12.1 Solving Two-Step Equations How would you solve 35x = 350? How would you solve x – 5 = 345? How would you solve 35x – 5 = 345? Isolate the variable – get it on one side of the equal sign by itself. Do outside of grouping symbols first. Do addition or subtraction inverse operations first. Do multiplication or division inverse operations second. Do inverse operations on both sides of the equal sign! Check your answer.

12.1 Solving Two-Step Equations 35x – 5 = 345 Add. Prop 35x – 5+ 5 = 345 + 5 Combine like terms 35x = 350 Div. Prop 35x ÷ 35 = 350 ÷ 35 x = 10 Check 35x – 5 = 345 35(10) -5 = 345 350 – 5 = 345 345 = 345 pp 696-697 ex 1-3 p 698: 1-7

12.1 Solving Two-Step Equations Be careful about inverse operation order. 0 = 9(k – 2/3) + 33 0 = 9(k – 2/3) + 33 0 = 9(k – 2/3) + 33 -33 -33 -33 -33 -33 = 9(k -2/3) -33 = 9(k – 2/3) +2/3 +2/3 -33 = 9k – 6 -32 1/3 = 9k +6 +6 ÷9 ÷9 -27 = 9k -3 16/27 = k ÷9 ÷9 -3 = k

12.2 Solving Multi-Step Equations Sometimes equations may be too complex to even solve in two steps. You need to take additional steps to simplify the problem. Too many variables and constants: 4x + 2 – 2x + 1 = 11 Combine like terms 2x + 3 = 11 Subtraction Prop. 2x + 3 – 3 = 11 - 3 Combine Like terms and Division Prop. 2x ÷ 2 = 8 ÷ 2 x = 4 Check 4x + 2 – 2x + 1 = 11 4(4) + 2 – 2(4) + 1= 11 16 + 2 – 8 + 1 = 11

12.2 Solving Multi-Step Equations Too complex – Distributive Property: 4(x + 2) = 12 Distributive Property 4x + 4 × 2 = 12 Simplify 4x + 8 = 12 Subtraction Prop. 4x + 8 – 8 = 12 – 8 Division Prop 4x ÷ 4 = 4 ÷ 4 x = 1 Check 4(x + 2) = 12 4(1 + 2) = 12 4(3) = 12 12 = 12 4(1) + 4(2) = 12 4 + 8 = 12

12.3 Solving Equations with Variables on Both Sides Some equations cannot be solved because the variable is on both sides of the equal sign. Get the variables on one side of the equal sign, and get the constants on the other side. 4x + 2 + 1 = 11 + 2x Subtraction Prop. 4x + 2 + 1 – 2x = 11 + 2x – 2x Combine like terms 2x + 3 = 11 Subtraction Prop. 2x + 3 - 3 = 11 - 3 Division Prop. 2x ÷ 2 = 8 ÷ 2 x = 4 Check 4x + 2 + 1 = 11 + 2x 4(4) + 2 + 1 = 11 + 2(4) 16 + 2 + 1 = 11 + 8 19 = 19

12.3 Solving Equations with Variables on Both Sides Get the variables on one side of the equal sign with no constants. ½x + 3 = 2½x – 7 Subtraction Prop. ½x + 3 - ½x = 2½x – 7 - ½x Simplify 3 = 2x - 7 Addition Prop. 3 + 7 = 2x – 7 + 7 Simplify 10 = 2x Division Prop. 10 ÷ 2 = 2x ÷ 2 5 = x Check ½(5) + 3 = 2½(5) – 7 2.5 + 3 = 12.5 - 7 5.5 = 5.5 Remember that fractions are division.

12.4 Inequalities Inequality: mathematical sentence made by placing an inequality sign (<, >, ≤, ≥) between two expressions that are not equal or may not be equal. 5 < 9 -9 > -10 6 + 4 ≤ 10 1 + -4 ≥ -3 Inequality signs: read symbol from left to right Less than: < Less than or equal: ≤ Greater than: > Greater than or equal: ≥ Algebraic Inequality: an inequality with a variable. Inequality: 3x < 9 x – 12 > -10 3x - 5 ≤ 10 -2x + 5 ≥ -3 Solution set: x < 3 x > 2 x ≤ 5 x ≤ 4 Solution set: the set of all numbers that you can substitute for the variable to make the inequality true.

12.4 Inequalities Graph of an inequality: set of points on a number line that show the solution of an inequality. Open circle: < or > Filled circle: ≤ or ≥ Darkened line: included values Equivalent inequalities: inequalities that have the same solution. These can be formed by using inverse operations. Compound inequality: the result of combining two inequalities. -1 ≤ x < 3

12.5 Solving Inequalities by Adding and Subtracting Subtraction Property of Equality: subtracting the same number from each side of an equation produces an equivalent equation. x + 7 = -10 → x + 7 = -10 - 7 – 7 x = -17 Check: x + 7 = -10 → -17+7 = -10 -10 = -10 Subtraction Property of Inequality : subtracting the same number from each side of an equation produces an equivalent inequality. x + 7 < -10 → x + 7 < -10 x < -17 … “like” -17.1 Check: x + 7 < -10 → -17.1+7 < -10 -10.1 < -10

12.5 Solving Inequalities by Adding and Subtracting Addition Property of Equality: adding the same number to each side of an equation produces an equivalent equation. -9 = y -12 → -9 = y – 12 +12 +12 3 = x Check: -9 = y -12 → -9 = 3 – 12 -9 = -9 Addition Property of Inequality: adding the same number to each side of an equation produces an equivalent inequality. -9 ≥ y -12 → -9 ≥ y – 12 3 ≥ x “like” 3 or 2.9 Check: -9 ≥ y -12 → -9 ≥ 3 – 12 -9 ≥ 2.9 – 12 -9 ≥ -9 -9 ≥ -9.1

12.6 Solving Inequalities by Multiplying and Dividing Solve: -3x = 12 24/y= -8 y/8 = 4 ÷-3 ÷-3 × y × y ×8 ×8 x = -4 24 = -8y y = 32 ÷-8 ÷-8 -3 = y -3x > 12 24/y ≥ -8 y/8 > 4 x < -4 24 ≥ -8y y > 32 -3 ≤ y Multiplication and Division Property of Inequalities: Divide or multiply both sides by the same positive number. Divide or multiply both sides by the same negative number, and reverse the inequality sign

12.7 Solving Multi-Step Inequalities Solve: x/5 – 15 < 10 Write inequality x/5 – 15 < 10 Add. Prop + 15 + 15 Combine like terms x/5 < 25 Mult. Prop × 5 × 5 x < 125 Solve: 42 ≤ y/-9 + 30 Write inequality 42 ≤ y/-9 + 30 Subt. Prop -30 -30 Combine like terms 12 ≤ y/-9 Mult. Prop × -9 × -9 REVERSE INEQUALITY -108 ≥ y

12.7 Solving Multi-Step Inequalities Solve: 3x – 8 – 5x > 2 Write inequality 3x – 8 – 5x > 2 Add. Prop + 8 + 8 Combine like terms -2x > 10 Div. Prop ÷ -2 ÷ -2 REVERSE INEQUALITY x < -5