(7) Prove the following statements: (a) If x, y € R with x < y, then x < (1 − t)x + ty < y when 0 < t <1. (b) If r, q are rational numbers, then r√√5 + q is irrational when r ‡ 0. (c) Between any two rational numbers there exists an irrational number of the form r√√5+q with r > 0. (d) Between any two real numbers there exists an irrational number of the form r√√5+q with r > 0.

Q: Show that if x > 0, then 1 + 1x − x² ≤ √1+x≤ 1 + ½ x. -

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Q: Solve for X such that 2XT - A = B + (CI)", where 21 3 -1 A =|-1 5, B = 2 2 -3 and C = -2 3 10 1. -7…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: d) Prove that −1 < x < 0 implies that |x³ − 2x + 1| < 1.26|x − 1].

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Q: (a) Explain with cases why, if a is a real number,  then a ≤ |a|. (b) Explain, if, for any real…

A:

Q: Exercises for Chapter 1. (a) Prove that √3 is irrational. (Hint: Example 1.3 should be useful.) (b)…

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Q: finD 1) 2) g(s (x) a) p(x) x?-2x+2 b.) g4)==3x t. %3D cl.) s(x) = x-1 4) Cx))

A: Given that px=x2−2x+2, qx=3x+1, rx=x and sx=x−1. 1 We have to find the prx. prx=pxUse…

Q: b) 0.6' Rational Irrational

A: Therefore, it represents the rational number. The answer is: rational

Q: (c) Prove that r| +11 36 T-1 2 +36 for all real I.

A: We have to prove that the following inequality holds: x-11x2+36≤x+1136 for all real x. We will use…

Q: 9. Two zeros of f(x) = x* – 7x³ +x² – 63x – 72 are 3i and 8. In what form are the other roots? A. 1…

A: Topic = Polynomial

Q: 3. Invent two interesting and original positive integers a and b. Let f(x) = ax +3. (a) (b) What is…

A: Given function is fx=ax+3. To find:  a  f10   (b). A number x such that fx=b Solution: Given…

Q: a. 3.2F> Fn+1 for n23 [Solutions and final ans

A: We need to prove 2Fn&gt;Fn+1 Now, we have formula for fibonacci sequence is Fn=Fn-1+Fn-2

Q: b) Prove that |x| ≤ 1 implies that |x² + 2x - 3| ≤ 4|x − 1].

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Q: 2. The set of all rational numbers in the interval (0, 1).

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Q: Show that V12 is irrational

A: A Rational Number is a number that can be written in terms of P/Q.  An irrational number can not be…

Q: Give the domain of s(x)=x2+2x√16−5xs(x)=x2+2x16-5x in interval notation

A: Given the functions(x)=x2+2x16−5x

Q: 1 where A and B are rational numbers. Find A + B

A: (2n)!2n+5!=(2n)!2n+52n+42n+32n+22n+12n!…

Q: 15. Given the equation 3x-4x -12 =0, how many roots are there? Rational or irrational?

A: Given equation  3x2-4x-12 = 0 ....... (1) Quadratic formula for the solution of the quadratic…

Q: 3. If the expression was placed in the form ax where a and b are real numbers, then which of the 2x…

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Q: How do you describe thr denominators of all terminating decimals? What are those which will yield…

A: Note: As per our guidelines, we are supposed to answer only one question. Kindly repost other…

Q: sin x+3 cosr dx. Determine the following rational function 1+cosx

A: We have to first do partial fraction of the given expression.

Q: Perform all indicated operations and reduce the rational expression x? 2x - 6 X -1 -3x -to simplest…

A: Given expression A(x)=x2x-1-2x-62÷-3x1-x2.

Q: 4. Show that Archimedean Principle is a consequence of the assertions that every interval (a, b)…

A:

Q: 10. Prove that the following are irrational numbers: (1) √3 (ii) 1 + √2 (iii) √3+ √2 (iv)

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Q: -6 14 (-7 15 , 18. Solve for X, if X³ = (). 0 2 Ans. X = -1 3

A: We are given that for a matrix X, such that X3=-614-715. We need to find the value of X. Let X =…

Q: 8z z + 3 VI €2.

A: To solve the given inequality.

Q: Prove that (z - x)(z - y) = (y - x)?. a:= is not a rational number. Hint: Express (x - x) and (z -…

A:

Q: _4. An expression in the form Vam where n is a positive integer and a is an element of the real…

A:

Q: 7. Prove that is rational if and only if 3x + 7 is rational.

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Q: (i) Show that √6 + √7 is an algebraic number.

A: We are answering only the first question as per the guidelines. Please repost the other question to…

Q: show that there doesnt exist a rational numbert such that t2 = 3

A: According to the given information, it is required to show that there does not exist any rational…

Q: If S is a nonempty subset of the negative rational numbers, then: 3x e S(Vy e S(x > y)). O true O…

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Q: a rational number y=10.01 b) Assume an integer, b=2 from 111 to 9 I assume from the range (0, ³/25).…

A: Given, X~Uniformx-32, x+32

Q: For any integer n > 1 find, Hint: Substitute x + √x² + 1 = = U 1 So " (x + √x² +1)″

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Q: Determine the union X ∪ Y. Express them in interval notation. X = {x/x>-17 } and Y = {x/x<equal-10…

A: Introduction: The number of elements carried by the operation and executing the results of a…

Q: [B] Consider the equation x1 + x2 + x3 = 25 where a1, x2, a3 E Z. 2/5 (B.1) How many solutions are…

A: Solution B: Given equation is x1+x2+x3=25 here x1, x2, x3∈Z B.1 Given x1≥2, x2≥0, x3≥5 remaining…

Q: If x > y and a are a negative real number, this always applies

A:

Q: The value of x which satisfy 3x + 1| < - is : 5.

A:

Q: How can I explain using (x0+ ε, x0-ε)? Explain why it is true that, if x0 is in (a,b), then there…

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Q: 23. Simplify the complex fraction કાકાસ x²+2x+3 '' + ; x+3 1 1 -+ x+5

A: Topic = Algebra