Chapter 5 Notes Algebra I

Section 5-1: Solving Linear Inequalities by Addition and Subtraction Inequality – Addition Property of Inequalities Words: Symbols:

Section 5-1: Solving Linear Inequalities by Addition and Subtraction Inequality – A mathematical sentence that contains < , > , < , or > Addition Property of Inequalities Words: Any number is allowed to be added to both sides of a true inequality Symbols: If a > b, then a + c > b + c If a < b, then a + c < b + c

Section 5-1: Solving Linear Inequalities by Addition and Subtraction Graphing inequalities on a number line: Put a circle on the _____________ point Fill the circle in if the sign is _______ or _______ Leave the circle open if the sign is _____ or ____ Shade the left side if the variable is ____________ to the number Shade the right side if the variable is ____________ to the number Ex) Graph x < -9 Ex) Graph 4 < x

Section 5-1: Solving Linear Inequalities by Addition and Subtraction Solve the following inequalities and graph the solution set on a number line. 1) x – 12 > 8 2) 22 > x – 8 3) x – 14 < -19

Section 5-1: Solving Linear Inequalities by Addition and Subtraction Subtraction Property of Inequalities Words: Symbols:

Section 5-1: Solving Linear Inequalities by Addition and Subtraction Subtraction Property of Inequalities Words: Any number is allowed to be subtracted from both sides of a true inequality Symbols: If a > b, then a – c > b – c If a < b, then a – c < b – c

Section 5-1: Solving Linear Inequalities by Addition and Subtraction Solve the following inequalities and graph the solution set on a number line. 1) x + 19 > 56 2) 18 > x + 8 3) 22 + x < 5

Section 5-1: Solving Linear Inequalities by Addition and Subtraction Solve the following inequalities and graph the solution set on a number line. 1) 3x + 6 < 4x 2) 10x < 9n – 1

Section 5-2: Solving Inequalities by Multiplication and Division Multiplication Property of Inequalities: Part 1 Words: Symbols:

Section 5-2: Solving Inequalities by Multiplication and Division Multiplication Property of Inequalities: Part 1 Words: Any positive number is allowed to be multiplied to both sides of an inequality Symbols (c > 0): If a > b, then ac > bc If a < b, then ac < bc

Section 5-2: Solving Inequalities by Multiplication and Division Multiplication Property of Inequalities: Part 2 Words: Symbols:

Section 5-2: Solving Inequalities by Multiplication and Division Multiplication Property of Inequalities: Part 2 Words: Any negative number is allowed to be multiplied to both sides of an inequality, as long as the sign gets flipped! Symbols (c < 0): If a > b, then ac < bc If a < b, then ac > bc

Section 5-2: Solving Inequalities by Multiplication and Division Solve 1) 2) 3) 4)

Section 5-2: Solving Inequalities by Multiplication and Division Division Property of Inequalities: Part 1 Words: Any positive number is allowed to be divided to both sides of an inequality Symbols (c > 0): If a > b, then If a < b, then

Section 5-2: Solving Inequalities by Multiplication and Division Division Property of Inequalities: Part 2 Words: Any negative number is allowed to be divided to both sides of an inequality, as long as the sign gets flipped! Symbols (c < 0): If a > b, then If a < b, then

Section 5-2: Solving Inequalities by Multiplication and Division Solve and graph the solution set 1) 4x > 16 2) -7x < 147 3) -15 < 5x 4) -20 > -10x

Section 5-3: Solving Multi-Step Inequalities Solve the following multi-step inequalities. Graph the solution set. 1) -11x – 13 > 42 2) 15 + 2x < 31 3) 23 > 10 – 2x

Section 5-3: Solving Multi-Step Inequalities Solve the following multi-step inequalities. Graph the solution set. 4(3x – 5) + 7 > 8x + 3 2(x + 6) > -3(8 – x)

Section 5-3: Solving Multi-Step Inequalities Translate the verbal phrase into an expression and solve Five minus 6 times a number n is more than four times the number plus 45

Section 5-3: Solving Multi-Step Inequalities, SPECIAL SOLUTIONS Solve the following inequalities. Indicate if there are no solutions or all real number solutions. 1) 9x – 5(x – 5) < 4(x – 3) 2) 3(4x + 6) < 42 + 6(2x – 4)

Day 1, Section 5-4: Compound Inequalities Compound Inequality – two inequalities combined into one with an overlapping solution set And vs. Or Graph Graph Inequality Inequality Intersection Union

Day 1, Section 5-4: Compound Inequalities (ANDs) Ex) Graph the intersection of the two inequalities and write a compound inequality for x > 3 and x < 7 Ex) Solve and graph -2 < x – 3 < 4

Day 1, Section 5-4: Compound Inequalities (ANDs) Solve the following inequalities and graph the solution set on a number line Ex) y – 3 > -11 and y – 3 < -8 Ex) 6 < r + 7 < 10

Day 2, Section 5-4: Compound Inequalities (ORs) Graph the following inequalities on the same number line: x > 2 or x < -1 Ex) Solve and graph the solution set for: -2x + 7 < 13 or 5x + 12 > 37

Day 2, Section 5-4: Compound Inequalities (ORs) Solve the following inequalities and graph the solution set a + 1 < 4 2) x < 9 or 2 + 4x < 10 Or a – 1 > 3

Day 1, Section 5-5 Absolute Value Inequalities (distance is “less than” What does mean? Absolute value inequalities that have _______ or _________ symbols are treated like _______ problems!! The solution set will be an _____________.

Day 1, Section 5-5 Absolute Value Inequalities (distance is “less than”) What does mean? The distance from zero on the number line is less than 3. Absolute value inequalities that have less than (<) or less than or equal (<) to symbols are treated like AND problems!! The solution set will be an intersection.

5-4: Steps for Solving Abs. Value Inequalities: or 1) Re-write as an AND problem with boundaries +c and –c. (Lose the abs. value symbols) 2) Solve the AND problem 3) Graph the solution Ex)

Day 1, Section 5-5 Absolute Value Inequalities (distance is “less than”) Solve the following absolute value problems. Graph the solution set. 1) 2) 3) 4)

Day 2, Section 5-5 Absolute Value Inequalities(distance is “greater than”) What does mean? Absolute value inequalities that have _______ or _________ symbols are treated like _______ problems!! The solution set will be a _____________.

Day 2, Section 5-5 Absolute Value Inequalities(distance is “greater than”) What does mean? The distance from 0 on the number line is greater than 3 Absolute value inequalities that have greater than (>) or greater than or equal to (>) symbols are treated like OR problems!! The solution set will be a union.

5-4: Steps for Solving Abs. Value Inequalities: or 1) Re-write as an OR problem. Split into 2 problems, one with +c and one with –c (Lose the abs. value symbols) ***YOU MUST FLIP SIGN ON –C PROBLEM 2) Solve the two problems 3) Graph the solution Ex)

Day 2, Section 5-5 Absolute Value Inequalities(distance is “greater than”) Solve the following absolute value inequalities 1) 2) 3) 4)

Section 5-6: Graphing Linear Inequalities Linear Equations vs. Linear Inequalities Plot any 2 points Draw a solid line through the 2 points Put in slope-int form first Plot any two points Determine whether to connect the points with a solid OR dashed line Shade ONE side of the line You must determine which side You shade the solution set!

Section 5-6: Graphing Linear Inequalities Steps for graphing linear inequalities: Plot 2 points on the line If the symbol is < or > connect with a dashed line If the symbol is < or >, connect with a solid line Shade above the line for y > mx + b (or >) 5) Shade below the line for y < mx + b (or <) Ex. Graph y < 2x – 4

Section 5-6: Graphing Linear Inequalities Graph the following linear inequalities Ex) Ex) y < x – 1

Section 5-6: Graphing Linear Inequalities Graph the following linear inequalities Ex) 3x – y < 2 Ex) x + 5y < 10