Absolute Value in Expressions and Equations

The absolute value of a number represents its distance from zero on the number line. Because distance cannot be negative, absolute value is always positive (or zero). We use the symbol $|x|$ to denote the absolute value of $x$.

Evaluating Absolute Value Expressions

When evaluating expressions that contain absolute value bars, treat the bars like parentheses. First, substitute the given value, then simplify the expression inside the absolute value bars, apply the absolute value, and finally perform the remaining operations.

Example: Evaluate $|x - 3| + |x + 2|$ when $x = 1$.

1. Substitute $x = 1$ into the expression: $|1 - 3| + |1 + 2|$
2. Simplify inside the absolute value bars: $|-2| + |3|$
3. Find the absolute values ($|-2| = 2$ and $|3| = 3$): $2 + 3$
4. Add the results together: $2 + 3 = 5$

Finding Distance on the Number Line

Absolute value is incredibly useful for finding the exact distance between any two points on a number line. The distance between two numbers, $a$ and $b$, is given by the formula: $\text{Distance} = |a - b|$ (Note: $|b - a|$ gives the exact same result!)

Example: Find the distance between $-7$ and $4$ on the number line.

1. Set $a = -7$ and $b = 4$.
2. Plug them into the distance formula: $|-7 - 4|$
3. Simplify inside the bars: $|-11|$
4. Find the absolute value: $|-11| = 11$

The distance between $-7$ and $4$ is $11$ units.

Solving Distance Equations

Sometimes you know the distance and a starting point, but you need to find the ending point. Because you can move in two directions on a number line (left or right), there are usually two answers!

Example: Find all values of $x$ on the number line that are a distance of $5$ from $-2$.

This means we are starting at $-2$ and moving $5$ units in both directions.

• Moving right (adding): $-2 + 5 = 3$
• Moving left (subtracting): $-2 - 5 = -7$

So, the values of $x$ are $3$ and $-7$. We can write this as an absolute value equation: $|x - (-2)| = 5$, which simplifies to $|x + 2| = 5$. Both $x = 3$ and $x = -7$ make this equation true!