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# Math 1050 Class notes Math 1050

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This 51 page Class Notes was uploaded by Samantha Williamson on Wednesday September 28, 2016. The Class Notes belongs to Math 1050 at East Carolina University taught by Sungwoo Ahn in Fall 2016. Since its upload, it has received 4 views. For similar materials see Exploration in Mathematics in Mathematics at East Carolina University.

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Date Created: 09/28/16

Chapter 6. The Real Numbers and their Representation 6.1 Real Numbers, Order, and absolute Value o Natural numbers i.e starts at 1 and goes on. (all positive numbers) o Whole Numbers Starts at 0 goes onto natural numbers (still all positive) o Integers All numbers are integers both positive and negative numbers o Rational numbers {x|x = n/m, where n and m are integers with m does not equal 0} is the set of rational numbers this includes fractions and decimals i.e (1/2, -2/3, 9/10, .12, 3.414) o Irrational numbers {x|x is a number on the number line that is not rational} is the set of irrational numbers i.e 3.14…… , 2, 3 + 1, Question does natural numbers or rational numbers have more numbers Rational numbers contain all sections EXCEPT irrational numbers BOTH OF SIDES OF THIS TABLE ARE REAL NUMBERS Example: Identify elements from (-5, -2/3, 0 2, 13/4, 5, 5.8) 1. natural Numbers {5} 2. Whole Numbers: {0,5} 3. Integers: {-5,0,5} 4. Rational Numbers: {-5, -2/3, 0,13/4,5,5.8} 5. Irrational: 2 6. Real numbers: (-5, -2/3, 0 2, 13/4, 5, 5.8) Order in the Real Numbers Suppose that a and b are two real numbers. Then one and only one pf the following is true A=b (meaning a is equal to b) 2 a<b (meaning a is less than b) a>b (meaning a is greater than b) ab (meaning b is greater than or equal to a) ab (meaning a is greater than or equal to b) Example Determine whether each statement is true or false 1. False because 6 is equal to 6 2. True because 19 is larger than 5 3. True because 15 is less than 20 4. False because 25 is less than 30 5. True because 12 is equal to 12 Additive inverse and Absolute Value For any nonzero real number x, there is exactly one number on the number line the same distance from 3 0 as x, but on the opposite side of 0. That is the additive inverse 1. –x 2. 0 3. 5 Double Negative Rule o For any real number x, the following is true -(-x)= x -(-3)= 3 -(-5)= 5 Absolute Value o The absolute value of a real number can be defined as the distance (nonnegative) between 0 and the number on the number line. |x| reads as the absolute value of x. Ex. |2|= 2, |-3|=3 Formal Definition of Absolute Value o For any real number x, the absolute value of x is defined as follows The absolute Value is always a positive 4 Annual rate of change in employment in percent A)Greatest change was Cut and sew because |-12.2| is the largest Least change was the child day care 6.2 Operations, Properties and application of real numbers o Adding Real Numbers Like signs: add two numbers with the same signs by adding their absolute values and use the same sign of the two numbers Unlike signs: add two numbers with different signs by subtracting their absolute value The sum is positive if the positive number has the greater absolute value The sum is negative if the negative number has the greater absolute value. 5 6.3 Rational Numbers and Decimal Representation 6.4 Irrational Numbers and Decimal Representation 6.5 Application of Decimals and Percent 6 6.2 operations, Properties and applications of real Numbers 6.2 Operations, Properties and application of real numbers o Adding Real Numbers Like signs: add two numbers with the same signs by adding their absolute values and use the same sign of the two numbers 3+5 =+8 (-5)+(-4)=-9 (ie 5+4=9 but because they are both negative the answer is negative) Unlike signs: add two numbers with different signs by subtracting their absolute value The sum is positive if the positive number has the greater absolute value The sum is negative if the negative number has the greater absolute value. o (-5)+4=-1 cause 5 is larger than 4 even tho it is negative 1. -9 2. -16 3. 3 4. 7 5. -4 6. 0 Definition of subtraction o For all real numbers a and b A-b= is the same as adding a+(-b) So following the same rule as adding 2 1. -2 2. -16 3. -3 4. 18 Product of Factors o Let a and b be real numbers then A x b is the product of a and b a and b are factors of ab ex. 12 factors are 6x2 3x4 12x1 3 1. 4x3=12 2. 4x2=8 3. 4x1=4 4. 4x0=0 5. 4x(-1)=-4 6. 4x(-2)=-8 7. -4x2=-8 8. -4x1=-4 9. -4x0=0 10. -4x-1=4 Multiplying real Numbers o Like Signs: multiplying two numbers with the same sign by multiplying their absolute values to find the absolute value of the product is The product is positive Unlike signs: Multiplying two numbers with different by multiplying their absolute values to find the absolute value of the product o The product is negative 4 1. -63 2. -70 3. 32 Note: the result of dividing two numbers is called their quotient. In the quotient a ÷b or a , where b≠0 b a 0 is undefined o a is called the numerator or dividend o b is called the denominator or divisor Dividing real numbers o Follows the same rules as the multiplication except that we need to divide two numbers 10 o Note if 2 =5,then10=5x2 a=c o In general, if b 1. -3 (diff signs) 2. 4 (same signs) 3. -20 *different signs 4. 0 Division by zero Division by zero is undefined so 3/0 is undefined 5 o Order of Operations If parentheses or square brackets are present Step 1: work separately above or below any fraction bar Step 2: use the rules below within each set of parentheses or square brackets Start with the innermost set and work outward If no parentheses or brackets are present Step 1: apply any exponents Step 2: do any multiplication or divisions in the order in which they occur, working from left to right Step 3: do nay additions or subtractions in the order in which they occur, working from left to right PEMDAS 12÷3x5÷6 = 1. 64 (2x2x2x2x2x2) 2. (-2)(-2)(-2)(-2)(-2)(-2)= 64 (because there is an even amount of numbers 26 3. -64 same meaning as –( ) 6 1. 5+6=11 2. 4 x (9) =36 a. 36+7-10 b. 43-10=33 3. (8-12)= -4 a. 2(-4)-11(4)/ -10-3 b. -8-44 / -13 c. -52/-13 = 4 4. (-8)(-3) –[4-(-3)] a. 24-[4+3] b. 24-7= 17 Properties of addition and multiplication o For real numbers a,b and c, the following properties hold Closure properties: a+b and ab are real numbers Commutative properties: a+b= b+a, ab=ba Associative properties: (a+b)+c = a+ (b+c), (ab)c = a(bc) 7 Identity properties: a+0= a=0+a (identity of addition), ax1=a or 1xa identity of multiplication. Properties of addition and multiplication o for real numbers a,b,and c , the following properties hold. Inverse properties: a + (-a) = 0 = (-a) +a additive inverse of a 1 1 a() ()= a a a Distributive property of multiplication with respect to addition A(b+c) = ab+ac 1. Closure property of addition 2. Associative property of addition 3. Identities property of addition 4. Inverse property of multiplication 5. commutative property of addition 6. distributive property of multiplication with respect to addition 8 application of real numbers o Hint: Gain (+), lose (-) Temp: above 0 (+), below 0 (-) Altitude: Above sea level (+), below sea level is (-) Q) Use a signed number to represent change in the percent returns, shown in the figure for the following years 1. 2002 to 2003 a. 40.13 increase 2. 2003 to 2004 a. 15.32 decrease 9 Example: the Record high temperature in the US was 134 degrees fer., recorded at Death Valley, California, in 1913. The record low was -80 degrees at Prospect Creek Alaska in 1971. How much Greater was the highest temperature than the lowest temperature? 134-(-80) = 214 10 6.3 Rational Numbers and decimals Representation Rational Numbers o {x|x is a quotient of two intergers, with denominator not 0} is the set of rational numbers o Note a rational number is said to be in the lowest terms if the greatest common factor of the numerator and the denominator is 1 Fundamental property of rational numbers o If a, b and k are integers with b≠ 0∧k ≠ , then the flowing is true o a∗k÷b∗k=¿ a*b 18*2/27*2 = 18/27 9*2/9*3 = 2/3 Greatest common Divider find the least common multiple 5 6 11 15 15 = 15 Example: adding and subtracting 4 1 5 1 1. 30 30= 30simplifie6¿ 173 69 2. 180−1200=¿ 3460 173*20 and 180*20 = 3600 207 69*3 and 1200*3 = 3600 2 3460− 207= 3253 3600 3600 3600 Multiplying Rational Numbers a∧c o If b are rational numbers then the following is d true a b∗c ac -> bd -> need to rewrite this into the d lowest form o Example: multiplying Rational Numbers 3 4∗7 21 = 10 40 5 18∗3 15 3 1 → simplify → 10 180 36 12 o or cancel out the multiples 5∗3 1∗1 o 18 6 10 changesit¿2 Definition of Division b≠0 o If a and b are real numbers, , then the following I true a =a÷b=a∗1/b b multiple by the reciprocal Example 3 .3 3 7 15∗15 1∗3 9 ÷ → → = 5 15 7 7 7 3 −4 3 7 ÷14→ Density Property of rational numbers o If r and t are distinct rational numbers, with r < t, then there exists a rational numbers s such that r<s<t 1 2∗9 ½ (2/3 + 5/6) -> ½ (4/6 +5/6) -> [can simplify 6 9/6 to 3/2] = 1*3/ 2*2 = ¾ 2918+320+2016+869+517+1427/ 6 = 1,353.5 The average number of the female workers is about 1,354 thousands. Decimal form of Rational Numbers o Rational numbers can be expressed as decimals. Decimal numbers have place values that are powers of 10 4 1. 0.375 2. 0.25 Repeating decimals: a decimal which continues indefinitely o .3636 (line over the 36) o .4545 (line over the 45) o .83 (line overt the 30 Decimal Representation of rational numbers o any rational number can be expressed as either a terminating decimal or a repeating decimal Criteria for terminating and repeating decimals a o A rational number in b in lowest term results in a terminating decimal if the only prime factor of the denominator is 2 of 5 (or both) 5 a o A rational number in b in lowest terms of repeating decimal if a prime other than 2 or 5 appears in the prime factorization of the denominator then they are repeating. 1. 7/23 terminate 2. 150=2*3 5^2 there is a 3 (repeating) 3. 2/25 5^2 terminate 6 437 1. 1000 1 41 2. 8 ∨ 5 5 Step one: let x= .085…. Step two: multiply by 100 b/c two digits are repeating Now you have 100x= 85.85….. Step three: 100x=85.85…. 99x=85 x=85/99 ex. 0.13…. 13/99 7 ex. 0.4….. 4/9 ex. 0.9999 = 1 85 ex. .8555….= 90 77/90 813 805 wx 8.131313… = 990 813-8 990 8 6.4 irrational numbers and decimal representation Note: every rational number has a decimal form that terminates or repeats. Every repeating or terminating decimal represents a rational number Irrational Numbers o {x|x is a number represented by a nonrepeating nonterminating decimal} is the set of irrational numbers Product Rule for square roots o For nonnegative real numbers a and b, the following is true. 1. a* b = ab ex 12=4*3 =4*3 = 23 Conditions necessary for the simplified form of a square root radical o A square root radical is in simplified form if the following three conditions are met 1. The number under the radical (radicand) has no factor that is a perfect square 2. The radicand has no fractions 2/3 2/√3 √ = √ 3. No denominator contains a radial o Example 27 1. 27 IS NOT A PERFECT SQUARE IT’S A PERFECT CUBE 3 2. √9∗3=√ √ √= 3 Quotient Rule for Square roots √25∗√9=5 ∗3 =5 1. 3 √3 √3 √3 2. 4 → 22= 2 √ √ √1 → 1 → 1√2 √2 3. √2 √ 2 √2 OVER ALL IT IS 2 2 1. (3+4) √6 =7 √6 2. √9∗2−√16∗2→√ √2−√16√2→3 √−4 √→ 3−4)√2=¿ 3 Pi from the circle, is equal to 3.145926 all Phi: golden ratio, is equal to 1.6180 or 1+ √ 2 E 2.718. called the natural exponential e^x or natural logarithmic functions logx-> ln 1. Use a calculator, then find the sum of the first twenty one terms , multiply by 4 2. Then it is about 3.18918… pi is about 3.1459 Golden ration: Phi o A rectangle that satisfies the condition that the ration of its lengthto its width is equal to the ration of the sum of its length and width to its length s called the Golden Rectangle. This ration is called the a+b a Golden Ratio here a =b=phi 4 after 144 we have F13: 89+144=233 F14:233+144=277 F16= 610 610 F15/F14= 377≈1.6180… 5 Irrational Number e 2.71828182845 It is also called “Euler’s number” and used as the base of the natural logarithm loge (x) = In x this can be obtained by The sum of the first seven terms Is about 2.728253968. Which agrees with e to fourth decimal places 6 6.5 applications of decimals and percent Addition and subtraction of Decimals To add or subtract decimal numbers, line up the decimal points in a column and perform the operation Example .46+3.9+12.58= 16.94 12.1-8.723= 3.377 Multiplication and division of Decimals Multiplication: multiply in the same manner as integers are multiplied. The number of decimal point in the product is the sum of the numbers of places to the right of the decimal points in the factors Division: Move the decimal point to the right the same number of places in the divisor and dividend so as to obtain a whole number in the divisor. Divide in the same manner as integers are divided. The number of decimals places to the right of the decimal point in the quotient in the same as the number of places to the right of the dividend. Example: 4.613 x 2.52= 11.62476 65.175/ 8.25= 7.9 Rules for Rounding Decimals 1. Locate the place to which the number is being rounded 2. Look at the next digit to the right of the place to which the number is being rounded 3. If this digit is less than 5 drop all digits to the right of the place to which the number is being rounded. Do not change the digit in the place to which the number is being rounded. (9.43 turns to 9.4) If the digit is 5 or greater add one to the digit in the place to which the number is being rounded. (9.45 turns to 9.5) Example: Round 3.917 to the nearest hundredth 3.92 Round 14.39646 to the nearest thousandth 14.3964 turns to 14.396 Percent 1% = 1/100 =0.01 Converting between decimals (a fraction) and percents to convert a percent to a decimal, drop the percent symbol and move the decimal point two places to the left inserting zeros as placeholders if necessary To convert a decimal to a percent, move the decimal two places to the right, inserting zeros as placeholders if necessary, and add the percent symbol To convert a fraction to a percent, convert the fraction to a decimal then convert the decimal to a percent Example: convert each percent to a decimal 3.4%-> 0.034 150% -> 1.5 Example: convert each decimal to a percent 0.13 -> 13% 2 2.3 -> 230% Convert each fraction to a percent 3 5 -> 0.6 -> 60% 14 -> .56 -> 56% 25 Example: find 18% of 250 250x .18 = 45 example: what percent of 500 is 75 Cross multiply and divide 75= x 500 100 500x=7500 x=15 Example: 38 is 5% of what number Cross multiply and divide 38 5 = x 100 3500/5=x 3 Q. How much of this amount (41.2 billion) was spent of pet food About 16.2 billion Finding percent Increase or Decrease To find the percent increase from a to b, where b>a, subtract a from b, and divide this result by a. Convert to a percent b−a a ∗100 To find the percent decrease from a to b, where b<a, subtract b from a and divide this result by a. convert to a percent a−b∗100 a Example: 4 An electronics store marked up a laptop computer from their cost of $1,200 to a selling price of $1,464. What was the percent markup? 22% increase The enrollment in a community college declined from 12,750 during one school year to 11,350 to the following year. Find the percent decrease to the nearest tenth. About 11% decrease 5 7.1 Linear Equations Linear Equations in One Variable o An equation in the variable x is linear if it can be written in the form Ax+B=C o Where A,B, and C are real numbers, with A 0. o Note) a linear equation in one variable is also called a first-degree equation, because the greatest power on the variable is one. o If the variable in an equation is replaced by a real number that makes the statement true, then that number is a solution of the equation. Example: find the solution of x-3=5 o X=8 Thus, 8 is a solution or {8} is the solution set Equivalent Equations are equations with the same solution set o Ex: 8x+1=17, 8x=16, x=2 Addition Property of Equality o For all real number A,B and C, the equations A=B and A+C=B+C are equivalent Multiplication Property of Equality o For all real number A, B and C where C does not equal zero the equations A=B and AC=BC are equivalent o Example solve 4x-2x-5=4+6x+3 (4-2)x-5=6x+(4+3) -> 2x-5=6x+7 2x-5 (+5) =6x +7+5 ->2x=6x+12 -4x+12 overall x=-3 Solving a Linear Equation in One Variable 1. Clear fractions. Multiply a common denominator 2. Simplify each side separately 3. Isolate the variable terms on one side 4. Transform so that the coefficient of the variable is 1 5. Check the solution into the original equation Example: solve each equation. Decide whether it’s a conditional equation, and identity or a contradiction o 5x-9=4(x-3) x=-3 (conditional) 2 o 5x-15=5(x-3) identity o 5x-15=5(x-4) contradiction no solution Solving for a Specified Variable o Transform the equation so that all terms containing the specified variable are on one side and all others are on the other side o Simplify each side separately o Divide both sides by the factor that is multiplied by the specified variable Solve the formula P=2L+2W for W o P-2L=2W P−2 L o W= 2 A mathematical model is an equation (or inequality) that describes the relationship between two quantities. A linear model is a linear equation o Example: If a range hood removes contaminants at a flow rate of F liters of air per second, then the percent P of contaminants that are also removed from the surrounding air can be modeled by the linear equation P=1.06F+7.18 Where 10F 75 what flow rate F must a range hood have to remove 50% of the contaminants of the air? The flow rate is about 40.4 L per second 3 4 7.1 Applications of Linear Equations Solving an Applied Problem 1. Read the problem carefully 2. Assign a variable to represent the unknown value 3. Write an equation 4. Solve the equation 5. State the answer. does it seem reasonable? 6. Check the answer Problem-solving Hint: Two unknown quantities. Then choose a variable to represent one of the unknowns and then represent the other quantity in terms of the same variable, using the information for the problem Example: Two outstanding major League pitches in recent years are Randy Johnson and Johan Santana. In 2004, they combined for a total of 555 strikeouts. Johnson had 24 more strikeouts than Santana. How many strikeouts did each pitcher have? Johnson had 290 strikeouts and Santana had 265 Example: a woodworking project calls for three pieces of wood. The longest piece must be twice the length of the middle sized piece, and the shortest piece must be 10 inches shorter than the middle sized pieces. If the three pieces are to be cut from a board 70 inches how long can each piece be? 1. Let x = the length of the middle side 2. Longest = 2x, shortest x-10 Mixture and interest Problems Percents often are used in problems involving mixing different concentrations of substance or different interest rates. In each case, to get the amount of pure substance of the interest, we multiply Mixture problems BASE * rate (%) = percentage Interest problems Principal * rate(%) = interest Example If a chemist has 40 liters of a 35% acid solution, how much pure acid is there? 2 40(amount of solution) *.35 (rate of concentration) = 14 (this is about 14 liters of pure acid) If $1,300 is invested for one year at 2% simple interest, how much interest is earned in one year? 1300(principal amount) *.02(interest rate)= $26 (interest) Ex) A chemist must mix 8 liters of a 40% acid solution with some 70% solution to obtain a 50% solution. How much of the 70% solution should be used? 40% (8) + (70%) (x) = 50% 8+x Amount of pure acid (8*.4) + x(.7) = (8+x)(.5) 3.2+.7x=4+.5x 32+7x = 40+5x 7x-5x=40-32 (a) 2x=8 (b) x=4 we need 4 liters of 70% acid Ex) After winning the state lottery, Theo Lieber has 40,000 to invest. He will put part of the money in an account paying 4% interest and the remaining into stocks paying 6% interest. His accountants tells him that the total annual income from these investments should be $2,040. How much should he invest at each rate? *Interest earned = (principal) * (interest rate) 4% interest + 6% interest (x)=2,040 40,000-x 3 Earned interest (40,000 –x)(.04) +(x)(.06)=2,040 (a) multiply by 100 (40,000-x)(4)+x*6=204,000 160000-4x+6x=204,000 2x=204,000-160,000 2x=44,000 x=22,000 18,000 goes into the 4% and 22,000 goes into the 6% Ex) for a bill totaling $5.65, a cashier received 25 coins consisting of nickels and quarters. How many of each denomination did the cashier receive? Let x= the number of nickels 25-x = the number of quarters Total value 5x+ 25(25-x)=565 5x+625-25x=565 -20x=565-625 x=-60/-20 x=+3 Distance= times * speed. Here, speed can be rate Motion problems: Distance, Rate, Time Relationship D=rt, r=d/t, t=d/r Example 4 The speed of sound is 1,088 feet per second at level at 32 degrees. In 5 seconds under these conditions, how far does sound travel 1088*5= 5440ft Example: the winnder of the first Indianapolis 500 race (in 1911) was Ray Harroun, driving a Marmon Wasp at an average speed of 74.59 per hour. How long di did it take the 500 mile course? Time =distance/speed 500/74.59 =6.7 hours Example: Greg Sabo can bike from home to work in ¾ hour/ by bus the trip takes ¼ hour. If the bus travels 20 mph fast than Greg rides his bike, how far is it to his workplace. Let x= distance Speed of bike =x/.75 Speed of bus =x/.25 x/.25-x/.75=20 (x*4/1 – x*4/3= 20) *3 x*4*3-x*4/3*3=20*3 12x-4x=60 8x=60 x=60/8=7.5 miles 5 7.5 Properties of Exponents and Scientific Notation Exponential expression o If a is a real number and n is a natural number, then the exponential expression a^n is defined as o a^n= a*a*a*a…. o the number a is the base and n is the exponent example: evaluate o 7^2=49 o (-2)^4=-16 o -4^2=16 o -8^4 = -4096 o (-8)^4 =4096 Product rule for exponents o If m and n are natural numbers and a is any real number then, o a^m* a^n= a^m+n o 3^4*3^7= 3^11 o y^3*y^8*y^2= y^13 o (5y^2)(-3y^4)= -15y^6 o (7p^3q)(2p^5q^2)= 14p^8q^3 Zero Exponent o If a is any nonzero real number than a^0=1 Negative Exponent o For any natural number n and any nonzero real number a, just make it a fraction with 1 above it A^-n=1/a^n o Example: 2−3 (5z)^-2 = 1/ 3^-1+4^-1 1/3+1/4= 7/12 b 2^3 = 8 3^2/2^3= 9/8 Quotient Rule of Exponents 2 o If a is any non zero number and m and n are integers then o 3^7-2=3^5 o p^6-3= p^4 o 7^4/7^6= 7^-2) 1/49 Power rules for Exponents o If a and b are real numbers and m and n are integers then Special Rules for negative exponents 3 o If a0 and b0 and n is and integer then Example 83 o 2 p)=8 p Scientific Notation o A number written in scientific notion when it is expressed in the form A a∗10 ,where1a|<10,∧nisaninteger o Ex: 8,200,000 8.2X10^6 .000072 7.2X10^-5 o Example: Convert each number from scientific notation to standard notation 6.93x10^5 693,000 4.7x10^-6 .00000047 4 7.4 Linear Inequalities Linear inequality in One Variable o A linear inequality in one variable can be written in the form Ax+b<C o The symbol < may be replaced by >, Addition Property of Inequality o A<B and A+C< B+C o Have the same solution Example: solve 7+3x>2x-5 o 3x>2x-12 o x>-12 Multiplication Property of Inequality o Let C 0 If C>0 then the inequalities have the same solution o If C< 0 then the inequalities have the same solution Division Proptery of Inequality o The same rule as multiplication property of inequality Solve o 3x<-18 o divide each side by 6 and the answer is x<-6 or (-,- 6) -4x8 o divide each side by -4 o x-2 Solving a linear Inequality in one variable a. Simplify each side separately b. Isolate the variable terns on one side i. Use the addition property of inequality (a- c)x<d-b c. Isolate the variable i. Use the multiplication property of inequality x<k,x>k, etc. ii. *note: reverse the direction of arrow when divide by a negative example: o 5(x-3)-7x4(x-3)+9 o x is less than or equal to negative 2 ex: o John has test grades of 86,88,78 on his first three tests in calculus, if he wants an average of at least 80 after his fourth test, what are the possible scores he can make on his fourth test? Let x= josh’s score on his fourth test 86+88+78+x/ 4 80 (x+252/4 80)4 x+252320 x320-252 x68 he must get a 68 or higher ex: o a rental company charges $15 to rent a chain saw, plus $2 per hour. Ken Pott can spend no more than 2 $35 to clear some logs from his yard. For how long can he rent the saw to stay within budget? Let H = the number of hours for a chainsaw 15+2h 35 2h35-15 =20 h20/2 =10 the max amount of hours he can have the chainsaw is 10 hours solve 43x-5<6 o 4=53x<5+5 o 93x<11 o divide it all my 3 o 3x<11/3 3

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