Solving Inequalities

Problem Statement

• Solve the inequality ( \frac{2}{3} > -4y - 8 \frac{1}{3} ).

Steps to Solve

Convert Mixed Numbers

1. Convert Mixed Number to Improper Fraction
   • Convert ( 8 \frac{1}{3} ) to improper fraction:
     • ( 8 \frac{1}{3} = \frac{25}{3} )
   • Rewriting the inequality:
     • ( \frac{2}{3} > -4y - \frac{25}{3} )

Eliminate Fractions

2. Multiply by 3 to Eliminate Fractions
   • Multiply both sides by 3:
     • ( 3 \times \frac{2}{3} > 3 \times (-4y - \frac{25}{3}) )
   • Simplify:
     • Left side: ( 2 )
     • Right side: ( -12y - 25 )
   • New inequality:
     • ( 2 > -12y - 25 )

Isolate the Variable

3. Move Constant Terms

   • Add 25 to both sides:
     • ( 2 + 25 > -12y )
   • Simplified to:
     • ( 27 > -12y )

4. Solve for y

   • Divide both sides by -12 (note the inequality swap):
     • ( \frac{27}{-12} < y )
   • Simplify fraction:
     • ( y > -\frac{9}{4} )

Solution Interpretation

• Final Solution: ( y > -\frac{9}{4} )

Additional Representations

• Mixed Number: ( y > -2 \frac{1}{4} )

Graphical Representation

• Number Line:
  • Draw a number line with points ( 0, -1, -2, -3 )
  • Plot ( -2 \frac{1}{4} ) with an open circle
  • Shade to the right of ( -2 \frac{1}{4} ) to represent all values greater than this.

Key Concepts

• Converting mixed numbers to improper fractions for easier manipulation.
• Eliminating fractions by multiplying through by the least common denominator.
• Sign change in inequality when multiplying or dividing by a negative number.