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Which of the following correctly expresses all of the values on the number line above?(\newline\)(A) $|x+3| \geq -1$ (\newline\)(B) $|x-1| \leq 3$ (\newline\)(C) $|x+1| \geq 3$ (\newline\)(D) $|x+3| \geq 1$ (\newline\)(E) $|x-3| \geq 1$

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Transformations of functions

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Q. Which of the following correctly expresses all of the values on the number line above?(\newline\)(A)

|x+3| \geq -1

(\newline\)(B)

|x-1| \leq 3

(\newline\)(C)

|x+1| \geq 3

(\newline\)(D)

|x+3| \geq 1

(\newline\)(E)

|x-3| \geq 1

Understand absolute value inequalities: Understand the concept of absolute value inequalities. $\newline$ The absolute value of a number is its distance from $0$ on the number line, regardless of direction. An inequality involving an absolute value can describe a range of numbers on the number line.
Analyze options: Analyze each option to determine if it correctly represents all values on the number line. $\newline$ (A) $|x+3| \geq -1$ : This inequality is always true because the absolute value is always non-negative, and thus it is always greater than or equal to any negative number.
Check option (B): Check option (B). $\newline$ (B) $|x-1| \leq 3$ : This inequality means that the distance between $x$ and $1$ is less than or equal to $3$ . This would include all $x$ values from $-2$ to $4$ . This seems to be a correct representation of all values on the number line within that range.
Check option (C): Check option (C). $\newline$ (C) $|x+1| \geq 3$ : This inequality means that the distance between $x$ and $-1$ is greater than or equal to $3$ . This would include all $x$ values less than or equal to $-4$ and greater than or equal to $2$ , which does not represent all values on the number line.
Check option (D): Check option (D). $\newline$ (D) $|x+3| \geq 1$ : This inequality means that the distance between $x$ and $-3$ is greater than or equal to $1$ . This would include all $x$ values less than or equal to $-4$ and greater than or equal to $-2$ , which does not represent all values on the number line.
Check option (E): Check option (E). $\newline$ (E) $|x-3| \geq 1$ : This inequality means that the distance between $x$ and $3$ is greater than or equal to $1$ . This would include all $x$ values less than or equal to $2$ and greater than or equal to $4$ , which does not represent all values on the number line.
Determine correct answer: Determine the correct answer based on the analysis. $\newline$ From the analysis, option (B) is the only inequality that correctly represents all values on the number line within the range of $-2$ to $4$ . Therefore, the correct answer is (B).

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