Question

Foreign Voltage In other countries, normal household voltage is 220 volts. It is acceptable for the actual voltage \(x\) to differ from normal by at most 8 volts. A formula that describes this is $$ |x-220| \leq 8 $$ (a) Show that a voltage of 214 volts is acceptable. (b) Show that a voltage of 209 volts is not acceptable.

Short answer

214 volts is acceptable; 209 volts is not acceptable.

Step-by-step solution

Understand the formula

The formula \(|x-220| \leq 8\) means that the voltage \(x\) can differ from 220 volts by at most 8 volts. This can be broken down into two inequalities: \(x - 220 \leq 8\) and \(220 - x \leq 8\) or equivalently \(212 \leq x \leq 228\).

Verify voltage of 214 volts

To verify if 214 volts is acceptable, substitute \(x = 214\) into the inequality: \(|214 - 220| \leq 8\). Calculate the absolute value, which gives \(|-6| = 6\). Since 6 is less than or equal to 8, 214 volts is acceptable.

Verify voltage of 209 volts

To verify if 209 volts is acceptable, substitute \(x = 209\) into the inequality: \(|209 - 220| \leq 8\). Calculate the absolute value, which gives \(|-11| = 11\). Since 11 is greater than 8, 209 volts is not acceptable.

Key concepts

absolute value

The concept of absolute value is central to this problem. Absolute value refers to the distance of a number from zero, regardless of direction. This means it is always a non-negative number. For example, the absolute value of both -5 and 5 is 5.
The mathematical notation for absolute value uses vertical bars, like this: \(|x|\).
In the formula given in the exercise, \(|x - 220| \leq 8)\), we are considering the distance of x from 220. This absolute value expression tells us how far x is from 220, no matter if it increases or decreases. So, \(|-6| = 6\), and \(|6| = 6\) both give a positive distance of 6.

inequalities

Inequalities are mathematical statements that show the relative size or order of two values. They are used to compare numbers or expressions.
In our exercise, the inequality is \(|x - 220| \leq 8)\), which reads as: the absolute difference between x and 220 must be less than or equal to 8. This effectively sets bounds for acceptable values of x.
When we interpret \(|x - 220| \leq 8)\), we can break it down into: \(x - 220 \leq 8\) and \(220 - x \leq 8\). This simplifies to the range: \(212 \leq x \leq 228\). So, the voltage x must be between 212 volts and 228 volts to be acceptable.

problem solving

Problem solving with absolute value inequalities can be straightforward if you follow the steps methodically.
Here's how to approach it:

• Step 1: Understand the formula. Know what the absolute value and inequality represent.
• Step 2: Break down the absolute value inequality into two separate inequalities.
• Step 3: Simplify the inequalities to find the range.
• Step 4: Substitute the given values into the formula to check if they fall within this range.

In this particular problem, after breaking down \(|x - 220| \leq 8)\), we found the range \(212 \leq x \leq 228\). We then tested the given voltages, 214 and 209, by substituting them into the original inequality to check for acceptability.

real-world application

Absolute value inequalities have key applications in the real world. They are essential in fields where tolerances and variations are critical.
Consider the exercise about voltage. Electrical engineers often need to ensure that the voltage supply stays within a safe and acceptable range. In this case, \(|x - 220| \leq 8)\) guarantees that the voltage does not deviate from 220 volts by more than 8 volts.
Similar applications can be found in manufacturing and quality control, where parts must meet precise dimensions or conditions. The principle is the same: keep variations within acceptable limits to achieve desired performance and safety.