Given that x is a positive integer, prove by contradiction that x + 1/x 2. (Put

Question

Given that x is a positive integer, prove by contradiction that x + 1/x 2.
(Put the statements in the correct order)

a. square of x-1, which is an integer, is a negative value which is a contradiction. Therefore x + 1/x 2.

b. (x-1) 2 < 0                                                     rewrite

c. x + 1/x < 2                                                   hypothesis

d. x2 + 1 < 2x                                          multiply both sides by x (given x is larger than 0)

e. (x2 + 1)/x < 2                                               rewrite

f. x2 -2x + 1 < 0                                              subtract both sides by 2x

Explanation / Answer

Correct order:

c. x + 1/x < 2                                                   hypothesis

e. (x2 + 1)/x < 2                                               rewrite

d. x2 + 1 < 2x                                          multiply both sides by x (given x is larger than 0)

f. x2 -2x + 1 < 0                                              subtract both sides by 2x

b. (x-1) 2 < 0                                                     rewrite

a. square of x-1, which is an integer, is a negative value which is a contradiction. Therefore x + 1/x 2.

Note: In proof by contradiction methode, we initially assume inverse of what we want to proove, and then finally show that there is a contradiction , which means our assumption was false.

Related Questions

Navigate

• Browse (All)
• Browse G
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.