Problem 88 - Precalculus Student Solutions Manual 5th

Question

Write each statement as an absolute value equation or inequality. \(q\) is no more than 8 units from 22.

Step-by-Step Solution

Answer

\[ |q - 22| \leq 8 \]

Step 1: Identify the key information

The problem states that the variable **q** is no more than **8** units from **22**. This means that the distance between **q** and **22** is at most **8** units.

Step 2: Express the absolute value

The absolute value of the difference between **q** and **22** represents the distance from **q** to **22**. This can be written as \(|q - 22|\).

Step 3: Set up the inequality

Since the distance is no more than **8** units, we need to set up an inequality that shows this. The absolute value expression \( |q - 22| \) must be less than or equal to **8**. Thus, the inequality is \(|q - 22| \leq 8\).

Key Concepts

absolute value equation

An absolute value equation represents the distance of a value from zero on a number line. The absolute value of a number is always non-negative. For instance, the absolute value of both 5 and -5 is 5, denoted as \(|5| = 5\) and \(|-5| = 5\). This concept is crucial in solving equations that describe distances. When dealing with absolute value equations, you often set up two separate equations to account for both positive and negative scenarios.
For example, solving \(|x - 3| = 7\) would involve:

• \(x - 3 = 7\)
• \(x - 3 = -7\)

These equations provide solutions within both a positive and negative distance from the reference point.

distance in algebra

Distance in algebra often involves the concept of absolute value. This concept helps measure how far a number is from another on a number line. Consider it as always taking the positive difference between two points.
In the problem, 'q' is no more than 8 units from 22, which means you are looking for a set of values that fit within an 8-unit radius of 22. This setup translates into an absolute value expression.
By stating \(|q - 22|\), we represent the distance of 'q' from 22. When we say this distance is at most 8, we are saying: \(|q - 22| <= 8\).

inequality setup

Setting up inequalities with absolute values involves understanding restrictions or bounds on possible values. The phrase 'no more than' suggests a maximum boundary.
In the context of the exercise, 'no more than 8 units from 22' becomes an inequality. You convert the absolute value of the difference into an inequality: \(|q - 22| <= 8\). This tells us that the expression inside the absolute value should lie within a range of -8 to 8.
Therefore, solving this includes writing the following inequality:

• \(-8 <= q - 22 <= 8\)

From this, solving for 'q' might involve breaking it into two inequalities:

• \(-8 <= q - 22\)
• \(q - 22 <= 8\)

Together, they define the range for 'q'.