Solving Inequalities Lesson Essential Questions What is an

< > Solving Inequalities Lesson Essential Questions: What is an inequality? How can you solve a linear inequality? < > < <

Activator (5 minutes)

Turn & Talk �What’s the difference between an equation and an inequality?

The Difference is… Equation: x + 7 = 13 Solution: x = 6 Inequality: x < 7 Solution: the value of x must be less than 7 List at least 7 numbers that are less than 7

What is an inequality? A way to represent a range of numbers for a solution set There are several different symbols that are used to represent inequalities

An inequality is like an equation, but instead of an equal sign (=) it has one of these signs: < : less than ≤ : less than or equal to > : greater than ≥ : greater than or equal to

Examples of Inequality Statements 5 7 means 9 2 means -3 4 means -3 is less than or equal to 4 8 8 means 8 is greater than or equal to 8 5 7 means 5 does not equal 7 5 is less than 7 9 is greater than 2

What are the solutions? For each problem below, state the solution or possible solutions. 1. x + 1 = 6 2. x + 1 > 6 3. y – 2 = 3 4. y – 2 < 3 5. x < 5 6. y > 7 7. x 9 8. y 2

Inequality Symbols Sort the words to the right into the appropriate inequality symbol column < > • less than • greater than • exceeds • less than or equal to • at most • greater than or equal to • no less than • no more than • at least • fewer than • more than

Inequalities Inequality symbols < • less than • fewer than > • greater than • more than • exceeds • less than or equal to • no more than • at most • greater than or equal to • no less than • at least

Important Words Sample Sentence is at least Bill is at least 21 years old. is at most At most 5 students dropped the course cannot exceed is less than is more than, is greater than is, is equal to earnings cannot exceed $1200. The speed exceeds 15 mph Spot's weight is less than 50 lb. Boston is more than 200 miles away. The ball is 12 lbs. Symbol Translation into inequality

Important Words Sample Sentence Inequality Translation is at least Bill is at least 21 years old. b > 21 is at most At most 5 students dropped the course < n < 5 earnings cannot exceed $1200. < r < 1200 The speed exceeds 15 mph > s > 15 Spot's weight is less than 50 lb. < w < 50 Boston is more than 200 miles away. > d > 200 cannot exceeds is less than is more than, is greater than is, is equal to The ball is 12 lbs. b = 12

Your turn (2 mins): �A number is less than 3. �A number is greater than or equal to 10. �A number has to be at least 5. �John cannot spend more than $40. �The auditorium seats a maximum of 450 people.

Your turn: �A number is less than 3. �A number is greater than or equal to 10. �A number has to be at least 5. �John cannot spend more than $40. �The auditorium seats a maximum of 450 people.

How do you graph inequalities on a number line?

Graphing Inequalities When we graph an inequality on a number line we use open and closed circles to represent the number. Symbol Meaning > Is greater than < Is less than Is greater than or equal to Is less than or equal to Graph

How to graph the solutions > Graph any number greater than. . . open circle, line to the right < Graph any number less than. . . open circle, line to the left Graph any number greater than or equal to. . . closed circle, line to the right Graph any number less than or equal to. . . closed circle, line to the left

x<5 means that whatever value x has, it must be less than 5. Try to name five numbers that are less than 5! Be Creative!

Numbers less than 5 are to the left of 5 on the number line. -25 -20 -15 -10 -5 0 5 10 15 20 25 • If you said 4, 3, 2, 1, 0, -1, -2, -3, etc. , you are right. • There also numbers in between the integers, like 2. 5, 1/2, -7. 9, π, etc. • The number 5 would not be a correct answer, though, because 5 is not less than 5.

X<5 • Notice that an Open circle (O) was used to graph X < 5 because the solution DOES NOT include 5.

x ≥ -2 means that whatever value x has, it must be greater than or equal to -2. Try to name five numbers that are greater than or equal to -2

Numbers Greater than -2 are to the Right of -2 on the number line. • If you said -1, 0, 1, 2, 3, 4, 5, etc. , you are right. • There also numbers in between the integers, like -1/2, 0. 2, 3. 1, 5. 5, etc. • The number -2 would also be a correct answer, because of the phrase, “or equal to”.

X ≥ -2 • Notice that a Closed circle ( ) was used to graph X ≥ - 2 because the solution DOES include – 2 because of the Greater than “or equal to”.

Graphing Inequalities Example # 1: p > -2 Step 1: Need a number line to graph on. Step 2: Decide whether we can include - 2 or not. (Open or Closed Circle? ) Step 3: Draw arrow to indicate where all the values greater than -2. (Arrow to Left or Right? )

Graphing Inequalities Example # 2: r -5 Step 1: Need a number line to graph on. Step 2: Decide whether we can include - 5 or not. (Open or Closed Circle? ) Step 3: Draw arrow to indicate where all the values less than or equal to -5. (Arrow to Left or Right? )

Graphing Inequalities Example # 3 n < 8 Step 1: Need a number line to graph on. Step 2: Decide whether we can include 8 or not. (Open or Closed Circle? ) Step 3: Draw arrow to indicate where all the values less than 8. (Arrow to Left or Right? )

Graphing Inequalities Example # 4 t -1 Step 1: Need a number line to graph on. Step 2: Decide whether we can include -1 or not. (Open or Closed Circle? ) Step 3: Draw arrow to indicate where all the values greater than or equal to -1. (Arrow to Left or Right? )

YOUR TURN X≥ 3 �Write the inequality to represent the number line X > -4 X<7 X ≥ -8

Quick Check What is the algebraic inequality for this graph? X>0 X 2. 5

TEST PREP

TEST PREP

Check Your Understanding a. Write the inequality statement down. b. Graph each inequality statement. x < 4 2. y 4 3. x > -3. 5 4. n > -2 1. 5. x -5 6. y 0 7. c > - 4 8. x 3. 5

INEQUALITY QUIZ: Graph the following on a number line: 1. x -5 2. x < 4 4. Explain the difference between x > 7 and x 7. 5. Write an algebraic inequality for the graph. 6. As required by law, the Smyrna Diner displays a sign stating “Maximum Occupancy: 225 persons. ” a Write an inequality to describe the situation. b. According to the rule, is it legal for 225 people to be in the diner?

How do you solve inequalities?

Adding and Subtracting �Is 4 < 10 ? �What happens if I add 1 to each side? Is 4 + 1 < 10 + 1 ? 5 < 11 TRUE � What happens if I subtract 1 from each side? Is 4 - 1 < 10 - 1 ? 3 < 9 TRUE

Solving Inequality Statements �Solving inequalities is very similar to solving equations which you already know how to do �REMEMBER THE RULES!! ◦ ◦ What you do to one side – you do to the other! Get the variable by itself! “Undo” the operation! Check your answer!

Solving Inequalities ● There a few special things to consider with inequalities: ● We need to look carefully at the inequality sign. ● We also need to graph the solution set on a number line.

Solve an Inequality w+5<8 -5 -5 All numbers less than 3 are solutions to this problem! w<3 -25 -20 -15 -10 -5 0 5 10 15 20 25

More Examples 8 + r ≥ -2 -8 -8 All numbers greater than-10 (including -10) -25 -20 -15 -10 r ≥ -10 -5 0 5 10 15 20 25

More Examples 2 x > -2 2 2 All numbers greater than -1 make this problem true! x > -1 -25 -20 -15 -10 -5 0 5 10 15 20 25

More Examples 2 h + 8 ≤ 24 -8 -8 2 h ≤ 16 2 All numbers less than 8 (including 8) -25 -20 -15 -10 2 h≤ 8 -5 0 5 10 15 20 25

YOUR TURN: 2 step Inequalities are just like 2 step Equations Remember that 1 st: add or subtract 2 nd: Multiply or Divide 3 rd: Graph on number line 2 x – 3 > 5

Solving Inequality Statements Let’s Practice Together! 1. 3. 2. 4.

Your Turn…. � Solve the inequality and graph the answer. x + 3 > -4 2. 6 d > 24 3. 2 x - 8 < 14 4. 2 c + 4 < 2 1. x > -7 2. d > 4 3. x < 11 4. c < -1

Which way is the sign? Inequality 9 > 4 12 > 7 24> 14 20> 10 ______ 4______ >2 -2 < -1 ______ -4 < -2 ______ 12 > 6 ______ 2 > 1 ______ -14 < -7 _______ Inequality Operation New Inequality Direction + 3 to both sides 12 > 7 stays same Multiply both sides by 2 24 > 14 stays same 20> 10 stays same - 4 from both sides stays same Divide both sides by 5 4 > 2 flip sign Divide both sides by -2 -2 < -1 Multiply both sides by 2 -4 < -2 stays same flip sign Multiply both sides by -3 12 > 6 stays same Divide both sides by 6 2 > 1 flip sign Multiply both sides by -7 -14 < -7 -9 < -2 stays same + 5 to both sides What operations cause the inequality sign to flip?

Multiply by a Negative �What happens if I multiply by – 1 to each side of 4 < 10? Is 4 (-1) < 10 (-1) ? Is - 4 < - 10 ? NO! - 4 is GREATER THAN – 10…so FLIP THE INEQUALITY! - 4 > - 10

Dividing by a Negative �What happens if I divide by – 2 to each side of 4 < 10? Is 4 < 10 ? NO!! - 2 is GREATER THAN – -2 -2 Is - 2 < - 5? 5…so FLIP THE INEQUALITY!! - 2 > - 5

Rule for multiplying or dividing by a negative. WRITE THIS DOWN!! When multiplying or dividing both sides by a negative number “flip” or “reverse” the direction of the inequality sign.

To Flip or Not to Flip that is the question! 3 x > -15 - 4 x < 20

Up to this point solving inequalities has been the same as solving equations…. Try this one: -2 x + 4 > 8

Let’s try another one…. 3 x – 5 ≤ 1

TURN AND TALK �When do you need to switch the inequality symbol? �NOW, JOT IT DOWN WHAT YOU AND YOUR BUDDY TALKED ABOUT IT!

CHECK YOUR UNDERSTANDING �Complete the 4 problems �Check your answers with your buddy �Come up to me to show me your problems for a classwork grade

Solving Inequality Statements

JOURNAL �Suppose you are going to tell someone how to solve an inequality like -4 x + 7 > 43. �What steps would you tell them? Why? �How would you check your solution to an inequality like -4 x + 7 > 43?

Equations vs Inequalities

JOURNAL �How is solving a linear inequality, like -4 x + 7 > 43 similar to solving a linear equation, like -4 x + 7 = 43? �How is it different?

TEST PREP

Homework

MATH GAMES �Rags to Riches: http: //www. quia. com/rr/438640. html �Genie Game: http: //www. mathplay. com/Inequality-Game. html �Countdown: http: //www. aaamath. com/g 725 inequalities. html �Arcade: http: //www. xpmath. com/forums/arcade. php ? do=play&gameid=87

Websites! �How do we solve Inequalities? ? ? http: //www. mccc. edu/~kelld/inequalities. htm http: //regentsprep. org/Regents/math/solvin/PSolv. In. htm http: //regentsprep. org/Regents/math/Solvin/Prac. Ineq. D. htm http: //www. math. com/school/subject 2/practice/S 2 U 3 L 4 Pract. html http: //www. nutshellmath. com/textbooks_glossary_demos/demos_content/alg_ solving_inequalities. html http: //www. math. com/school/subject 2/practice/S 2 U 1 L 4 Pract. html http: //www. aaamath. com/g 725 -inequalities. html