Solving and Graphing Linear Inequalities Chapter 5 Solving and Graphing Linear Inequalities

5.1 Solve Inequalities with Addition and Subtraction. I can solve inequalities using addition and subtraction. CC.9-12.A.REI.3

Graphing Inequalities on a Number Line ** the graph shows all possible solutions of the inequality

Solving an Inequality: Solving a linear inequality in one variable is much like solving a linear equation in one variable. Isolate the variable on one side using inverse operations. Solve using addition: Add the same number to EACH side.

Solving Using Subtraction: Subtract the same number from EACH side.

Using Subtraction: Graph the solution. -5 -4 -3 -2 -1 0 1 2 3 4 5

Using Addition: Graph the solution. -5 -4 -3 -2 -1 0 1 2 3 4 5

Homework: p. 301 #1, 2, 3-27odd, 29, 32, 36

5.2 Solve Inequalities Using Multiplication and Division I can solve inequalities using multiplication and division. CC.9-12.A.REI.3

Solving Inequalities Using Multiplication: Multiply each side by the same positive number.

Solving Inequalities Using Division: Divide each side by the same positive number.

Multiply or Divide by a Negative Number: NOTE: When you multiply or divide each side of an inequality by a negative number, you must reverse the inequality symbol to maintain a true statement.

Solving by multiplication of a negative # Multiply each side by the same negative number and REVERSE the inequality symbol.

Solving by dividing by a negative # Divide each side by the same negative number and reverse the inequality symbol.

Example: You are shopping for bicycles Example: You are shopping for bicycles. The type you want costs at least $185. You have saved $97. Find the possible amounts of money you need to save to buy the bicycle you want. LSowatsky This Photo by Unknown Author is licensed under CC BY-SA

Homework: p. 308 #1, 2, 3-35 odd, 36, 37, 39

5.3 Solve Multi-Step Inequalities I can solve multi-step inequalities. CC.9-12.A.CED.1, CC.9-12.A.CED.3, CC.9-12.A.REI.3

Multi-Step Inequalities: Contains more than one operation Use inverse operations to undo each operation in reverse order Don’t forget to reverse the inequality symbol when multiplying or dividing by a negative number. LSowatsky

Example: Solve 5m - 8 > 12 Graph the solution.

Example: Solve 12 - 3a > 18 Graph the solution.

Example: Solve the inequality. Graph the solution.

Example: Solve the inequality. Graph the solution.

Homework: p. 314 #1, 2, 3 – 31 odd, 34, 35, 37 - 39

5.4 Solve Compound Inequalities I can solve compound inequalities. CC.9-12.A.CED.1, CC.9-12.A.CED.3, CC.9-12.A.REI.3

Difference between and and or AND means intersection -what do the two items have in common? OR means union -if it is in one item, it is in the solution A B A B

● ● Example: Graph x < 4 and x ≥ 2 a) Graph x < 4 o o 3 4 2 o 3 4 2 o b) Graph x ≥ 2 3 4 2 ● ● c) Combine the graphs d) Where do they intersect?

● ● Example: Graph x < 2 or x ≥ 4 a) Graph x < 2 o o 3 4 2 o 3 4 2 o b) Graph x ≥ 4 3 4 2 ● 3 4 2 ● c) Combine the graphs

3) Which inequalities describe the following graph? -2 -1 -3 o y > -3 or y < -1 y > -3 and y < -1 y ≤ -3 or y ≥ -1 y ≥ -3 and y ≤ -1

Example: Solve and graph the solution. 2x + 3 < 9 or 3x – 6 > 12 LSowatsky

Example: Graph 3 < 2m – 1 < 9

Homework: p. 326 #1, 2, 3 – 35 odd, 38, 40, 44

5.5 Solve Absolute Value Equations I can solve absolute value equations CC.9-12.A.CED.1, CC.9-12.A.CED.3, CC.9-12.F.IF.7b

Absolute Value: Definition – distance from the origin Ex: |3| = 3 |-5| = 5

Absolute Value Equations If |x| = 3, what do you know about x? Remember: Absolute Value is a distance. x has a distance of 3 from zero. If x is 3 ‘steps’ from zero on the number line, what could the value of x be?

Solving an Absolute Value Equation The equation is equivalent to the statement ax + b = c or ax + b = -c

Example: Solve

Example: Solve

Example: Solve, if possible.

Absolute Deviation: The absolute deviation of a number x from a given value is the absolute value of the difference of x and the given value. Absolute deviation =

Example: A volleyball league is preparing a two minute radio ad to announce tryouts. The ad has an absolute deviation of 0.05 minute. Find the minimum and maximum acceptable times the radio ad can run.

Homework: p. 335 #1, 2, 3–35odd, 38, 39, 42, 43, 45- 47

Graph Absolute Value Functions (time permitting): This Photo by Unknown Author is licensed under CC BY-SA

An absolute value function is a piecewise function that consists of two rays and is V-shaped. The corner point (turning point) of the graph is the vertex. This Photo by Unknown Author is licensed under CC BY

To Graph: To graph an absolute value function, it is useful to make a table of values. You can also use symmetry to get more values. Be sure to graph the vertex.

Parent Graph:

Graph and compare with the parent function.

Graph and compare to the parent function.

Graph and compare to the parent function.

Homework: p. 339 #1-7

5.6 Solve Absolute Value Inequalities I can solve absolute value inequalities. CC.9-12.A.CED.1, CC.9-12.A.CED.3

What does really mean? All the numbers whose distance from zero is greater than 4. -4 4 or On this slide, convince students about the less than -4 piece by asking them how far -5, -6, -20 etc. are from zero. Then ask if that distance is greater than 4 from zero. *Notice that you need to have two inequalities to represent the distances that are greater than 4 from zero. © 2011 The Enlightened Elephant

What does really mean? -4 4 and All the numbers whose distance from zero is less than 4. -4 4 and *Notice that you need to have two inequalities to represent the distances that are less than 4 from zero. On this slide, convince students about the greater than -4 piece by asking them how far -3, -2 are from zero. Then ask if that distance is less than 4 from zero. However, since these inequalities must happen at the same time, it should be written as © 2011 The Enlightened Elephant

Absolute Value Inequalities Rules: If AND -c < ax + b < c OR ax + b < -c or ax + b > c

You Try! Solve and graph. or Final answer: -3 4 © 2011 The Enlightened Elephant

You Try! Solve and graph. Final answer: -2 6 © 2011 The Enlightened Elephant

You Try! Solve and graph. Final answer: -3 2 © 2011 The Enlightened Elephant

You Try! Solve and graph. Final answer: NO SOLUTION! Absolute values are distances, which cannot be negative. You can choose any value for x and the left side of the inequality will always be positive. Positive numbers are NEVER less than negative numbers. © 2011 The Enlightened Elephant

Homework: p. 343 #1, 2, 3 – 31 odd, 35, 36

5.7 Graph Linear Inequalities in Two Variables I can graph linear inequalities in two variables. CC.9-12.A.CED.3, CC.9-12.A.REI.12

Inequalities in Two Variables: The solution of an inequality in two variables x and y is an ordered pair (x, y) that produces a true statement when the values of x and y are substituted into the inequality. The graph is the set of points that shows all solutions of the inequality.

Check whether the ordered pairs are solutions of 2x - 3y ≥ -2. a Check whether the ordered pairs are solutions of 2x - 3y ≥ -2. a. (0, 0) b. (0, 1) c. (2, -1)

2x – 3y ≥ -2 What it looks like: Every point in the shaded region is a solution of the inequality and every other point is not a solution. 3 2 1 -3 -2 -1 1 2 3 4 -1 -2 -3 2x – 3y ≥ -2

Steps to graphing a linear inequality: Sketch the graph of the corresponding linear equation. Use a dashed line for inequalities with < or >. Use a solid line for inequalities with ≤ or ≥. This separates the coordinate plane into two half-planes. Half-plane: created by the boundary line; only one half-plane contains the solutions

Steps cont. Test a point in one of the half planes to find whether it is a solution of the inequality. If the test point is a solution, shade its half plane. If not shade the other half plane.

Sketch the graph of y – 2x < 4

Sketch the graph of 2x – y ≥ 1

Sketch the graph of x + 8y >16

Sketch the graph y < 6.

Homework: p. 351 # 1, 2, 3 – 51 odd, 53, 54

Review: P.361 #17 – 34 (chapter test from book)

Chapter Test