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!