Opener Complete the following…show work whenever work is necessary. Give an example of the commutative property of multiplication. 4( ¼ ) = 1 is an example of what property? Put grouping symbols around the expression below so that the value of the expression is -8. 3 + 1 – 2 • 4 – 32 + 2 4. Classify the following numbers as rational, irrational, integer, whole, or natural. a) 2.3451…. c) -34 b) 4.65 d) 0 5. Evaluate 5 – 2(3-1) + 5(-2)

Solving Equations and Inequalities

Challenge Question Can you solve the following equation? 3(2x – 4) – 5(2 – 3x) + 5 = ½(8 – 6x) + 3(2 – x) x = 1

1.6 Intro to Solving Equations Objectives: Write and solve a linear equation in one variable Solve a literal equation for a specified variable

Steps to Solving Equations Simplify both sides of the equation - Get rid of parenthesis - Combine like terms on each side of the equation 2. If variables on both sides of the equation, move the smallest variable term to the other side using the opposite operation (addition/subtraction) 3. Undo Addition or Subtraction – Isolate the variable term 4. Undo Multiplication or Division using the opposite operation 5. Check

Example 1 Solve. 3x – 8 = 5x - 20 3x – 8 = 5x - 20 -3x -3x + 20 + 20 12 = 2x 2 2 6 = x CHECK: 3(6) – 8 = 5(6) - 20 18 – 8 = 30 - 20 10 = 10

Example 2 2(3x – 4) + 2 = ½ (2x – 12) – 5 6x – 8 + 2 = x – 6 – 5 Step 1 6x – 6 = x – 11 Step 1 - x -x Step 2 5x – 6 = -11 +6 +6 Step 3 5x = -5 5 5 Step 4 x = -1

Guided Practice Solve: 1. 59 + a = -123 2. 9x – 5 + 3x = 25 3. 2(3x – 3) = 4 4. 5x – 4 = 12 – 2x 5. ½ x + ¾ = ¼ x – 5 6. 6x + 5 = 2x – 19 7. 3t -5 = 7t + t – 15 8. 2(x – 3) + 3(2 – 4x) = 60

Solving Literal Equations Literal Equations are basically formulas that mainly have variables with very few numbers in them. If there are multiple terms with the variable you are solving for in the equations, you must get those terms on the same side of the equation and everything else on the other. Factor out the variable you are solving for… Isolate the desired variable by using opposite operations - undo addition/subtraction - undo multiplication/division

Example 3 A Solve P = for n. 1+ ni P(1 + ni) = A Multiply by (1+ ni). P + Pni = A distributive property Pni = A – P Subtract P. Divide by Pi. n = A – P Pi

Guided Practice V = lwh ; h 2. A = ½ d1 d2 ; d1 3. Y = kx ; x 4. A = p + prt ; r z 5. A = p + prt ; p

Solving Inequalities Inequalities differ from equations in one major way…they have an infinite number of solutions where equations usually have just one The solutions are a graphically displayed on a number line Solve inequalities the same way you solve an equation…except if you multiply or divide by a negative number you must flip the inequality symbol and graph the solution WHEN GRAPHING ALWAYS READ THE INEQUALITY VARIABLE FIRST AND FOLLOW THE RULES BELOW: - if less than then shade left - if greater than then shade right If equal to then you color in the circle…if not then leave the circle open

Example 4 5 – 2x > 11 -5 -5 -2x > 6 -2 -2 x < -3 X is less than or equal to -3 Closed circle at 03 and shaded to the left

Guided Practice Solve and graph the solutions: 2x + 4 < 12 2. 4 – 5x > 24 3. 5 + 3(3-2x) < 2

Compound Inequalities A compound inequality is an inequality with two parts tied together with an “and” or an “or” To Solve a compound inequality…solve each one individually then graph to find the solution… - an “or” you just graph where it the inequalities indicate…and that’s the solution - an “and” you graph both, and the solution is only where both graphs overlap; if there is no overlap, there is no solution to the inequality

Example 5 3x – 1 > 2 or 2x + 6 < 2 + 1 +1 - 6 -6 3x > 3 2x < -4 3 3 2 2 x > 1 or x < -2 See graph on board

Example 6 4x + 6 > -10 and -3x – 5 > -17 - 6 -6 +5 +5 4x > -16 -3x > -12 4 4 -3 -3 x > -4 and x < 4 Graph and Look for overlap

Guided Practice -6(x + 2) > 12 and 17 – 2x < 111 10x – 4 < 6 or 4x -12 < 8