Math with Mr. J: Solving One-Step Inequalities

Introduction

• Topic: Solving one-step inequalities
• Key Concept: Similar to one-step equations; isolate the variable using inverse operations.
• Difference from Equations: Inequalities have infinite solutions, while equations have one.
• Important Rule: Flip the inequality symbol when multiplying or dividing both sides by a negative number.

Example 1: ( y + 7 < 8 )

• Objective: Isolate ( y )
• Operation: Subtract 7 (inverse of +7)
  • ( y + 7 - 7 < 8 - 7 )
  • Result: ( y < 1 )
• Solution: Any number less than 1
• Verification: Plug ( y = 0 ), and it satisfies ( 0 + 7 < 8 )

Example 2: ( \frac{x}{5} \geq 3 )

• Objective: Isolate ( x )
• Operation: Multiply by 5 (inverse of division by 5)
  • ( 5 \times \frac{x}{5} \geq 3 \times 5 )
  • Result: ( x \geq 15 )
• Solution: Any number 15 or greater
• Verification: Plug ( x = 20 ), and it satisfies ( \frac{20}{5} \geq 3 )

Example 3: ( 14 \geq n - 11 )

• Objective: Isolate ( n )
• Operation: Add 11 (inverse of -11)
  • ( 14 + 11 \geq n - 11 + 11 )
  • Result: ( 25 \geq n )
• Solution: Any number 25 or less
• Verification: Plug ( n = 20 ), and it satisfies ( 14 \geq 20 - 11 )

Example 4: ( -6r < 36 )

• Objective: Isolate ( r )
• Operation: Divide by -6 (inverse of multiplication by -6)
  • Important Step: Flip inequality sign
  • ( r > -6 )
• Solution: Any number greater than -6
• Verification: Plug ( r = -2 ), satisfies ( -6(-2) < 36 )
• Note: Flipping sign necessary for correct solutions

Summary

• Solving one-step inequalities involves isolating the variable similar to solving equations.
• Must remember to flip the inequality symbol when multiplying/dividing by negative numbers.
• Infinite solutions differentiate inequalities from equations.
• Additional resources available for further understanding why the inequality sign is flipped.