91Ó°ÊÓ

In mathematics, distance is a measure of the space between two points. It tells us how far apart objects or numbers are from each other. When we talk about the distance between a number and a fixed point on the number line, we can use absolute value to represent this distance.

For example, if we want to find the distance between the variable \(x\) and the number 7, we write this as \(|x - 7|\). Here, the absolute value symbol \(|...|\) ensures that the distance is always a positive number, as distance cannot be negative.

In our exercise, when we say \(x\) is less than 2 units from 7, we mean that the distance between \(x\) and 7 is less than 2. Mathematically, this is represented as \(|x - 7| < 2\).

Inequalities are mathematical statements that compare two values using the symbols <, >, ≤, or ≥. These symbols show whether one value is less than, greater than, less than or equal to, or greater than or equal to another value.

There are different types of inequalities, such as linear and quadratic inequalities. In our exercise, we are dealing with a linear inequality because it involves a variable raised to the first power.

When we work with inequalities involving absolute values, we express the range of values that satisfy the inequality. For example, the inequality \(|x - 7| < 2\) means that \(x\) can be any number that is within 2 units of 7. This can be written as two separate inequalities:

• \(x - 7 < 2\)
• \(x - 7 > -2\)

Combining these gives us a compound inequality:
\(-2 < x - 7 < 2\).
Adding 7 to each part of the inequality, we get:

\(5 < x < 9\). This is the same as saying that \(x\) is between 5 and 9.

Absolute value is a mathematical concept that represents the distance of a number from zero, without considering which direction on the number line it is. The absolute value of a number \(x\) is written as \(|x|\).

For instance, the absolute value of both 3 and -3 is 3, because both numbers are three units away from zero. Mathematically:

• \(|3| = 3\)
• \(|-3| = 3\)

When solving inequalities with absolute values, we can split the inequality into two separate inequalities. For the inequality \(|A| < B\), where \(A\) and \(B\) are expressions, we write:

• \(A < B\)
• \(A > -B\)

Combining these, we get a compound inequality: \(-B < A < B\).

In the example given in the exercise, \(|x - 7| < 2\) was understood to mean the distance between \(x\) and 7 is less than 2. We solved it by writing the compound inequality \(-2 < x - 7 < 2\), then simplified this to find the range 5 to 9. Thus, \(x\) is between 5 and 9, which is our final answer.