Chapter P.2, Problem 50E

Solution

Write the given statement using absolute value notation.

  $\left|y+2\right|\ge 4$

Given information:

Statement is:

“The distance between y and $-2$ is greater than equals to 4.”

Formula used:

1. Absolute value is representation on either side of the expression given by two parallel line.

Absolute value x means:

  $\left|x\right|=\left\{\begin{array}{l}x\text{ if }x\text{ is positive}\\ -x\text{ if }x\text{ is negative}\end{array}\right.$

For example, if the distance between x and y is a this is represented by absolute value notation as:

  $\left|x-y\right|=a$

2. a greater than equals to b is represented as:

  $a\ge b$

Calculation:

To write the given statement using absolute value notation first note that for any x :

  $\left|x\right|=\left\{\begin{array}{l}x\text{ if }x\text{ is positive}\\ -x\text{ if }x\text{ is negative}\end{array}\right.$

Now, note that the distance between x and y is a this is represented by absolute value notation as:

  $\left|x-y\right|=a$

Here, the given statement is:

“The distance between y and $-2$ is greater than equals to 4.”

As, the representation of a greater than equals to b is represented as:

  $a\ge b$

Thus, the distance between y and $-2$ is greater than equals to 4 is represented by:

  $\begin{array}{c}y-\left(-2\right)\ge 4\\ y+2\ge 4,\text{ or}\\ -\left(y-\left(-2\right)\right)\ge 4\\ -\left(y+2\right)\ge 4,\end{array}$

This can be represented in absolute value notation as:

  $\begin{array}{c}y+2\ge 4,\text{ or}\\ -\left(y+2\right)\ge 4\\ \Rightarrow \left|y+2\right|\ge 4\end{array}$

Thus, it can be represented by absolute value notation as:

  $\left|y+2\right|\ge 4$

Thus, the absolute value notation of the given statement is:

  $\left|x-4\right|=3$