Problem 1. The absolute value of a real number x, denoted by [x], is defined { Let x, y ER and a ≥ 0. Prove the following statements. (a) x≤ a if and only if -a ≤ x ≤a. (b) |xy| = |x|y|- (c) |x+y| ≤ |x| + |y|. as |x| = = x if x ≥ 0 -x if x < 0.

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

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Q: The function f (x) = ! takes the value of -1 at x = -1 and 1 at x = 1. The number is between -1 and…

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Q: An interval a ≤ x ≤ b means that the endpoints a and b are not regarded as points belonging to the…

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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: Which x satisfies both Ix - 31 &lt; 2 and Ix - 51 &lt; 1?

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Q: A8. (a) State (without proof) the Intermediate Value Theorem. (b) Use the Intermediate Value Theorem…

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Q: Find the indicated set if given the following 1.A= (x| x2-2) B- (x x<8) C= ( -1<xs9 (a) BUC (b) 8nc

A: For the given sets

Q: Suppose that the functions s and t are defined for all real numbers x as follows. s(x)=2x+5 t(x)=3x…

A: Step 1: Given functions: s(x)=2x+5, t(x)=3xwe have to find the expressions: (s⋅t)(x), (s+t)(x), and…

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Q: 10. State the domain of each of the following using interval notation. 9-4x b) g(x)= Jx+10 x+2 a)…

A: In this question, we need to find the domain of the given functions.

Q: 10. Σ = {(1,3), (-∞, 2), (0, ∞0), (-∞,4)}

A: This is the question from the topology.

Q: '-x+b, f(x) = { 1, if x −3(and x ‡ b) −3, For what value(s) of b does f have a jump discontinuity at…

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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: 11 – |x| X 2 a. f(0.5) = b. f(-5.5) = c. f(-3.5) = d. f(6) =

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

A: We have to prove thatWe know that

Q: 1. Let P (x) be the statement "x = x²." If the domain consists of the integers, give True/False…

A: We have given a statement ,  P(x) : x = x2 Here , domain consists of the integers , that is x ∈ ℤ.…

Q: Suppose that x is a negative real number and that x < 1/x. Prove that x < 1.

A: Step 1: Step 2: Step 3: Step 4:

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: The function f is defined by f (x) = –x² – 3 for all real numbers x. Which of the following…

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

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Q: 1. Let C be a contradiction (i.e. a statement that's always false) and let X be any statement. Use a…

A: Here is the solution

Q: Q3 (Recall the notation: For any set S CR and any k eR, kS = {ks : sE S}.) (a) Someone has claimed…

A: We will use the fact that the set of integers is countable and every subset of a countable set is…

Q: Question 4. Describe the real numbers r for which r < x² – 12 < 3x.

A: Given:         x&lt;x2−12&lt;3x

Q: Prove or disprove the following statement: "For all real numbers x, y, [xy] = [x][y]." (3 marks)

A: Introduction: Two important functions are frequently used in mathematics and computer programming.…

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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Q: 5. Function fand g are defined by formulas as shown below. f(x) = x³ and g(x) = x- 1, for all real…

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Q: 10. Prove the following inequaltities for any positive real numbers x,y: xy³≤ x² + y4 (ii) xy³+x³y…

A: Given Data: We have to prove the following inequaltities for any positive real numbers x,y : i…