Solving Absolute Value Equations: Step-by-Step Tutorial

📚 What are Absolute Value Equations?

Absolute value equations are equations where the variable is inside an absolute value symbol. The absolute value of a number is its distance from zero on the number line, so it's always non-negative. Because of this, absolute value equations often have two possible solutions.

📜 A Little History

The concept of absolute value has been used implicitly for centuries, but the modern notation using vertical bars $|x|$ became popular in the 19th century. It's a fundamental concept in real analysis and has applications in various fields like physics and engineering.

🔑 Key Principles for Solving

• ➕ Isolate the Absolute Value: Before doing anything else, get the absolute value expression alone on one side of the equation. For example, in $2|x - 3| + 1 = 9$, isolate $|x - 3|$ first.
• ↔️ Consider Both Positive and Negative Cases: Since the absolute value makes a number positive, you need to consider both the positive and negative possibilities inside the absolute value. If $|x| = a$, then $x = a$ or $x = -a$.
• 🧩 Solve Each Equation Separately: Solve the two resulting equations for the variable.
• ✅ Check Your Solutions: Plug each solution back into the original equation to make sure it's valid. Absolute value equations can sometimes have extraneous solutions (solutions that don't actually work).

🪜 Step-by-Step Solution

Let's solve the equation $|2x - 1| = 5$.

1. ➕Set up two equations: $2x - 1 = 5$ and $2x - 1 = -5$.
2. ➗Solve the first equation: $2x - 1 = 5$ \Rightarrow $2x = 6$ \Rightarrow $x = 3$.
3. ➖Solve the second equation: $2x - 1 = -5$ \Rightarrow $2x = -4$ \Rightarrow $x = -2$.
4. ✅Check solutions: Plug $x = 3$ and $x = -2$ back into the original equation. Both work!

✍️ Examples in Action

Example 1: Solve $|x + 4| = 7$

• ➕ Set up two equations: $x + 4 = 7$ and $x + 4 = -7$
• ➗ Solve: $x = 3$ and $x = -11$

Example 2: Solve $|3x - 2| = 10$

• ➕ Set up two equations: $3x - 2 = 10$ and $3x - 2 = -10$
• ➗ Solve: $3x = 12 \Rightarrow x = 4$ and $3x = -8 \Rightarrow x = -\frac{8}{3}$

Example 3: Solve $|2x + 5| = -1$

• 🛑 No Solution: Absolute value cannot be negative.

📝 Practice Quiz

Solve the following absolute value equations:

1. $|x - 2| = 3$
2. $|2x + 1| = 7$
3. $|3x - 4| = 5$
4. $|4x + 2| = 6$
5. $|x + 5| = 2$
6. $|2x - 3| = 9$
7. $|5x + 1| = 4$

Solutions are at the end of this article.

💡 Tips and Tricks

• ⚠️ Beware of Negative Values: An absolute value expression can never equal a negative number. If you see $|something| = -number$, there is no solution.
• 🔍 Isolate Carefully: Make sure to isolate the absolute value before splitting into two equations.
• ✍️ Always Check: Check your answers in the original equation!

🌍 Real-World Applications

Absolute value is used to represent distances or magnitudes, so absolute value equations appear in physics (e.g., calculating error margins), engineering (e.g., tolerances in manufacturing), and computer science (e.g., measuring differences in data).

🔑 Conclusion

Solving absolute value equations involves isolating the absolute value, considering both positive and negative cases, and carefully checking your solutions. With practice, you'll master this important algebraic skill!

🔑 Quiz Solutions

1. $x = 5, -1$
2. $x = 3, -4$
3. $x = 3, -\frac{1}{3}$
4. $x = 1, -2$
5. $x = -3, -7$
6. $x = 6, -3$
7. $x = \frac{3}{5}, -1$