7. Algebra I.pdf

NUMBER SYSTEMS The natural numbers are the numbers we count with: The real numbers can be represented as points on the number 1, 2. 3, 4, 5, 6, , 27, 28,.....

NUMBER SYSTEMS The natural numbers are the numbers we count with: The real numbers can be represented as points on the number 1, 2. 3, 4, 5, 6, , 27, 28,.. line. All rational numbers aTe real, but the real number line has Real numbers., many points that are "between" rational numbers and aTe called The whole numbers are the numbers we count with and zero: -! 0, 1, 2, 3, 4, 5, 6,. ilTotional. Ex: 0, To, v'3 - 9, 0.12112111211112. 1.25 2 3 etc. The imaginary numbers are square roots of negative numbers. Irrationals -1, -2, -3,.. The integers are the numbers we count with, their negatives. They don't appear on the real number line and are written in v'5~ ~~ Integers and zero:... , -3, -2, -1, 0, I, 2, 3,. terms of i = FI. Ex: J=49 is imaginary and equal to iV49 or 7i. )'t2+3 - - Whole Dwnbers -The positive integers are the natural numbers. -The negotfve integers are the "minus" natural numbers: The complex numbers are all possible sums of real and etc. Natural numbers -1, -2. -3, -4,. imaginary numbers; they are written as a + bi. where a and bare real and i = FI is imaginary. All reals are complex (with The rcmonal numbers are all numbers expressible as ~ b = 0) and all imaginary numbers are complex (with a = 0). fractions. The fractions may be proper (less than one; Ex: k) or Venn Diagram of Number Systems The Fundamental Theorem of Algebra says that every improper (more than one; Ex: ft). Rational numbers can be polynomial of degree n has exactly n complex roots (counting positive (Ex: 5.125 = ¥-) or negative (Ex: - ~). All integers are multiple roots). rational: Ex: 4 = y. I SETS A set Is ony collecllon-finite or infinite-- Usually, move variables to the side that had the larger variable N-o~ut j" 9 = 3 and 7(X + 4) = 2 and Vi = 5 ore not linear 8~O:::l g coefficient to begin with. Equation should look like a::1: = b. Use the same procedure as for equalities, except flip the N ~.!!! g) :c in one vW'iable will always have (a) exactly Linear eCJlUlh'ons Add 8x to both sides to get 5x + 8x - 18 = 77 or inequality when mliltiplying or dividing by a negative @(J)~Oaa u one real number solution, (b) rlO solutions, or (c) all real 13x-18=77. number. Ex: -x > 5 is equivalent to x < -5. ~;~~ ~ c 'C"'E,~ 0 "0 numbers as solutions. Add 18 to both sides to get 13x = 77 + 18 or 13x = 95. -The inequality may have no solution if it reduces to an ~.>:: (; SttS c 5. Divide both sides by the variable's coefficient. Stop if a = U. impossible statement. Ex: :£ + 1 > x + 9 reduces to 1 > 9. 84 Sto c( ~ FINDING A UNIQUE SOLUTION Divide by Ij to get x = ~. -The inequality may have all real numbers as solutions if it Ex: J(-t -3) TT=9-(x- ~). 6. Check the solution by plugging into the original equation. reduces to a statement that is always true. Ex: 5 - x 2: 3 - x Does ~ (2.~~ - 3) + ~ = 9 - (~- ~)? Yes! Hooray. reduces to 5 ~ 3 and has infintely many solutions. 1. Get rid of fractions outside parentheses. Solutions given the reduced Inequality and the condition: Multiply through by the LCM of the denominators. DETERMINING IF A UNIQUE a>O aO b b means" < -b or " > b. If there are infinitely many solutions or no Solve the three equntions lOx + I = 7I + 3. solutions, check two numbers of large Ex: IIOI + II < 7x + 3 is eqUivalent to -lOx-l=7x+3. and 7x+3=0 to find If b < 0, then magnitude. positive and negative. 7.e + 3 ~ 0 and -7x - 3 < lOx + 1 < 7x + 3. potentinl boundary points. The three points are. lal ::; b is impossible. -Be especially careful if the equation contains Thus the equations 7x + 3 ~ 0, not surprisingly. ~. - At, and - ~. Testing the 1"1 > b means a could be anything. three boundary pOints and a point from each of variables both inside and outside the -7x - 3 < lOx + I. and IOI + I < 7x + 3 the four intervals gives the solution - ~ $ x < ~. absolute value bars. Keep track of which must all hold. SolVing the equations. we see that GRAPHING ON THE REAL.: NUMBER LINE.. The real number line is a pictorial x ::; a: Shaded closed ray: everything to the GRAPHING ABSOLUTE OTHER COMPOUNO representation of the real numbers: every left of and including a. VALUE STATEMENTS INEQUALITIES number corresponds to a point. Solutions to one· o a Absolute value is distance. Ix - al = b means Intersection: Inequalities joined by AND. Both vorioble equations and (especiallyi inequalities I I that lhe distance between a and x Is b. (or all) of the inequalities must true. may be graphed on the real number line. The x > '" Shaded open ray: everything to the Thraughout. b must be non-neg olive. Ex: Ix - II < 4 is really x > -3 AND x < 5. idea is to shade in those ports of the line that right of (and not including) a. An open circle Ix - al = b: The distance from a to x is b. Plot Equivalently, it is {x: x > -3} n {x: x < 5}. represent solutions. around the point a represents the not· two points: x = a + b, and x = 0 - b. The graph is the intersection of the graphs of Origin: A special point representing O. By included endpoint. a-b 0 a a+b both inequalities. Shade the portions that I I I would be shaded by both if graphed convention, points to the left of the origin represent negative numbers, and points to the oI a¢ b b independently. right of the origin represent positive Union: Inequalities joined by OR. At least one numbers. x < a: Shaded open ray: everything to the left Ix - al < b; I.e - "I ::; b: of the inequalities must be true. Ray: A half-line; everything to the left or the of (and not including) a. Open circle around a. Ex: Ix - 11 > 4 is really x < -3 OR x > 5. The distance from a to x is less than (no right of a given point. The endpoint may Equivalently, it is{x, x < -3} U {x : x > 5}. oa more than) b; or x is closer than b to a. The graph is the union of the graphs of the or may not be included. I ; Plot the interval (open or closed) individual inequalities. Shade the portions Interval: A piece of the line; everything a - b < x < ,,+ b (or" - b::; x ::; a + b.) x '# a: Everything is shaded except for a.. that would be shaded by either one (or both) between two endpoints. which mayor around which there is an open circle. if graphed independently. may not be included. a-b 0 a a+b -Endpoints may disappear. Ex: x > 5 OR Open (ray or interval): Endpoints not o I a : Q '­.I I : x ~ 6 just means that x > 5. The point 6 is included. b b no longer an endpoint. Closed (ray or interval): Endpoints included. a