Nae Period ALGEBRA II Chapter B ad A Notes Solvig Iequalities ad Absolute Value / Nubers ad Fuctios SECTION.6 Itroductio to Solvig Equatios Objectives: Write ad solve a liear equatio i oe variable. Solve a literal equatio for a specified variable. A equatio is a stateet that two expressios are equal. To solve equatios, the Properties of Equality, listed below, ay be used. PROPERTIES OF EQUALITY Reflexive Property a = a Subtractio Property If a = b, the a c = b - c Syetric Property If a = b, the b = a Multiplicatio Property If a = b, the ac = bc Trasitive Property If a = b, ad b = c, the a = c Divisio Property If a = b, the a/c = b/c, b 0 Additio Property If a = b, the a + c = b + c Substitutio Property If a = b, the a ca be placed with b. Ters are separated by additio ad subtractio sigs. Like ters have the sae for of the variable x. Ex. Solve each equatio. a. 5x + 5 = x b, ⅓ x = x + c. x x 6 You ca solve equatios by graphig. Set y = to the left side of the equatio ad y = to the right side of equatio ad where the two graphs itercept, the x-value of that coordiate is the solutio. Ex. Solve each equatio by graphig. Give your aswers to the earest hudredth ( decial places). a. 0.x +. =.56 x.5 b. 0.75x +. =.6 Literal equatios You just rearrage the equatio to solve for aother variable. For exaple, the distace forula d = rt is solved for d (distace). Now solve it for t (tie).
Ex. Solve each literal equatio for the idicated variable. a. P = l + w for w b. I = P( + rt) for t c. A hb b for h d. ax + b = cx + d for x Ex. Aelia has a job baby-sittig for a eighbor. She is paid $0 plus $.50 for each hour o the job. If Aelia wats to ear $0 to buy a ew sweater, how ay hours would she eed to work? Ex 5 Terry has two differet jobs i sellig isurace. Oe job pays hi $75 per week plus $5 for each policy sold. The other pays hi $5 per week plus $8 for each policy sold. How ay policies would Terry have to sell to ake the sae total salary i either job? Review scietific otatio: Soe ubers are very large or very sall. If we covert the ito scietific otatio, they becoe easy to apply basic ath applicatios. A uber writte i scietific otatio is o the for c 0, where c 0 ad is a iteger. May calculators autoatically switch to scietific otatio to display large or sall ubers. Try ultiplyig 980,000,000 by 80. Most calculators will display the aswer i scietific otatio. You see which eas Ex. 6 Write i decial otatio. 7 6 a.. 0 b. 6.00 0 Ex. 7 Write each uber i scietific otatio. a. 0.000855 b. 75,000,000
SECTION.7 Itroductio to Solvig Iequalities Objectives: o Write, solve, ad graph liear iequalities i oe variable. o Solve ad graph copoud liear iequalities i oe variable. There are Properties of Iequality to help solve iequalities. They are listed as follows: Properties of Iequality For all real ubers a, b, ad c, where a * b: Additio Property a + c b + c Subtractio Property a c b c Multiplicatio Property If a b, ad c 0, the ac bc. (c is a positive uber). If a b, ad c 0, the ac bc. (c is a egative uber). a b Divisio Property If a b, ad c 0, the. c c a b If a b, ad c 0, the. c c *, >, or < ca also be used. Ay value of a variable that akes a iequality true is called a There are usually sets of ubers. Oe exaple is, x <. The solutios ca also be show o a graph. Ex. Solve each iequality, ad graph the solutio o a uber lie. a. 5x > 0 b. 5x > 0 c. x + 7 < d. 0 x e. (x 5) < 8x + f. (x ) < 5(x + ) Ex. Use a graphig calculator to solve for x. x < A is a pair of iequalities joied by ad or or. A atheatical ad eas that the solutios have to be i both equatios. A atheatical or eas that the solutios ca be i oe or the other or i both. The solutio of a ad copoud iequality usually looks like ad ca be cobied as a < x < b. The solutio of a or copoud iequality usually looks like
Ex. Graph each copoud iequality o a uber lie. a. x > ad x < b. x > ad x > c. x > or x < d. x > or x > e. x < ad x < f. x < ad x > g. x < or x < h. x < or x > Ex. Graph the solutio of each copoud iequality o a uber lie ad write the solutio i set otatio. a. 9x 8 ad (x+ 6) b. 6x < or (x + ) 9 c. x 7 < 5x + 8 or ½(6 x) 0 SECTION.8 Solvig Absolute-Value Equatios ad Iequalities Objective: Write, solve, ad graph absolute-value equatios ad iequalities i atheatical ad real-world situatios. Geoetric defiitio of the absolute value of x is Algebraic defiitio: Let x be ay real uber. The absolute value of x deoted by x, is give by the followig: If x is a positive uber, x 0, the x = x. If x is a egative uber, x 0, the x = x. Absolute-Value Equatios (=) If a 0 ad x a, the x a or x a. Sice x is distace, it is always a positive uber. The solutio to the x-value iside the absolute-value, however, ca be egative.
Ex. Solve each equatio. Note that these equatios cotai the = sig. a. x 5 b. x 8 c. x 5 Ex. Check the above equatios with a graphic calculator. (Set y = x 5 ad y =. x=the itersectios.) a. x 5 * b. x 8 * c. x 5 You ca see that there are o solutios for the equatio x =. Absolute-Value Iequalities If a 0, ad x a, the a x a. ( ad) If a 0, ad x a, the x a or x a. The sig < ca be replaced with or the sig > ca be replaced with. Ex. Solve each iequality. Graph the solutio o a uber lie. If the equatio has o solutio, write o solutio. a. x 5 7 b. x 5 5 c. x 5 d. 9x e. x Ex. Match each stateet o the left with a stateet or setece o the right.. x + = a. x < ad x > 6. x + < b. x = or x = 6. x + < c. x > or x < 6. x + > d. There is o solutio 5. x + > e. The solutio is all real ubers. 6. x + = f. Noe of the above 5
Objectives: Idetify ad use Properties of Real Nubers. CHAPTER Nubers ad Fuctios SECTION. Operatios with Nubers Evaluate expressios by usig the order of operatios. Nuber Sets Natural ubers,,,... Whole ubers 0,,,,... Itegers...,,,, 0,,,,... Ratioal ubers p, where p ad q are itegers ad q 0 q Irratioal ubers ubers whose decial part does ot repeat or teriate Real ubers all ratioal ad irratioal ubers Ex. Draw a Ve Diagra that show the relatioship betwee the various uber sets. Ex. Classify each uber i as ay ways as possible. a. 0 b. 5 c.. d. 0 e. 6 For every real-uber there is a correspodig poit o the uber lie. Likewise, for every poit o the uber lie, you ca assig a real-uber coordiate. Ex. Graph each pair of ubers o a uber lie. a..5 ad b. / ad 7 c. 7 ad 7 Soe of the ost iportat properties are used with the real-ubers. They are fudaetal i the operatios of additio ad ultiplicatio i the real-ubers. These two operatios are liked by the Distributive Property. Properties of Additio ad Multiplicatio For all real ubers a, b, ad c: Additio Multiplicatio Closure a + b is a real uber. ab is a real uber. Coutative a +b = b + a ab = ba Associative (a + b) + c = a + (b + c) (ab)c = a(bc) Idetity There is a uber 0, s. t. There is a uber such that a + 0 = a, ad 0 + a = a. a = a ad a = a. Iverse For every real uber a, For every ozero real uber a, there is a real uber a there is a real uber /a, such that s. t. a + a = 0. a (/a) =. Distributive a(b+c)= ab + ac ad (b + c)a = ba + ca Ex. State the property that is illustrated i each stateet. All variables represet real ubers. 6
a.. + x = x +. b. 0 = x + ( x) c. 7 = 7 d. (x ) = x Whe a expressio ivolves oly ubers ad operatios, you ca evaluate the expressio by usig the: Order of Operatios (PEMDAS) or (Please excuse y dear Aut Sally). P Perfor operatios with the ierost groupig sybols usig the order of operatios.. E Perfor operatios idicated by expoets (powers).. M D Perfor ultiplicatio ad divisio i order fro left to right.. A S Perfor additio ad subtractio i order fro left to right. Ex. 5 Evaluate each expressio by usig the order of operatios. ( a. 5² + 6 b. ( ) (5 )² 6 + c. ) ( ) d. 5 Objectives: Evaluate expressios ivolvig expoets. Siplify expressios ivolvig expoets. SECTION. Properties of Expoets a is called a power of a. a is, called the ad is called the Properties of Iteger Expoets Let a be a real uber. If is a atural uber, the a = a a a a a, ties. *If a is ozero, the a 0. If is a atural uber, the a, also a. a a o 0 *I the expressio a, a ust be ozero because 0 is udefied. Ex. Evaluate each expressio. a. 0 7 b. ³ c. 5 d. 8 5 e. 0 7 Product of Powers a a a Power of a Power a Power of a Quotiet Properties of Expoets Let a ad b be ozero real ubers. Let ad be itegers. Quotiet of Powers a a a ab a b a Power of a Product a b a b Ex. Evaluate each expressio. 7
a. ³² b. 5 c. 5 d. e. A expressio with ratioal (fractios) expoets ca be represeted i a equivalet for that ivolves the radical sybol. Defiitio of Ratioal Expoets For all positive real ubers a: Ex. Evaluate each expressio. a. If is a ozero iteger, the a a If ad are itegers ad 0, the a a a a. 6 b. 7 c. 6 d. 5 5 e. 9 Ex. Check your aswers for Ex. probles with the calculator. You do eed to show your work for full credit. Ex. 5 Siplify each expressio, assuig that o variable equals zero. Write your aswer with positive expoets oly. a. z 5 z b. a b 8 c. xy(x y ) d. x y 5 x y e. a b 5 b a 8