5. In Algebra you are taught that for any real numbers a > 0 and a that the inequality -a

Q: . Prove that for any two nonnegative real numbers, their arithmetic mean is always greater than or…

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Q: 3.Determine the intersection X ∩ Y. Express the  in interval notation. X = {x/ x>equal 8} and Y =…

A: We have: X=x  x≥8=[8,∞)Y=x  x<7=(-∞,7)

Q: 2. Show that (2) (¹)+(2-1). Can you provide = an interpretation of this equality?

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Q: can apply the definitions of even/oddness, absolute value to prove statements that involve cases.…

A: Answer — Exploring different cases and relation between x and 2                      a)…

Q: Consider the statement Vxy (y <x) Let x be all natural numbers Let y = x-1 Is the statement true or…

A: We know that the set of natural numbers starts from 1 and it is an infinite set. This set is…

Q: 3. If x >0 show that |(₁ + 2) ¹/³² - ( ₁ + 1 ² - √ ² ) | ≤ 3 ²². )1/3 S Use this inequality to…

A: We recall the binomial theorem for fractional index.

Q: 2. Find p and q such that 7x+1 5 |5x-1|≤4 implies p ≤: <q

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Q: 6. Prove that the statement "If x is a real number such that x² > 4 then x > 2" is false.

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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…

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Q: We assume that x0. Therefore there exists a natural number n such that n> Since z is fixed we can…

A: Suppose that x and y are 2 distinct real numbers. We have to prove that, there exists a rational…

Q: If Xo and X, are two numbers with Xo F(X1) O B. F(XO) = F(X1) O C. None of the above O D. F(X0) <…

A: It depends on the conditions; If F is a increasing function then option D is correct. If F is…

Q: .18 Let x, a and 8 be real numbers. Show that the statements |x-al < 8 and a -8 < x < a +8 are…

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Q: 27) (x- 1)(x + 1) s0 A) {1} B) [-1, 1] C) (-1, 1) D) (-, -1]U[1, 0)

A: As per the guidelines when multiple question are posted in single pic  then we are supposed to solve…

Q: 5) Suppose that W1 < R" and W2 4 R". Prove (W1 n W2) < R"

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Q: Qas: Denolop fex) the ouries in Seales, it defined as: -1<えく0 S -3x , 32,f o<ん<1

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Q: Prove if this is true of false. We know |x| = x if x ≥ 0, and |x| = -x, if x < 0. Then, for all…

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Q: 9. Prove or disprove: The inequality 2 ≥ x + 1 is true for all positive real numbers x.

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Q: Prove that for every real number x, 3x - 6 ≤ x iff |x-1| ≤

A: The objective is to prove that , where  is a real number.The inequality  is equivalent to .

Q: 3. Let r be a real number. Prove by contrapositive that if r+4x <35, then z < 3.

A: Note: Proof by contrapositive: If we want to prove that "if A, then B". Then we use contrapositive…

Q: If Xo and X, are two numbers with Xo < X, then the true one is

A: Answer:-  Option D is the correct answer.

Q: Prove that −3 ≤ x ≤ 2 implies that |x² + x − 6| ≤ 6|x − 2|.

A: We have to prove thatWe know that

Q: Prove the following:  If a <b, then -b < -a

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Q: 8. Give two examples of Fatou's Lemma, where one is an inequality. equality and the other a strict

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Q: Show that the nondegenerate interval [2, 5] is uncountable.

A: Here first we have to show that  is not countable.If possible let,  is countable ,so A can be…

Q: 2. Definition. Let r be a non-negative positive real numbel A square root of r is the Iinique…

A: Note: Square of any real number must be non-negative and addition or subtraction of any two real…

Q: Determine the intersection X ∩ Y. Express the in interval notation. X = {x/ x>equal 8} and Y =…

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Q: a) Prove that |x| ≤ 2 implies that |x² − 4| ≤ 4|x − 21.

A: To prove the given statement, let's break it down into a few steps: Starting with the assumption…

Q: Let x, y, z E Z be such that both odd, and z is even. Prov

A: Introduction: Even and odd numbers are distinguished by the following characteristics: Even Number:…

Q: 5) For each + ive integer i, where i is equal to three, then find the union and intersection of…

A: 5 Given that Ai=x∈R|−1i&lt;x&lt;1i=−1i,1i. a We have to find the A1∪A2∪A3 and A1∩A2∩A3. Here,…

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

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Q: The interval notation for the set (xlx ≤4 or x>-2} is. [-2,4] A O (-2,4] BO R .CO DO [-2,4) EO…

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Q: Find U₁A, and A; if for every positive integer i, d) A, (i, co), that is, the set of real numbers x…

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Q: Using only the Axioms and Elementary Properties of the real numbers, prove Cauchy's Inequality: for…

A: xy≤12x2+y2 For all real numbers x,y.

Q: Prove the following statement. For every real number x, if x − [x] ≥ 1/1/12 then [2x] = 2[x] + 1.…

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Q: 2. Define the absolute value of x E R. then find all real numbers x such that |x + 1 < |x – 1).

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