Question

Make notes for this One-Variable Equations & Inequalities

Answer 1

Certainly! Here are concise notes on One-Variable Equations and Inequalities:

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### One-Variable Equations

**Definition:** An equation involving only one variable where the objective is to find the value of that variable that makes the equation true.

#### General Form:
- \( ax + b = c \)
- Where \( a, b, c \) are constants, and \( x \) is the variable.

#### Solving Steps:
1. **Isolate the variable**: Use inverse operations (addition, subtraction, multiplication, division).
2. **Simplify both sides**: Combine like terms if necessary.
3. **Check your solution**: Substitute back into the original equation to ensure both sides are equal.

#### Example:
- Solve \( 2x + 3 = 11 \)
- Subtract 3: \( 2x = 8 \)
- Divide by 2: \( x = 4 \)

### One-Variable Inequalities

**Definition:** An inequality compares two expressions involving one variable and determines the range of values that satisfy the inequality.

#### General Forms:
- \( ax + b < c \)
- \( ax + b \leq c \)
- \( ax + b > c \)
- \( ax + b \geq c \)

#### Solving Steps:
1. **Isolate the variable**: Same as equations, use inverse operations to get the variable on one side.
2. **Flip inequality sign**: When multiplying or dividing by a negative number, flip the inequality sign.
3. **Express the solution**: Either as an interval or using set notation.

#### Example:
- Solve \( 3x - 5 > 4 \)
- Add 5: \( 3x > 9 \)
- Divide by 3: \( x > 3 \)
- Solution: \( (3, \infty) \)

### Graphing Inequalities:

- **Open Circle**: Represents \( < \) or \( > \) (not inclusive).
- **Closed Circle**: Represents \( \leq \) or \( \geq \) (inclusive).
- **Shading**: Indicates the solution set. Shade to the left for \( < \) or \( \leq \), and to the right for \( > \) or \( \geq \).

### Key Points to Remember:

- Make sure to perform the same operation on both sides of the equation/inequality.
- Always check your final solution by substituting it back.
- When representing your solution graphically, be mindful of whether the endpoints are included.

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These notes cover the basic principles and steps for solving one-variable equations and inequalities. Let me know if you need further details or examples!