1. Let a be a positive real number. In Part (1) of Theorem 3.25, we proved that for each real number x, jxj < a if and only if a < x < a. It is important to realize that the sentence a < x < a is actually the conjunction of two inequalities. That is, a < x < a means that a < x and x < a. ? (a) Complete the following statement: For each real number x, jxj a if and only if . . . . (b) Prove that for each real number x, jxj a if and only if a x a. (c) Complete the following statement: For each real number x, jxj > a if and only if .

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cdchavezq cdchavezq · Answer 1 · 2020-05-05T06:14:36+02:00

Answer:

(a)

if [tex]x \geq 0[/tex] then [tex]x \geq a[/tex] and if x<0 then [tex]-x > a[/tex]

(b)

That is straightforward from what you showed on theorem 3.25

(c)

Following the same ideas from (a) x>a or -x > a.

Step-by-step explanation:

Remember how we define the absolute value of a number.

(a)

In general

[tex]|x| = x \,\,\,\text{if} \,\,\,\, x \geq 0 \\|x| = -x \,\,\,\text{if} \,\,\,\, x < 0[/tex]

Therefore if [tex]|x| \geq a[/tex] you have two cases, if [tex]x \geq 0[/tex] then [tex]x \geq a[/tex] and if x<0 then [tex]-x > a[/tex]

(b)

That is straightforward from what you showed on theorem 3.25

(c)

Following the same ideas from (a) x>a or -x > a.