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# Absolute Value Inequalities

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Practice Absolute Value Inequalities
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Wireless Roaming

Credit: Robert Scoble
Source: http://www.flickr.com/photos/scobleizer/5492884560/

Do you have a cell phone or tablet that uses WiFi when you’re at home and the cellular network (e.g. 3G, 4G) when you’re away? How does your device know which network to connect to? How does it roam from one network to another or, even within one cellular network, from one cell tower to another? Your mobile device in fact uses an absolute value inequality to help it make the decision.

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To illustrate this process, let’s suppose you are a cellular user who is traveling back and forth in a straight line, shown as AB in the figure below. From your current position ($x$), a cell tower is located 500 meters away and a WiFi access point is situated 600 meters in the opposite direction.

Credit: Laura Guerin
Source: CK-12 Foundation

Whenever the signal power received from the WiFi access point is higher than that received from the cell tower, your phone or tablet will connect to the WiFi interface. Otherwise, it will use the cellular network. The process of switching a wireless handheld device’s connection interface from one access point to another is called handover or handoff.

There are times, however, when you might want to override a handover. Why? Because although WiFi coverage is much smaller in area compared to the cellular network, it’s usually cheaper. By using a WiFi access point instead of a cellular tower to download data, you can save a significant amount of money.

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Suppose you allow your device to make the connection choice itself—just how does it decide? Distance from the tower or access point isn’t the only factor that goes into its decision. The signal strength transmitted by each connection option is also part of the calculation.

Using the scenario introduced in the figure above, let’s assume the cell tower to your left transmits a signal power that is four times the power transmitted by the WiFi to your right. To find the range of $x$ over which you will connect to WiFi, your phone or tablet would solve the following absolute value inequality: $|x-500|>2|600-x|$. What is that range?

1. [1]^ Credit: Robert Scoble; Source: http://www.flickr.com/photos/scobleizer/5492884560/; License: CC BY-NC 3.0
2. [2]^ Credit: Laura Guerin; Source: CK-12 Foundation; License: CC BY-NC 3.0

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