Absolute Value

🎯 Absolute Value

Part 1 of 5 — Distance From Zero

Topics in This Part

🔑 Key Concept: The absolute value of a number is its distance from $0$ on the number line. Distance is never negative, so absolute value is never negative.

What Absolute Value Means

We write absolute value with two straight bars: $\lvert n\rvert$, read as "the absolute value of $n$."

Picture the number line. The absolute value of a number is how far it is from $0$ — and direction doesn't matter.

$\lvert -5\rvert = 5 \qquad \lvert 5\rvert = 5$

Both $-5$ and $5$ sit $5$ units away from zero, so both have absolute value $5$.

💡 Shortcut: To take the absolute value, just drop the negative sign if there is one. $\lvert -12\rvert = 12$ and $\lvert 12\rvert = 12$.

Concept Check 🎯

Evaluate 🧮

Find each absolute value. (Distance from zero is never negative.)

1) $\lvert -6\rvert = \,?$ 2) $\lvert 15\rvert = \,?$ 3) $\lvert 0\rvert = \,?$

Why It's Always Positive (or Zero)

There is exactly one number whose absolute value is $0$: the number $0$ itself.

$\lvert 0\rvert = 0$

Every other number — positive or negative — has a positive distance from zero. So:

$\lvert n\rvert \ge 0 \quad \text{for every number } n$

⚠️ Watch out: $\lvert -3\rvert$ does not equal $-3$. Taking the absolute value of a negative number gives a positive result: $\lvert -3\rvert = 3$. The bars are not just "fancy parentheses."

In Part 2 we'll handle trickier expressions like $-\lvert -4\rvert$, where a sign sits outside the bars.

True or False? 🔽

Decide whether each statement is true or false.

🎯 Absolute Value

Part 2 of 5 — Evaluating Expressions & Comparing

🔑 The Order Rule: Treat the absolute-value bars like a grouping symbol (like parentheses). Simplify what's inside first, take the absolute value, and only then apply any sign or operation sitting outside.

Signs Outside the Bars

A negative sign outside the bars is applied after the absolute value.

Worked Example: $-\lvert -4\rvert$

1. Inside the bars: $-4$
2. Take the absolute value: $\lvert -4\rvert = 4$
3. Apply the outside negative: $-(4) = -4$

$-\lvert -4\rvert = -4$

Worked Example: $\lvert -4\rvert + \lvert -1\rvert$

Each absolute value first, then add:

$\lvert -4\rvert + \lvert -1\rvert = 4 + 1 = 5$

Worked Example: $\lvert 3 - 10\rvert$

Simplify inside first, then take the absolute value:

$\lvert 3 - 10\rvert = \lvert -7\rvert = 7$

⚠️ Common trap: $\lvert 3 - 10\rvert \ne \lvert 3\rvert - \lvert 10\rvert$. The left side is $7$; the right side is $3 - 10 = -7$. Always finish the inside before opening the bars.

Concept Check 🎯

Evaluate Each Expression 🧮

Simplify inside the bars first, then handle anything outside.

1) $\lvert -5\rvert + \lvert -2\rvert = \,?$ 2) $\lvert 4 - 11\rvert = \,?$ 3) $-\lvert 6\rvert = \,?$

Comparing With Absolute Value

To compare expressions, evaluate each one to a single number first, then compare those numbers.

Example: Which is larger, $\lvert -7\rvert$ or $\lvert 5\rvert$?

$\lvert -7\rvert = 7 \qquad \lvert 5\rvert = 5 \qquad 7 > 5$

So $\lvert -7\rvert$ is larger — even though $-7 < 5$ as plain numbers!

💡 A bigger absolute value means farther from zero — regardless of the original sign.

Compare 🔽

Evaluate each side, then choose the correct symbol.

🎯 Absolute Value

Part 3 of 5 — Distance Between Two Points

🔑 The Big Idea: The distance between two numbers $a$ and $b$ on the number line is $\lvert a - b\rvert$. Absolute value guarantees the distance comes out positive, no matter which number you subtract first.

Distance Between Two Numbers

How far apart are $3$ and $10$ on the number line? Count: $7$ units. The formula says the same thing:

$\lvert 10 - 3\rvert = \lvert 7\rvert = 7 \qquad\text{or}\qquad \lvert 3 - 10\rvert = \lvert -7\rvert = 7$

Either order gives $7$ — that's the whole point of the absolute value.

Worked Example: distance between $-4$ and $2$

$\lvert -4 - 2\rvert = \lvert -6\rvert = 6 \qquad\text{or}\qquad \lvert 2 - (-4)\rvert = \lvert 2 + 4\rvert = \lvert 6\rvert = 6$

The two points are $6$ units apart.

💡 Subtracting a negative becomes adding: $2 - (-4) = 2 + 4 = 6$. Watch those double signs!

Concept Check 🎯

Find the Distance 🧮

Use $\lvert a - b\rvert$ to find how far apart each pair is.

1) Distance between $4$ and $9 = \,?$ 2) Distance between $-3$ and $5 = \,?$ 3) Distance between $-7$ and $-1 = \,?$

Real-World Distance: Temperature Swings

Absolute value shows up any time we care about size of change, not direction.

🔑 Example: The temperature drops from $\;7^\circ\text{F}$ to $\;-5^\circ\text{F}$ overnight. How big was the change?

$\lvert 7 - (-5)\rvert = \lvert 7 + 5\rvert = \lvert 12\rvert = 12$

The temperature changed by $12$ degrees. The drop is negative ($-12$), but the size of the swing is $12$.

Temperature Change 🔽

A diver descends and a thermometer is read. Pick the size of each change.

🎯 Absolute Value

Part 4 of 5 — Opposites & Equations Like $\lvert x\rvert = 5$

🔑 Two Solutions: Because $\lvert 5\rvert = 5$ and $\lvert -5\rvert = 5$, an equation like $\lvert x\rvert = 5$ has two answers: $x = 5$ or $x = -5$. Absolute value usually points to a pair of numbers.

Opposites Share an Absolute Value

Two numbers are opposites if they're the same distance from zero on opposite sides — like $-6$ and $6$. Opposites always have the same absolute value:

$\lvert -6\rvert = \lvert 6\rvert = 6$

So if you know an absolute value, there are usually two numbers that produce it.

⚠️ Special case: $\lvert x\rvert = 0$ has only one solution, $x = 0$, because only zero sits at zero. And $\lvert x\rvert = -2$ has no solution — absolute value can't be negative.

Concept Check 🎯

Solve 🧮

Each equation has two solutions. Enter the positive solution.

1) $\lvert x\rvert = 8 \;\Rightarrow\; x = \,?$ (positive answer) 2) $\lvert x\rvert = 1 \;\Rightarrow\; x = \,?$ (positive answer)

3) How many solutions does $\lvert x\rvert = 0$ have? $\,?$

Two Points, One Distance

Think of $\lvert x\rvert = 5$ as asking: "Which points are exactly $5$ units from zero?"

$\cdots\; -5 \;\bullet\; \cdots \;\; 0 \;\; \cdots \;\bullet\; 5 \;\cdots$

There are two such points: one to the left ($-5$) and one to the right ($5$). That picture is why these equations have two answers.

💡 Pattern to remember: for any positive number $k$, $\lvert x\rvert = k \quad\Longrightarrow\quad x = k \ \text{ or } \ x = -k.$

This idea — a single distance pointing to two locations — carries straight into Algebra, where you'll solve harder absolute-value equations the same way.

Match the Solutions 🔽

🎯 Absolute Value

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) read absolute value as distance from zero, (2) evaluate expressions with signs inside and outside the bars, (3) find the distance between two numbers, and (4) solve equations like $\lvert x\rvert = k$. Let's put it all together.

Quick Reference

⚠️ Three things never to forget: absolute value is never negative; $\lvert x\rvert = 0$ has one solution; $\lvert x\rvert = (\text{negative})$ has none.

Mixed Practice 🎯

One More Set 🧮

1) $-\lvert -3\rvert - \lvert 2\rvert = \,?$ 2) Distance between $-6$ and $7 = \,?$ 3) The positive solution of $\lvert x\rvert = 14$ is $\,?$

Exit Quiz ✅

Answer all three to finish the lesson.