91Ó°ÊÓ

When measuring the distance between two points on a coordinate plane, we use the distance formula. This formula helps us find the straight-line distance between any two points. It is especially useful in geometry and algebra.

For two points \((x_1, y_1)\) and \((x_2, y_2)\), the distance \((d)\) is calculated using:

\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

However, if the y-coordinates of the points are the same (as in the given problem where both are 2), the formula simplifies significantly. You only need to find the absolute difference between the x-coordinates:

\[d = |x_2 - x_1|\]

This makes calculation easier and faster, especially useful for quick checks and straightforward problems.

The absolute value of a number represents its distance from zero on the number line, regardless of direction. It is always a non-negative number. In equations, the absolute value symbol is used to denote this operation:

\[|a| = a\] if \((a \geq 0)\)

\[|a| = -a\] if \((a < 0)\)

In this problem, the distance is presented as an absolute value equation. When the equation \(|d + 3| = 6\) is solved, it means that the quantity \((d + 3)\) is 6 units away from zero. This creates two possible linear equations:

\[d + 3 = 6\]
\[d + 3 = -6\]

These equations represent the possible values of \((d + 3)\) given that both must satisfy the distance requirement.

Solving equations, especially those involving absolute values, is fundamental in algebra. Let's solve \(|d + 3| = 6\):

• From \((d + 3 = 6)\), subtract 3 from both sides to find \(d = 3)\).
• From \((d + 3 = -6)\), subtract 3 from both sides to find \(d = -9)\).

So the two solutions are \(d = 3\) and \(d = -9\). Always check your solutions by substituting them back into the original equation to ensure they satisfy the distance condition. Therefore, 3 and -9 are both valid solutions, giving us the full set of possible values for \(d\).

Understanding to solve such equations equips students to handle more complex problems in algebra and beyond.