. For each inequality, graph the boundary line: < or >: dashed line 3 or 2: solid line 2. For each inequality7 pick a test point off the line and plug into the inequality: If True, shade the region containing the test point. If False. shade the other region. 3. The final solution is the intersection of the above shadings. Law of Exponents O. b0=1forb#0 1- bm _ bn : bm+n bm _ 2. W = bm " 3. (13171)" = bmn Multiply Polynomials 1. Distribute. 2. Combine like terms. Factor by Grouping 1. Group the first two terms and last two terms. 2. Factor GCF from each group (the binomial inside the parenthesis must be the same). 3. Factor out the common binomial. Factor Trinomial: Reverse Process of FOIL 1. Consider the last sign: + —+ 01 terms have same sign. — —-> OI terms have opposite signs. 2. Solve the mystery of OI by finding the appropriate factors of FL that add up to the middle coefficient. 3. Write as an appropriate product of binomials that give the proper OI terms as found in Factor Completely 1. Factor GCF. 2. Other strategies: 4 terms: Factor by grouping. 3 terms: Factor trinomial. 2 terms: Sum of squares: A2 + B2 prime Difference of squares: A2 — B2 = (A — B)(A + B) Sum of cubes: A3 + B3 = (A + B)(A2 — AB + B2) Difference of cubes: A3 — B3 = (A — B)(A2 + AB + Bg) 3. Repeat (2) until none applies. Solve Non-linear Equations 1. Set to 0. 2. Factor completely. 3. Solutions are zeros of each factor: —B :0 :— Am—l—B -—> w A ...
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