Solving Linear Equations and Linear Inequalities

Introduction

• Linear equations and inequalities involve constants and variables.
• Linear Equations use the equal sign (=) (e.g., 2x + 1 = 5).
• Linear Inequalities use inequality signs (>, <, ≥, ≤) (e.g., 2x + 1 > 5).

Key Concepts

Solving Linear Equations

1. Goal: Find the value of the variable by isolating it.
2. Maintain equality by performing the same operation on both sides.

Types of Linear Equations

• Linear Equations in One Variable
  • Example: Solve 2x + 1 = 5 → x = 2
• Combining Like Terms
  • Example: 2x - 4 = 5x → x = 3
• Distributing Coefficients
  • Example: 2(x + 1) = 5 → x = 1.5
• Fractions
  • When dealing with fractions, consider the equation's structure:
    • Keep the fraction until the final step if it's a coefficient.
    • Get rid of fractions early if both coefficients and constants are fractions.
• Negative Numbers
  • Negative x Negative = Positive
  • Positive x Negative = Negative

Linear Equations in Two Variables

• Example: If 2x + 5y = 1 and x = 3, solve for y to get y = -1.

Using Linear Equations to Evaluate Expressions

• Solve the equation first or find a relationship to evaluate other expressions.
• Example: If 2x + 1 = 5, evaluate 8x + 4 → result is 20.

Solving Linear Inequalities

• Process similar to linear equations, but pay attention to inequality signs.
• No Reversal Needed
  • Example: 2x + 1 > 5 → x > 2
• Reversal Needed
  • If dividing by a negative, reverse the inequality sign.
  • Example: -2x > 4 → x < -2

Number of Solutions

• One Solution: Equation can be rewritten as x = a.
• No Solution: Equation reduces to a ≠ b when constants differ.
• Infinitely Many Solutions: Equation reduces to a form like x = x.

Practice Problems

• Evaluate a linear expression (e.g., If 6x + 10 = 24, find 3x + 5).
• Solve a linear equation in one variable.
• Identify values that do not satisfy an inequality.
• Determine conditions for no solution in an equation.

Tips

• Always treat both sides equally when solving equations.
• Reversal of inequality signs is necessary when multiplying or dividing by negative numbers.
• Use systematic steps to avoid errors and manage time efficiently on tests.