Question
Lesson 6 absolute value equations
Answers
GPT-4o mini
Sure! An absolute value equation is an equation that contains an absolute value expression. The general form of an absolute value equation is:
\[
|x| = a
\]
where \( a \) is a non-negative number (i.e., \( a \geq 0 \)). The absolute value \( |x| \) represents the distance of \( x \) from 0 on a number line, regardless of direction.
### Solving Absolute Value Equations
To solve an absolute value equation, you can use the following steps:
1. **Isolate the absolute value expression**: If the absolute value expression is not already isolated, move all terms to one side of the equation.
2. **Set up two separate equations**: The equation \( |x| = a \) can be rewritten as two separate equations:
- \( x = a \)
- \( x = -a \)
3. **Solve each equation**: Solve both equations to find the possible solutions.
4. **Check for extraneous solutions**: Substitute each solution back into the original equation to ensure it satisfies the equation.
### Example 1: Simple Absolute Value Equation
Solve the equation:
\[
|x - 3| = 5
\]
**Step 1**: Set up the two equations:
\[
x - 3 = 5 \quad \text{and} \quad x - 3 = -5
\]
**Step 2**: Solve each equation:
1. \( x - 3 = 5 \)
- \( x = 8 \)
2. \( x - 3 = -5 \)
- \( x = -2 \)
**Step 3**: The solutions are \( x = 8 \) and \( x = -2 \).
### Example 2: Absolute Value with a Variable Expression
Solve the equation:
\[
|2x + 4| = 10
\]
**Step 1**: Set up the two equations:
\[
2x + 4 = 10 \quad \text{and} \quad 2x + 4 = -10
\]
**Step 2**: Solve each equation:
1. \( 2x + 4 = 10 \)
- Subtract 4 from both sides: \( 2x = 6 \)
- Divide by 2: \( x = 3 \)
2. \( 2x + 4 = -10 \)
- Subtract 4 from both sides: \( 2x = -14 \)
- Divide by 2: \( x = -7 \)
**Step 3**: The solutions are \( x = 3 \) and \( x = -7 \).
### Example 3: Absolute Value with Multiple Terms
Solve the equation:
\[
|x^2 - 1| = 3
\]
**Step 1**: Set up the two equations:
\[
x^2 - 1 = 3 \quad \text{and} \quad x^2 - 1 = -3
\]
**Step 2**: Solve each equation:
1. \( x^2 - 1 = 3 \)
- Add 1 to both sides: \( x^2 = 4 \)
- Take the square root: \( x = 2 \) or \( x = -2 \)
2. \( x^2 - 1 = -3 \)
- Add 1 to both sides: \( x^2 = -2 \)
- This has no real solutions since you cannot take the square root of a negative number.
**Step 3**: The solutions are \( x = 2 \) and \( x = -2 \).
### Key Points to Remember
- \(|x| = a\) has solutions \(x = a\) or \(x = -a\) (for \(a \geq 0\)).
- Always check for extraneous solutions by plugging them back into the original equation.
- Recognize that some absolute value equations can lead to no real solutions.
Feel free to ask if you have more questions or need further clarification!
\[
|x| = a
\]
where \( a \) is a non-negative number (i.e., \( a \geq 0 \)). The absolute value \( |x| \) represents the distance of \( x \) from 0 on a number line, regardless of direction.
### Solving Absolute Value Equations
To solve an absolute value equation, you can use the following steps:
1. **Isolate the absolute value expression**: If the absolute value expression is not already isolated, move all terms to one side of the equation.
2. **Set up two separate equations**: The equation \( |x| = a \) can be rewritten as two separate equations:
- \( x = a \)
- \( x = -a \)
3. **Solve each equation**: Solve both equations to find the possible solutions.
4. **Check for extraneous solutions**: Substitute each solution back into the original equation to ensure it satisfies the equation.
### Example 1: Simple Absolute Value Equation
Solve the equation:
\[
|x - 3| = 5
\]
**Step 1**: Set up the two equations:
\[
x - 3 = 5 \quad \text{and} \quad x - 3 = -5
\]
**Step 2**: Solve each equation:
1. \( x - 3 = 5 \)
- \( x = 8 \)
2. \( x - 3 = -5 \)
- \( x = -2 \)
**Step 3**: The solutions are \( x = 8 \) and \( x = -2 \).
### Example 2: Absolute Value with a Variable Expression
Solve the equation:
\[
|2x + 4| = 10
\]
**Step 1**: Set up the two equations:
\[
2x + 4 = 10 \quad \text{and} \quad 2x + 4 = -10
\]
**Step 2**: Solve each equation:
1. \( 2x + 4 = 10 \)
- Subtract 4 from both sides: \( 2x = 6 \)
- Divide by 2: \( x = 3 \)
2. \( 2x + 4 = -10 \)
- Subtract 4 from both sides: \( 2x = -14 \)
- Divide by 2: \( x = -7 \)
**Step 3**: The solutions are \( x = 3 \) and \( x = -7 \).
### Example 3: Absolute Value with Multiple Terms
Solve the equation:
\[
|x^2 - 1| = 3
\]
**Step 1**: Set up the two equations:
\[
x^2 - 1 = 3 \quad \text{and} \quad x^2 - 1 = -3
\]
**Step 2**: Solve each equation:
1. \( x^2 - 1 = 3 \)
- Add 1 to both sides: \( x^2 = 4 \)
- Take the square root: \( x = 2 \) or \( x = -2 \)
2. \( x^2 - 1 = -3 \)
- Add 1 to both sides: \( x^2 = -2 \)
- This has no real solutions since you cannot take the square root of a negative number.
**Step 3**: The solutions are \( x = 2 \) and \( x = -2 \).
### Key Points to Remember
- \(|x| = a\) has solutions \(x = a\) or \(x = -a\) (for \(a \geq 0\)).
- Always check for extraneous solutions by plugging them back into the original equation.
- Recognize that some absolute value equations can lead to no real solutions.
Feel free to ask if you have more questions or need further clarification!