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Consider the set

A=\{x \in \mathbb{R}:|x-1|<6\}

. a)

x \in A

if and only if ...

Oct 6, 2024

Solution by Steps

step 1

To determine the set A = \{ x \in \mathbb{R} : |x - 1| < 6 \} , we start by rewriting the absolute value inequality: |x - 1| < 6 implies -6 < x - 1 < 6

step 2

Adding 1 to all parts of the inequality gives us: -6 + 1 < x < 6 + 1 , which simplifies to -5 < x < 7

step 3

Therefore, we can express this as

x \in A

if and only if -5 < x < 7

step 4

The interval notation for set

A

A = (-5, 7)

step 5

In terms of the question's blanks, we can fill them as follows:

x \in A

if and only if

6

is strictly less than

x

and

x

is less than

7

step 6

For part b, the interval notation for

A

A = (-\infty, 7)

Answer

A = (-5, 7)

Key Concept

Understanding absolute value inequalities and interval notation.

Explanation

The solution involves rewriting the absolute value inequality to find the range of

x

that satisfies the condition, leading to the interval

(-5, 7)