Question

U.S. Voltage In the United States, normal household voltage is 110 volts. It is acceptable for the actual voltage \(x\) to differ from normal by at most 5 volts. A formula that describes this is $$ |x-110| \leq 5 $$ (a) Show that a voltage of 108 volts is acceptable. (b) Show that a voltage of 104 volts is not acceptable.

Short answer

A voltage of 108 volts is acceptable because it falls within the range of 105 to 115 volts. A voltage of 104 volts is not acceptable because it is outside of this range.

Step-by-step solution

Understanding the inequality

The inequality \( |x-110| \leq 5 \) means that the actual voltage \(x\) can vary by up to 5 volts from the normal voltage of 110 volts. This translates to the range \(110 - 5 \leq x \leq 110 + 5\), or \ 105 \leq x \leq 115\.

Checking if 108 volts is acceptable

Substitute \(x = 108 \) into the inequality \( |x-110| \leq 5 \). We get \ |108-110| \leq 5 \, which simplifies to \ |-2| \leq 5 \. Since \ 2 \leq 5\ is true, 108 volts is acceptable.

Checking if 104 volts is acceptable

Substitute \(x = 104 \) into the inequality \( |x-110| \leq 5 \). We get \ |104-110| \leq 5 \, which simplifies to \ |-6| \leq 5 \. Since \ 6 \leq 5 \ is false, 104 volts is not acceptable.

Key concepts

absolute value

The concept of absolute value is fundamental in understanding voltage ranges and inequalities.
The absolute value of a number is its distance from zero on the number line, without considering direction.
For example, the absolute value of -3 is 3, and the absolute value of 3 is also 3. We write this as \(|-3| = 3\) and \(|3| = 3\).
When dealing with problems involving absolute values, it simplifies to always think in terms of distance.
In the original problem, we use absolute value to express that the voltage can be a certain distance away from 110 volts, in this case, up to 5 volts.

inequalities

Inequalities help us express ranges of acceptable or unacceptable values.
An inequality like \(|x-110| \leq 5\) tells us that the value of \(x\) can be up to 5 volts away from 110 volts, either more or less.
Understanding how to work with inequalities is key to solving such problems.
We can split the absolute value inequality into two separate inequalities:
\(x - 110 \leq 5\) and \(x - 110 \geq -5\) which simplifies to \(105 \leq x \leq 115\).
This range means that any voltage between 105 and 115 volts is acceptable.

voltage range

Voltage range in our example defines the span of acceptable voltages.
In the exercise, the acceptable voltage range is determined by the inequality \(|x-110| \leq 5\).
We use the concept of voltage range to check if specific voltages are within accepted limits.
For instance, 108 volts falls within the range \(105 \leq x \leq 115\).
On the other hand, 104 volts does not fall within this range, making it unacceptable.
Knowing the acceptable voltage range is crucial for ensuring electrical devices operate correctly.

household voltage

Household voltage in the United States typically is 110 volts.
The exercise shows how actual voltage may vary slightly while still being acceptable for household use.
Electric appliances are designed to work within a safe voltage range, usually specified in terms of absolute values.
For the U.S. standard, any voltage between 105 and 115 volts (as calculated) is acceptable.
This ensures the safety and optimal operation of household appliances and helps prevent damage caused by voltage fluctuations outside this range.