Ma Lesson 18 Section 1.7

Transcription

1 Ma Lesson 18 Section 1.7 I Representing an Ineqality There are 3 ways to represent an ineqality. (1) Using the ineqality symbol (sometime within set-bilder notation), (2) sing interval notation, and (3) sing a nmber line graph. The following table illstrates all three ways. Notice that interval notation looks like an ordered pair, sometimes with brackets. When writing the ordered pair, always write the lesser vale to the left of the greater vale. A parenthesis next to a nmber illstrates that x gets very, very close to that nmber, bt never eqals the nmber. A bracket next to a nmber means it can eqal that nmber. With or a parenthesis is always sed, since there is not an exact nmber eqal to or. Table 1.4 on page 174 of the textbook also illstrates the 3 ways to represent an ineqality where a < b. Ex 1: Write this ineqality in interval notation and graph on a nmber line. {x x > 1} 1

2 Ex 2: Write the set of nmbers represented on this nmber line as an ineqality and in interval notation. ( ] Ex 3: Write the following as an ineqality and graph on a nmber line. (,5] Examine the nmber line below ] ( In set-bilder notation, it wold be represented as { x x 6 or x > 1} and in interval notation, it wold be represented as (, 6] (1, ). II Solving a Linear Ineqality in One Variable Solving linear ineqalities is similar to solving linear eqation, with one exception. Examine the following. 5 < 10 Add 6 to both sides: 1 < 4 Tre Mltiply both sides by 2: 10 < 20 Tre Divide both by 5: 1 < 2 Tre However, try mltiplying by 2 : 10 < 20 False Divide by 5 : 1< 2 False. This leads to the following properties of Ineqalities 1) The Addition Property of Ineqality If a < b, then a + c < b + c a - c < b - c 2) The Positive Mltiplication Property of Ineqality If a < b and c is positive, then ac < bc a b < c c 3) The Negative Mltiplication Property of Ineqality If a < b and c is negative, then ac > bc a b > c c 2

3 Ex 4: Solve each ineqality. Write the soltions with the ineqality symbol, in interval notation, and graph the soltions on a nmber line. a) 3( x + 2) 4(5 x) b) 6( x 4) 3( x + 2) > c) ( a + 3) a ( a 3) It the variables drop ot of a linear ineqality, the soltion is either all real nmbers (except for any that may not be in the domain) or there is no soltion. 3( x + 2) < 3x + 7 3x + 6 < 3x < 7 The reslt above is always tre, 6 is less than 7. The soltion is { x x is a real nmber} or R or (, ). 3( x + 2) < 3x + 2 3x + 6 < 3x < 2 Six is never less than 2. The reslt is false. The soltion is or no soltion. 3

4 III Solving Compond Ineqalities When solving an ineqality sch as 12 < 3x + 3 < 2, the goal is to isolate the x in the middle. Sch an ineqality is called a compond ineqality and means the same as 12 < 3x + 3 and 3x + 3 < 2. The soltion will be the nmbers that, when sbstitted in 3x + 3, yield between 12 and 2. 4 < 2x 1< 5 Begin by adding 1 to the left, middle, and right. Example: 3 < 2x < 6 Divide the left, middle, and right by 2. 3 < x < 3 2 Any nmber between -1 ½ and 3 makes the ineqality statement tre. Ex 5: Solve each compond ineqality. Write the soltions sing the ineqality symbols, in interval notation, and graph the soltions on a nmber line. a) 1< 3x 2 < 12 b) 2 < 6 3x < 3 IV Solving Absolte Vale Ineqalities Absolte Vale Ineqalities: < c or c, if c 0 The ineqality < c indicates all vales less than c nits from the origin. Therefore < c is eqivalent to the compond ineqality c < < c. There is a similar statement for c. c c 4

5 Absolte Vale Ineqalities: > c, c, if c > 0 The ineqality > c indicates all vales more than c nits from the origin. Therefore > c is eqivalent to the ineqality statement < c or > c. There is a similar statement for c. c c To help yo keep the two cases straight in yor head, I recommend thinking of a nmber line. If the absolte vale is greater than a positive nmber c, it is greater than that many nits away from zero. -c c If the absolte vale is less than a positive nmber c, it is within that many nits of zero. -c c Ex 6: Solve each. Write soltions sing interval notation and graph the soltions on a nmber line. Hint: Always isolate the absolte vale before writing an ineqality withot the absolte vale. a) x + 4 < 6 b) 3x x + 2 c) < 1 3 5

6 Ex 7: Solve each ineqality. Write the soltions sing interval notation and graph the soltions on a nmber line. a) 2 x > 5 x b) c) 3x > 7 V Applied Problems Ex 8: Mary wants to spend less than $600 for a DVD recorder and some DVDs. If the recorder of her choice costs $425 and DVDs cost $7.50 each, how many DVDs cold Mary by? 6

7 Ex 9: The percentage, P, of US voters who sed pnch cards or lever machines in national elections can be modeled by the formla P = 2.5x where x is the nmber of years after In which years will fewer than 35.7% of US voters se pnch cards of lever machines? Ex 10: A college provides its employees with a choice of two medical plans shown in the following table. Plan 1: $100 dedctible payment 30% of the remaining payments Plan 2: $200 dedctible payment 20% of the remaining payments For what size hospital bills is plan 2 better for the employee than plan 1? (Assme the bill is over $200.) Ex 11: The room temperatre in a pblic corthose dring a year satisfies the ineqality T 71 < 3 where T is in degrees F. Express the range of temperatres withot the absolte vale symbol. 7