91Ó°ÊÓ

When solving absolute value equations, it's important to understand the basic definition of absolute value, which is the distance of a number from zero on the number line. In the given problem \(|x - 5| = 3\), this means that the expression \(x - 5\) can be equal to either 3 or -3, since both \(3\) and \(-3\) are three units away from zero.

To solve such equations, we need to set up and solve two different equations:

• \(x - 5 = 3\)
• \(x - 5 = -3\)

For the first equation, solving \(x - 5 = 3\) involves isolating the variable \(x\) by adding 5 to both sides. This gives us:
\[ x = 8 \]
For the second equation, solving \(x - 5 = -3\) involves the same step of isolating the variable \(x\) by adding 5 to both sides, leading to:
\[ x = 2 \]
Therefore, the solutions to the equation \( |x - 5| = 3 \) are \( x = 8 \) and \( x = 2 \). These are the points where the distance from the expression inside the absolute value to zero equals 3.

Understanding absolute value as the 'distance from zero' helps to visualize solutions. The absolute value \(|a|\) of a number \(a\) indicates how far \(a\) is from zero on a number line, regardless of the direction.

In the equation \(|x - 5| = 3\), we are looking for all \ x \ values such that the expression \(x - 5\) is 3 units away from zero. This means:

• \