Solving Absolute Value Equations

TL;DR

Absolute value represents a number's distance from zero, so an absolute value equation usually has two possible solutions. To solve them, first isolate the absolute value expression, then set up two separate equations: one with the positive value and one with the negative. Always check your solutions to make sure they're valid.

1. The Mental Model

Think of absolute value as asking "how far away from zero is this number?" Since you can go left (negative) or right (positive) from zero, there are usually two numbers that are the same distance away. For example, both 5 and -5 are 5 units away from zero.

2. The Core Material

When you see an absolute value equation like |x| = 5, it's asking: "What number(s) are 5 units away from zero?" The answers are x = 5 and x = -5.

The key to solving these equations is to remember that the expression inside the absolute value bars could be either positive or negative to get the final positive result.

How to Solve Absolute Value Equations

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1. Isolate the Absolute Value: Get the absolute value expression by itself on one side of the equation. This means moving any other numbers or terms away from it.
2. Split into Two Equations: Once the absolute value is isolated, you'll create two separate equations.
   • Equation 1: The expression inside the absolute value equals the positive value on the other side.
   • Equation 2: The expression inside the absolute value equals the negative value on the other side.
3. Solve Each Equation: Solve both of these new equations independently.
4. Check Your Solutions: Plug each solution back into the original absolute value equation. Sometimes, a solution you find won't actually work (this is called an extraneous solution).

graph TD
    A["Start"] --> B["Isolate the absolute value expression"]
    B --> C{{"Isolate successful?"}}
    C -- "No" --> B
    C -- "Yes" --> D["Split into two equations:"]
    D --> E["1. Expression = Positive Value"]
    D --> F["2. Expression = Negative Value"]
    E --> G["Solve Equation 1"]
    F --> H["Solve Equation 2"]
    G --> I["Check solution 1 in original equation"]
    H --> J["Check solution 2 in original equation"]
    I --> K["Valid Solution(s)"]
    J --> K
    K --> L["End"]

Special Cases:

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• Absolute Value Equals a Negative Number: If you end up with something like |x| = -3 after isolating the absolute value, there is no solution. Absolute value always represents a distance, and distance can't be negative.
• Absolute Value Equals Zero: If you have |x| = 0, there's only one solution: x = 0. This is because 0 is the only number that's 0 units away from zero.

3. Worked Example

Let's solve 3|2x - 1| + 5 = 14.

1. Isolate the Absolute Value:

   • Subtract 5 from both sides:
     3|2x - 1| = 14 - 5
3|2x - 1| = 9
   • Divide by 3:
     |2x - 1| = 9 / 3
|2x - 1| = 3

2. Split into Two Equations:

   • Equation 1: 2x - 1 = 3
   • Equation 2: 2x - 1 = -3

3. Solve Each Equation:

   • For Equation 1:
     2x - 1 = 3
2x = 4
x = 2
   • For Equation 2:
     2x - 1 = -3
2x = -2
x = -1

4. Check Your Solutions:

   • Check x = 2:
     3|2(2) - 1| + 5 = 14
3|4 - 1| + 5 = 14
3|3| + 5 = 14
3(3) + 5 = 14
9 + 5 = 14
14 = 14 (This works!)
   • Check x = -1:
     3|2(-1) - 1| + 5 = 14
3|-2 - 1| + 5 = 14
3|-3| + 5 = 14
3(3) + 5 = 14
9 + 5 = 14
14 = 14 (This also works!)

So, the solutions are x = 2 and x = -1.

4. Key Takeaways

• Absolute value measures a number's distance from zero, always resulting in a non-negative value.
• Most absolute value equations will have two distinct solutions.
• Always isolate the absolute value expression before splitting the equation.
• You create two equations by setting the inside expression equal to both the positive and negative versions of the other side.
• Remember to check all your solutions in the original equation to catch extraneous solutions.
• If an isolated absolute value equals a negative number, there's no solution.
• If an isolated absolute value equals zero, there's only one solution.

Common Mistakes to Avoid:
- Don't split the equation before isolating the absolute value expression.
- Forgetting to create the second equation (the negative case).
- Incorrectly distributing a negative sign when solving the second equation.
- Not checking solutions, which can lead to including extraneous solutions.

5. Now Try It

Solve the equation |4x + 8| - 7 = 9. Show all your steps, including isolating the absolute value, setting up two equations, solving each, and checking your final answers.
Success looks like correctly finding two valid solutions for x.