Algebra foundations #6 - Questions and Answers

Question 1 Is 0 a rational, irrational, whole, integer, natural or real number? Answer 1 We can say that 0 is a real number. O is also a rational number. The set of rational numbers include all the integers and whole numbers. Hence, 0 is also an integer and a whole number, however, it is not a natural number as natural numbers begin from 1 and go till infinity. Answer 2 0 is a rational, whole, integer and real number. Question 2 What are the symbols for sum, difference, product, and quotient? Answer Well, there are simple symbols, such as a + b, ab, axba÷b But, there are also other symbols Δα For example, there is a symbol for difference used in some equations Ay. For instance Ду X2 X1 is the same as Y2 X1 10 There is also a symbol for sum En n=0 This is the symbol for summing a function of n denoted as a capital sigma. The bottom number denoted as n is the starting number. The top number is the ending number. You then plug in n for each number up to 10 and add them up. 10 In is the symbol for product denoted as a capital pi. In this case, you have the same n=1 rules for summing apply to products, but you multiply instead of add. Question 3 What is -12? An integer, rational, or real number? Answer 1 Integers are, for example, {...-3,-2,-1,0,1,2,3,4,...} Rational number is any number that is or can be made into a fraction. Irrational number can't be made into a fraction, for example: π, √2 Real number is any number that is rational or irrational - any number on the number line. This means that -12 is an integer, but it is also a rational number because it can be made into a fraction, and it is real as it can be found on the number line. Answer 2 -12 is an integer, rational, and real number. Question 4 Translate twelve less than three times a number, x in algebraic expression. Answer 1 Step 1 Given Twelve less than three times a number x Step 2 3x - 12 Answer 2 Explanation: "9 less than" means you are subtracting 9 from something. "9 less than 12 is 3" would be written as 12 - 9 =3 In the case the something is three times an unknown number. Call the number x. Three times the number is 3 x x = 3x "9 less than this" is written as 3x - 9 Answer 3 First, let's identify the different components in the phrase: "Three times a number, x" indicates that we need to multiply the number x by 3. This can be represented as 3x. "Twelve less than" means that we need to subtract 12 from the result obtained in step 1. This can be represented as 3x - 12. Therefore, the algebraic expression that represents the phrase "twelve less than three times a number, x" is given by 3x - 12. Let's summarize the steps again: Identify the number as x. Multiply the number by 3: 3x. Subtract 12 from the result: 3x - 12. The final algebraic expression is 3x 12. Answer 4 The statement "Translate twelve less than three times a number, x" can be represented algebraically as: 3x-12 In this expression, we multiply the number x by 3 and then subtract 12 from the result. Therefore, the algebraic expression that represents the given statement is 3x - 12. Question 5 Write the expression as an algebraic expression in x for x>0. sin (tan¯¹x) Answer 1 sin (tan¯¹x) Now let t = tan 12 ⇒ tan (t) = x x ⇒ sin (t) = √1+x2 (because perpendicular tant = perpendicular = x and base = 1then hypotenuse base = √1 + x2 also sint = perpendicular hypotenuse x ⇒ t = sin-1 √1+x2 x So, tan x 1x = sin-1. √1+x2 ⇒ sin (tanz) = sin (sin) x √1+x2 Answer 2 Explanation: sin²(0) + cos²(0) = 1 We divide both sides by sin²(0) so we have 1 + cot²(0) = csc²(0) 1 Or, 1 + 1 tan²(0) sin²(0) Taking the least common multiple we have tan²(0) + 1 1 = tan²(0) sin²(0) Inverting both sides we have tan²(0) sin²(0) = tan²(0) + 1 So we say that 0 = arctan (x) tan²(arctan(x)) sin²(arctan(x)) = tan²(arctan(x)) +1 Knowing that tan (arctan(x)) = x sin²(arctan(x)) = x2 x²+1 So we take the square root of both sides sin (arctan (x)) = ± V2 2 +1 = ± |x| √x²+1 Checking the range of the arctangent, we see that during it the sine is always positive so we have x sin (arctan(x)) = √x²+1 Answer 3 Answer: if you were to write the expression shown above as an algebraic equation in x without using trig or inverse trig functions then the correct answer choice would be letter D) I D.V 49.9s