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The concept of absolute value is fundamental when working with distances on a number line or between any two points. Absolute value refers to the magnitude or the "distance" a number is from zero on a number line, regardless of direction.
It is always a non-negative number.

• For a positive number or zero, the absolute value is simply the number itself.
• For a negative number, the absolute value is the number without the negative sign.

When expressing an inequality involving absolute value, such as \[ |x + 2| \geq 4, \] it means the distance between the value of the expression \(x+2\) and zero, is at least 4. This leads to two possible scenarios:

• \( x + 2 \geq 4 \)
• \( x + 2 \leq -4 \)

By solving these, we determine that:

• \( x \geq 2 \)
• \( x \leq -6 \)

Thus, the expression \(|x + 2| \geq 4\) describes a region on the number line that represents numbers at least 4 units away from \(-2\).

Coordinate geometry, also known as analytic geometry, involves using a coordinate system to investigate geometric shapes and their properties. A coordinate system such as the Cartesian plane uses pairs of numbers to uniquely define the position of a point.

• The horizontal axis is called the x-axis.
• The vertical axis is referred to as the y-axis.
• Every point on this plane can be described by an ordered pair \((x, y)\).

In exercises like ours, often one-dimensional aspects of coordinate geometry apply. Here, instead of pairs, we deal with single numbers which are sufficient to locate points on a line, also known as a coordinate line or number line. Point \(A\) at \(-2\) and point \(B\) at \(x\) means we are moving along this line to find various distances, encapsulated in expressions like \( |x + 2| \geq 4 \).

Calculating the distance between points is a key concept in both everyday measurement and more advanced fields like physics or engineering. On a coordinate line, this is simplified to the absolute value of the difference between the two points' coordinates.

• Consider two points: \(A\) and \(B\).
• If \(A = -2\) and \(B = x\), the formula used to calculate the distance is:

\[ d(A, B) = |x - (-2)| = |x + 2| \]This formula assures that the calculated distance will always be positive or zero, perfectly aligning with the definition of absolute value.
This type of problem generally expresses constraints like \(|x + 2| \geq 4\) to ensure that conditions, such as minimum required distance, are met.
By visualizing this on a number line, you can easily identify that the magnitude of distance dictated by this inequality creates a range or specific zone above or below certain points.