7th Grade Math Review (Part 2)

Solving Equations

Single-Step Equations

• Equation: ( \frac{x}{2} = -14 )
  • Multiply both sides by 2
  • Solution: ( x = -28 )

Two-Step Equations

• Equation: ( 3x - 6 = 15 )
  • Add 6 to both sides
  • Result: ( 3x = 21 )
  • Divide by 3
  • Solution: ( x = 7 )

Variables with Negative Coefficients

• Equation: ( -4x + 2 = 26 )
  • Subtract 2 from both sides
  • Result: ( -4x = 24 )
  • Divide by -4
  • Solution: ( x = -6 )

Combining Like Terms

• Equation: ( 5x + 2x - 7 = 42 )
  • Combine like terms:
    • ( 5x + 2x = 7x )
  • Result: ( 7x - 7 = 42 )
  • Add 7 to both sides
  • Result: ( 7x = 49 )
  • Divide by 7
  • Solution: ( x = 7 )

Distributive Property

• Equation: ( 5(x - 3) = 15 )
  • Distribute 5:
    • ( 5x - 15 = 15 )
  • Add 15 to both sides
  • Result: ( 5x = 30 )
  • Divide by 5
  • Solution: ( x = 6 )

Distributive Property with Simplification

• Equation: ( -2(3y - 6) - 7 = 16 )
  • Distribute -2:
    • ( -6y + 12 - 7 = 16 )
  • Combine constants:
    • ( -6y + 5 = 16 )
  • Subtract 5 from both sides
  • Result: ( -6y = 11 )
  • Divide by -6
  • Solution: ( y = \frac{-11}{6} )

Inequalities

Basics of Inequalities

• Treat inequalities like equations.
• Flip inequality sign when multiplying/dividing by a negative.

Example Problem

• Equation: ( -5x - (-9) \leq -6 )
  • Simplify double negative to positive:
    • ( -5x + 9 \leq -6 )
  • Subtract 9 from both sides
  • Result: ( -5x \leq -15 )
  • Divide by -5 (flip inequality)
  • Solution: ( x \geq 3 )

Example Problem with Division

• Equation: ( \frac{x}{-6} \geq 14 )
  • Multiply by -6 and flip inequality
  • Result: ( x \leq -84 )

Two-Step Inequality

• Equation: ( 9x - 4 \geq -5 )
  • Add 4 to both sides
  • Result: ( 9x \geq -1 )
  • Divide by 9
  • Solution: ( x \leq \frac{-1}{9} )

Evaluating Expressions

• Given: ( a = -5, b = -2, x = 4 )
• Expression: ( 8a - 3b + 18 )
  • Substitute values:
    • ( 8(-5) - 3(-2) + 18 )
  • Simplify:
    • ( -40 + 6 + 18 = -16 )
  • Result: ( -16 )

Simplifying Expressions

• Expression: ( 6x + 4x - 8x - 3y - 2y )
  • Combine like terms:
    • ( 6x + 4x - 8x = 2x )
    • ( -3y - 2y = -5y )
  • Result: ( 2x - 5y )

Distributive Property in Expressions

• Expression: ( 5(x - 4) - (x + 7) )
  • Distribute 5 and -1:
    • ( 5x - 20 - x - 7 )
  • Combine like terms:
    • ( 5x - x = 4x )
    • ( -20 - 7 = -27 )
  • Result: ( 4x - 27 )