Solving Absolute Value Equations
Absolute value equations involve finding the value(s) of a variable that make an absolute value expression equal to a specific value. Understanding how to solve these equations is crucial for solving more complex algebraic problems.
Outline
- Introduction to Absolute Value Equations
- Solving Basic Absolute Value Equations
- Solving Absolute Value Equations with Multiple Terms
- Examples and Practice Exercises
- Homework
- Revision
- YouTube Video Explanation
Examples
Example 1
Solve the equation \( | x | = 4 \).
The absolute value equation \( | x | = 4 \) has two solutions: \( x = 4 \) and \( x = -4 \).
Example 2
Solve the equation \( | x - 3 | = 5 \).
To solve \( | x - 3 | = 5 \), set up two equations: \( x - 3 = 5 \) and \( x - 3 = -5 \). Solving these gives \( x = 8 \) and \( x = -2 \).
Example 3
Solve the equation \( | 2x + 1 | = 7 \).
To solve \( | 2x + 1 | = 7 \), set up two equations: \( 2x + 1 = 7 \) and \( 2x + 1 = -7 \). Solving these gives \( x = 3 \) and \( x = -4 \).
Example 4
Solve the equation \( | 3x - 2 | = 8 \).
To solve \( | 3x - 2 | = 8 \), set up two equations: \( 3x - 2 = 8 \) and \( 3x - 2 = -8 \). Solving these gives \( x = \frac{10}{3} \) and \( x = -2 \).
Example 5
Solve the equation \( | x/2 + 4 | = 3 \).
To solve \( | x/2 + 4 | = 3 \), set up two equations: \( x/2 + 4 = 3 \) and \( x/2 + 4 = -3 \). Solving these gives \( x = -2 \) and \( x = -14 \).
Example 6
Solve the equation \( | 5 - x | = 2 \).
To solve \( | 5 - x | = 2 \), set up two equations: \( 5 - x = 2 \) and \( 5 - x = -2 \). Solving these gives \( x = 3 \) and \( x = 7 \).
Exercises
Exercise 1
Solve the equation \( | x + 5 | = 9 \).
Solutions are \( x = 4 \) and \( x = -14 \).
Exercise 2
Solve the equation \( | 4x - 1 | = 7 \).
Solutions are \( x = 2 \) and \( x = -1.5 \).
Exercise 3
Solve the equation \( | 3x + 2 | = 11 \).
Solutions are \( x = 3 \) and \( x = -\frac{13}{3} \).
Exercise 4
Solve the equation \( | x/3 - 2 | = 5 \).
Solutions are \( x = 21 \) and \( x = -9 \).
Exercise 5
Solve the equation \( | 2 - x | = 6 \).
Solutions are \( x = -4 \) and \( x = 8 \).
Exercise 6
Solve the equation \( | 6x + 3 | = 15 \).
Solutions are \( x = 2 \) and \( x = -3 \).
Homework
Complete the following problems for additional practice:
- Solve \( | x - 7 | = 10 \)
- Solve \( | 2x + 4 | = 8 \)
- Solve \( | 3x - 5 | = 12 \)
- Solve \( | x/4 + 6 | = 5 \)
- Solve \( | 7 - x | = 3 \)
Revision
Review these important points:
- To solve \( | A | = B \), consider both \( A = B \) and \( A = -B \).
- Ensure to check your solutions in the original equation to confirm they work.
- Practice with various problems to gain confidence in solving absolute value equations.
Video Explanation
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