Absolute Value Calculator

A Complete Guide On Absolute Valute Calculator:

Imagine you’re five miles from home. That could mean you’re five miles north, or five miles south. But either way, your distance from home is still five miles. That’s what absolute value is all about—distance, no matter the direction. An absolute value equation, like $|x − 3| = 7$, asks: what numbers are exactly 7 units away from 3?

In this article, we’ll explore how to solve these kinds of equations, how to use the Symbolab Absolute Value Equation Calculator to check your work, and why this idea, distance without direction, shows up everywhere, from measuring how far you are from a high score in a game to figuring out if your guess was close on a math quiz.

What Is Absolute Value?

Let’s go back to the number line, the one you probably first saw in middle school, stretching left and right from zero. On that line, the number 5 is five steps to the right of zero. And −5? It’s five steps to the left. But either way, both are the same distance from zero. That distance, from zero to a number, no matter which direction, is what we call absolute value.

In math terms, the absolute value of a number is written with two vertical bars, like this: $|x|$.

It’s just asking: “How far is x from zero?”

So:

• $|5| = 5$, because 5 is five units from 0.
• $|−5| = 5$ too, because even though it’s on the opposite side, it’s still five units away.

Here’s the formal definition:

• If x is positive or zero, then $|x| = x$.
• If x is negative, then $|x| = −x$ (because flipping the sign makes it positive—aka a distance).

Think of it like this: absolute value doesn’t care which way you went, just how far you ended up from the starting point. And if you’re a visual learner, picture those numbers on a number line: both 5 and −5 are sitting the same number of spaces from zero, just on different sides.

Absolute value is a way to express that symmetry in numbers.

Real-world example: You’re trying to meet a friend outside school, and they text you, “I’m two blocks away.” It doesn’t matter if they’re coming from the north side or the south side, what matters is they’re two blocks away. That’s absolute value. It gives us the distance, no matter the direction.

What Is an Absolute Value Equation?

Now that we know absolute value means distance from zero, let’s take it one step further: what happens when that absolute value is part of an equation?

An absolute value equation is just what it sounds like—an equation that includes an absolute value expression. It’s usually written in the form:

$|expression| = value$

The goal? To figure out which values of the variable make that equation true.

Let’s look at a few different kinds of absolute value equations—and how each one plays out, both mathematically and in real life.

1. |expression| = positive number → two solutions

Example: $|x − 4| = 6$

This is asking: “What numbers are exactly 6 units away from 4?”

The answer: $x = 10$ and $x = −2$, because both are six steps from 4 on the number line.

Real-world example:

Imagine your school is 4 miles from home. You’re told your GPS shows you’re exactly 6 miles away from school. Where could you be?

You could be 6 miles to the east of the school or 6 miles to the west. Either way, you’re 6 miles away. That gives you two possible locations.

2. |expression| = 0 → one solution

Example: |x + 3| = 0

This equation is asking: “What value makes the expression inside the absolute value equal to zero?”

The only answer is $x = −3$.

Real-world example:

You’re meeting a friend at the movie theater. You both check your phones and realize you’re exactly at the same location—your distance from each other is zero. There’s only one point where that happens: when you’re standing in the same spot. So there’s only one solution.

3. |expression| = negative number → no solution

Example: $|x − 2| = −5$

This equation is asking: “What number is 5 units away from 2... in a negative direction?”

But that’s not possible—absolute value represents distance, and distance can’t be negative.

Real-world example:

Let’s say your fitness tracker shows how far you’ve walked from your starting point. If it told you you’re “−5 steps away” from your starting line, you’d probably laugh—it doesn’t make sense. You can walk backward, sure, but your distance from the start is still a positive number. Negative distance just doesn’t exist, so this kind of equation has no solution.

So to sum it up:

• $|expression|$ = positive number → two solutions (like being either side of a location)
• $|expression| = 0$ → one solution (you’re at the exact spot)
• $|expression|$ = negative number → no solution (you can’t be a negative distance away)

Solving Absolute Value Equations: Step-by-Step

When you see an equation like $|A| = B$, you’re being asked: what values of A are a distance of B units from zero? Solving absolute value equations is really about solving distance problems, just in math language. Let’s break this into three main cases, and we’ll walk through each type of equation with a real-life example.

Case 1: $|A| = B$, where $B > 0$

When the absolute value equals a positive number, there are two possible answers because a number can be that distance in either direction.

Math Example:

$|x − 4| = 6$

Step 1: Split into two equations:

→ $x − 4 = 6$

→ $x − 4 = −6$

Step 2: Solve both:

→ $x = 10$ or $x = −2$

Final Answer: $x = 10$ or $x = −2$

Real-World Example:

Your school is located at mile marker 4 on a long, straight road. You’re told that you’re exactly 6 miles away. That could mean you’re at mile 10 (6 miles past the school), or at mile −2 (6 miles in the opposite direction). Same distance, different directions—two correct answers.

Case 2: $|A| = 0$

When the absolute value equals zero, there's only one possible solution—because the only number that’s zero units away from something is that number itself.

Math Example:

$|x + 3| = 0$

Step 1: Set the inside equal to 0:

→ $x + 3 = 0$

→ $x = −3$

Final Answer: $x = −3$

Real-World Example:

You’re playing a game where you have to get your character to the center of a target at position −3. The game says you’re 0 units away from it. That can only mean one thing: you’re already standing on −3. There’s only one way that’s possible when the expression inside equals zero.

Case 3: $|A| = B$, where $B < 0$

This is the trick question. You can’t have an absolute value equal a negative number, because distance is always zero or positive. So these have no solution.

Math Example:

$|2x − 5| = −4$

Final Answer: No solution

Real-World Example: Your fitness app tracks how far you’ve moved from your starting point. Today, it tells you that you’re −4 meters from where you started. That’s not possible. You might be back at zero (you didn’t move), or you could be 4 meters away, but there’s no such thing as negative distance. The app must be broken, or the equation has no answer.

### Recap: Building Blocks of Solving Absolute Value Equations

• Positive number on the right? → Two answers
• Zero on the right? → One answer
• Negative number? → No answer at all

Case 4: Absolute Value on Both Sides — $|A| = |B|$

So far, we’ve looked at equations where the absolute value is on just one side. But what happens if both sides of the equation have absolute values? Something like:

$|2x − 1| = |x + 3|$

Now we’re comparing two distances. The question becomes: when are these two expressions the same distance from zero?

And just like before, we split it into two separate equations—because there are still two possible ways for the absolute values to be equal:

The expressions are equal.

→ $2x − 1 = x + 3$

The expressions are opposites. → $2x − 1 = −(x + 3)$

Let’s solve them.

Math Example:

$|2x − 1| = |x + 3|$

Case 1:

$2x − 1 = x + 3$

→ Subtract x from both sides: $x − 1 = 3$ → Add 1 to both sides: $x = 4$

Case 2:

$2x − 1 = −(x + 3)$

→ Distribute the negative: $2x − 1 = −x − 3$

→ Add x to both sides: $3x − 1 = −3$

→ Add 1 to both sides: $3x = −2$

→ Divide by 3: $( x = -\frac{2}{3})$

Final Answer: $x = 4$ or $( x = -\frac{2}{3})$

Real-World Example:

You and your friend are walking toward each other on opposite ends of a long hallway. Somewhere in between, there's a vending machine—and you're both trying to figure out where it is. You’re told that your distance from the vending machine is exactly the same as your friend’s. So what are the possibilities?

You could both be standing exactly the same number of steps away from the machine, maybe you're at one end of the hallway and your friend is at the other, or maybe you're both close to it from opposite sides. Either way, the distances are equal.

That’s what $|A| = |B|$ means in the world of absolute value: we’re looking for the points where two things are equally far from something, whether they’re on the same side or opposite sides.

Common Mistakes to Avoid While Solving Absolute Equations

Absolute value equations can be surprisingly sneaky. They look straightforward, but it’s easy to miss a sign, forget a case, or assume you’re done when there’s still more to check. Here are some of the most common mistakes students make—and how to steer clear of them.

1. Forgetting to Write Both Cases

This is the big one. Remember, $|A| = B$ (when B is positive) has two solutions:

One where $A = B$ And one where $A = −B$

What goes wrong:

You solve just one version—maybe $x − 3 = 7$—but forget to also solve $x − 3 = −7$.

How to avoid it:

As soon as you see an absolute value, pause and ask: "Am I writing both cases?" Even jotting a quick note like "two cases!" in the margin can help.

2. Dropping the Absolute Value Too Early

Sometimes students try to skip the case-splitting and just remove the bars like they’re parentheses. But absolute value doesn’t work that way.

What goes wrong:

You treat $|x − 5| = 3$ like $x − 5 = 3$, and miss the negative case entirely.

How to avoid it:

Treat absolute value bars like a red flag: they mean stop and think. Don’t drop them until you’ve written your two separate equations.

3. Not Checking for Extraneous Solutions

This mistake usually shows up with more complex problems—especially when variables appear on both sides of the equation or after multiple steps.

What goes wrong:

You solve both cases, but one of your answers doesn’t actually work when you plug it back in. Maybe it makes the absolute value equal a negative number, or creates an equation that’s false.

How to avoid it:

Always plug your answers back into the original equation. Not the simplified one. If it doesn’t make the equation true, it’s not a real solution—it’s extraneous.

4. Assuming All Absolute Value Equations Have Two Answers

It’s true that $|A| = B$ usually has two solutions—but not always.

What goes wrong:

You treat $|x + 1| = 0$ like it has two answers, when really there’s only one: $x = −1$.

How to avoid it:

Check the number on the right side.

a. If it’s positive → two solutions

b. If it’s zero → one solution

c. If it’s negative → no solution

This quick scan can save you from over-solving (or overthinking).

5. Ignoring the Absolute Value Altogether

It happens more often than you’d think—students sometimes get so used to solving regular equations that they overlook the absolute value bars completely.

What goes wrong:

You solve $|x − 2| = 8$ like it’s just $x − 2 = 8$, and stop at $x = 10$.

How to avoid it:

Circle or highlight absolute value symbols in the problem. Anything that makes them jump off the page and reminds you: "This is not a regular equation."

Solving absolute value equations isn’t just about doing the steps, it’s about thinking carefully. Each solution tells you something about distance, direction, and possibility. If you slow down, write out your cases, and double-check your work, you’ll catch mistakes before they catch you.

Using the Symbolab Absolute Value Equation Calculator

You’ve practiced the steps. You’ve done the math. But now you want to check your answer—or better yet, see the solution unfold in real time. That’s exactly what the Symbolab Absolute Value Equation Calculator is for. It’s not just about getting the right number, it’s about understanding the process, and having a little backup when you need it.

Step 1: Enter the Equation

At the top of the Absolute Value Equation Calculator page, you’ll see a large input bar. You can enter your equation in several ways:

1. Type it directly using your keyboard: The calculator understands $abs()$ as absolute value, just like vertical bars. Example: $abs(2x + 3) = abs(x - 2)$

2. Use the on-screen math keyboard: Tap the keyboard icon to insert absolute value symbols, variables, and math operators. This is especially helpful on tablets or touchscreens.

3. You can also type it out in words like: “absolute value of $2x + 3$ equals absolute value of $x - 2$”—Symbolab will interpret it correctly.

4. Upload a handwritten equation or screenshot: If you’ve written your equation by hand or have it on paper, click the camera icon and upload a clear photo or screenshot.

Once your equation is in, you're ready to move on to solving it—just click “Go.”

Step 2: See the Final Answer Immediately

Symbolab will show the solution at the top, right after solving:

For example, with $∣2x+3∣=∣x−2∣$, the solution appears as: $x=−5$ or $x = -\frac{1}{3}$

Both the fractional form and decimal approximation are provided.

Example: $x=−5$ or $x = -\frac{1}{3}$​ is also shown as $x = -5$ and $x = -0.333…$

This helps you quickly check if your answer is correct, especially useful when comparing to your own work or test prep.

Step 3: Review the Step-by-Step Solution

Click “One step at a time” (if not already expanded). Each step is labeled, explained, and easy to follow. You can click any section to expand the explanation, trace how the expression was simplified, and understand how each answer was found—or excluded.

Step 4: Check the Graph

Scroll down, and you’ll see a graph of the equation. Use the graph to connect the numbers to their meaning. You’ll then see the equation come to life.

Step 5: Try More Examples or Ask for Help

Scroll further and you’ll find:

1. A section with Absolute Value Examples to try out.

2. A chat box where you can ask Symbolab for help on a specific step or term you don’t understand.

3. A button to save your work to a Notebook if you’re using a Symbolab account.

No matter how you input it, typed, drawn, or snapped from your notes, the Symbolab Absolute Value calculator gives you a clean, step-by-step solution. It's like having a patient math tutor walking through the problem with you, explaining each move.

Real-World Applications of Absolute Value Equations

Absolute value equations aren’t just for solving math problems—they help us measure how far something is from a target, no matter the direction. Here’s where you might see them in everyday life:

1. Navigation & Distance: Find how far a point is from a location, regardless of direction.

   Example: $|x − 3| = 7$ means you are 7 miles from mile marker 3, whether east or west.

2. Sports & Scoring: Measure how far off a score is from the target or record.

   Example: $|46 − 50| = 4$ shows you're 4 points away from the winning score.

3. Engineering & Manufacturing: Keep measurements within a specific tolerance range.

   Example: $|x − 10| ≤ 0.2$ means a part must be within 0.2 mm of 10 mm.

4. Temperature Control: Maintain safe operating ranges in heating or cooling systems.

   Example: $|x − 75| ≤ 3$ means the temperature must stay within 3°F of 75°F.

5. Error Checking: Measure how far a guess or estimate is from the correct value.

   Example: $|18 − 22| = 4$ means your answer was 4 units off.

6. Gaming & Leaderboards: Show how close a player is to beating a high score.

   Example: $|8200 − 8500| = 300$ means you’re 300 points away from the top score.

Absolute value equations aren’t just about solving for x—they’re about understanding distance, precision, and possibility. Whether you're figuring out how far off your answer is, checking tolerances in a science lab, or measuring differences in real life, this kind of math shows up more than you might expect. The key is knowing how to break these problems down, avoid common pitfalls, and think through both sides of the equation, literally.

With tools like the Symbolab Absolute Value Equation Calculator, you don’t have to figure it out alone. Keep practicing, keep asking questions, and remember: every step forward is part of the learning process.