KEY CONCEPTS TO LEARN:

1. Solve Equations  
   • Intuition 2x = 10, x = 5
   • Algebra Rules
     • Eventually, too complex to use just intuition
     • Must resort to rules of algebra
       • Do same operation on both entire sides
         • generally inverse operations to “undo” complexity
       • Isolate variable, x
         • x = number e.g. x = 5
         • x “in terms of” y, x = 2y – 3
   • What to do given different equation presentations?
     • Base case: 2-step equation 2x + 10 = 20
       • Subtract 10 to “undo” +10 on same side as x, 2x = 10
       • Divide by 2 to “undo” multiply by 2, 2x/2 = 10/2, x = 5
2. Solve (more complex) Equations  
   • • Same Variable on both sides. e.g. 5x + 10 = 4x – 5
       • Add/subtract variables to isolate x on one side
       • e.g. 5x + 10 – 4x = 4x – 5 – 4x, x + 10 = -5
     • Parentheses e.g. 2(x + 5) = -(x – 5)
       • Distribute to get rid of parentheses
         • a(b + c) = ab + ac  distributive property
         • 2x + 2•5 = -x – (-5) becomes 2x + 10 = -x + 5
         • * Be careful of signs * (huge source of mistakes)
     • Denominators/Fractions in equation e.g. 3/x = 10
       • Multiply by the denominator(s) on both sides to get rid of the denominators/fractions e.g. x(3/x) = x(10), 3 = 10x
         • 1st step after simplification if not already simplified
         • maybe find common denominator if fraction add/subtract
         • maybe cross multiply if proportion equation a/b = c/d
         • goal is to get an equation w/o fractions/denominators
         • multiply by the entire denominator. e.g. 3/(x+1) = 10
           • (x+1)(3/(x+1) = (x+1)10, 3 = 10(x+1)
     • Exponents, squares, or square roots
       • Isolate the radical or squared term with the variable
       • e.g. (x + 1)² – 6 = 10, (x + 1)² = 16
       • Apply the inverse function (undo a square with a square root etc.)
       • e.g. (x + 1) = ±4
3. Check your result
   • Plug your answer back into the equation
     • make it a habit to avoid mistakes
     • almost half of all mistakes result from bad habits, not bad math knowledge