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Solving Absolute Value Equations: A Step-by-Step Guide
Absolute value equations are a type of equation that involves the absolute value of a variable. The absolute value of a number is its distance from zero, regardless of direction. This means that the absolute value of a number is always positive, even if the number itself is negative.
For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5. We write the absolute value of a number using vertical bars: |5| = 5 and |-5| = 5.
Solving Absolute Value Equations
To solve absolute value equations, we need to consider both the positive and negative values of the expression inside the absolute value bars. Here's a step-by-step guide:
Step 1: Isolate the Absolute Value
The first step is to isolate the absolute value expression on one side of the equation. This means getting rid of any constants or coefficients that are outside the absolute value bars.
Step 2: Set Up Two Equations
Once the absolute value expression is isolated, we need to set up two equations. The first equation will be the original equation, but without the absolute value bars. The second equation will be the same as the first, but with the expression inside the absolute value bars multiplied by -1.
Step 3: Solve Each Equation
Solve each of the two equations separately. This will give you two possible solutions to the original equation.
Step 4: Check Your Solutions
It's important to check your solutions by plugging them back into the original equation. This will help you identify any extraneous solutions (solutions that don't actually work).
Example 1
Solve the equation: |x + 2| = 5
Step 1: Isolate the Absolute Value
The absolute value expression is already isolated on the left side of the equation.
Step 2: Set Up Two Equations
Equation 1: x + 2 = 5
Equation 2: x + 2 = -5
Step 3: Solve Each Equation
For Equation 1: x = 3
For Equation 2: x = -7
Step 4: Check Your Solutions
Plugging x = 3 into the original equation, we get |3 + 2| = 5, which is true.
Plugging x = -7 into the original equation, we get |-7 + 2| = 5, which is also true.
Therefore, the solutions to the equation |x + 2| = 5 are x = 3 and x = -7.
Example 2
Solve the equation: 2|3x - 1| - 4 = 6
Step 1: Isolate the Absolute Value
Add 4 to both sides of the equation: 2|3x - 1| = 10
Divide both sides by 2: |3x - 1| = 5
Step 2: Set Up Two Equations
Equation 1: 3x - 1 = 5
Equation 2: 3x - 1 = -5
Step 3: Solve Each Equation
For Equation 1: x = 2
For Equation 2: x = -4/3
Step 4: Check Your Solutions
Plugging x = 2 into the original equation, we get 2|3(2) - 1| - 4 = 6, which is true.
Plugging x = -4/3 into the original equation, we get 2|3(-4/3) - 1| - 4 = 6, which is also true.
Therefore, the solutions to the equation 2|3x - 1| - 4 = 6 are x = 2 and x = -4/3.
Practice Problems
Here are some practice problems for you to try:
- |x - 3| = 7
- 4|2x + 5| + 1 = 17
- |x/2 - 1| = 3
Remember to follow the steps above to solve these equations. Good luck!