1. > < ≥ ≤ Signs of Inequality Term Definition Example Greater than x > 2 x is greater than 2 < Less than x < 1 x is less than 1 ≥ Greater than or equal to x ≥ 2 x is greater than or equal to 2 ≤ Less than or equal to x ≤ 1 x is less than or equal to 1 On a number line: Open Circles mean > or < Shaded Circles mean ≥ or ≤

2. Graphing Inequalities greater than X is ____________ -1 Simple Inequalities Examples greater than a) X > -1 X is ____________ -1 less than X is ____________ 3 b) X < 3 greater than or equal to c) X ≥ 1 X is ____________ 1

2. Graphing Inequalities b) -4 < a ≤ 4 Compound Inequalities Examples b) -4 < a ≤ 4 c) a ≤ 1 OR a ≥ 3 d) -2 ≤ c < 3

3. Linear inequalities are inequalities with more than one term Solving Linear Inequalities Linear inequalities are inequalities with more than one term Ex: 2x + 5 > 11 We solve linear equalities exactly the same as linear equations EXCEPT The direction of the inequality changes when multiplying or dividing by negative numbers

3. Ex 1: -3x + 5 > 26 - 5 -5 -3x > 21 ÷ -3 ÷ -3 x < -7 Solving Linear Inequalities Examples Ex 1: -3x + 5 > 26 - 5 -5 -3x > 21 ÷ -3 ÷ -3 THE INEQUALITY CHANGED DIRECTION x < -7 Check: try x = -10 -3(-10) + 5 = 30+5 = 35 > 26 CHECK!

3. Solving Linear Inequalities Examples Ex 2: 2x + 10 ≥ 4x + 20 - 4x - 4x . -2x + 10 ≥ 20 - 10 - 10 -2x ≥ 10 ÷ -2 ÷ -2 x ≤ -5 THE INEQUALITY CHANGED DIRECTION Check: try x = -10 2(-10) + 10 ≥ 4(-10) + 20 -10 ≥ -20 CHECK!

Example 1 3. Solving Linear Inequalities Word Problems Examples Your cell phone plan costs $35 plus $0.05 per text message. What is the maximum number of messages, t, you can send to spend less than $50? What are we trying to find? Maximum number of messages for bill to cost less than $50 What do we know? t = text messages $35 plus $0.05 per text Total cost < $50

3. 35 + .05t < 50 -35 -35 .05t < 15 ÷ .05 ÷ .05 t < 300 Example 1 3. Solving Linear Inequalities Word Problems Examples Your cell phone plan costs $35 plus $0.05 per text message. What is the maximum number of messages, t, you can send to spend less than $50? How do we solve it? Did we answer the original question? 35 + .05t < 50 -35 -35 Check: try t = 299 35 + .05(299) = 49.95 < 50 CHECK! .05t < 15 ÷ .05 ÷ .05 t < 300 300 messages

3. X Y -3 2 -1 Ex 1: Graph this linear inequality: Y > x - 3 Graphing Linear Inequalities 2 Variables! Ex 1: Graph this linear inequality: Y > x - 3 X Y Treat it as an equation and graph the line by plotting two ordered pairs The line is solid if “or equal to” and dotted if just greater or less than Read from Y perspective Shade under if less than Shade over if greater than -3 2 -1 TEST AN ORDERED PAIR IN THE SHADED AREA TO CHECK!!!

3. X Y 5 2 3 Ex 2: Graph this linear inequality: Y ≤ -x + 5 Graphing Linear Inequalities 2 Variables! Ex 2: Graph this linear inequality: Y ≤ -x + 5 Treat it as an equation and graph the line by plotting two ordered pairs The line is solid if “or equal to” and dotted if just greater or less than Start where x is Shade under if less than Shade over if greater than X Y 5 2 3 TEST AN ORDERED PAIR IN THE SHADED AREA TO CHECK!!!

3. X Y 6 2 2 Ex 3: Graph this linear inequality: 2x + y > 6 Graphing Linear Inequalities 2 Variables! Ex 3: Graph this linear inequality: 2x + y > 6 X Y Treat it as an equation and graph the line by plotting two ordered pairs The line is solid if “or equal to” and dotted if just greater or less than Read from Y perspective Shade under if less than Shade over if greater than 6 2 2 TEST AN ORDERED PAIR IN THE SHADED AREA TO CHECK!!!

4. X Y X Y -3 5 2 -1 2 3 Ex 1: Graph this linear system: Y > x – 3 Graphing Linear Systems Y > x - 3 Y ≤ -x + 5 X Y X Y -3 5 2 -1 2 3 TEST AN ORDERED PAIR IN THE SHADED AREA TO CHECK!!! Graph each inequality The common shaded area is the solution