6th Grade Equations & Inequalities 2015-12-01 www.njctl.org - - PDF document

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6th Grade

Equations & Inequalities

2015-12-01 www.njctl.org

Slide 3 / 138 Table of Contents

Click on a topic to go to that section.

· Equations and Identities · Tables · Solving an Equation for a Variable · Solving One Step Addition & Subtraction Equations · Glossary & Standards · Determining Solutions of Equations · Solving One Step Multiplication & Division Equations · Writing Simple Inequalities · Solutions to Simple Inequalities · Graphing Solution Sets to Simple Inequalities · Writing Equations

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Equations and Identities

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Slide 5 / 138 Equations and Identities

What is an equation? How is it different than an identity? Discuss in your groups.

Slide 6 / 138 Equations and Identities

An equation is created when two expressions are set equal to one another such that they are equal for some values of their variables, but not for all. If they are equal for all values of their variables, then that is an identity, not an equation. So, not all mathematical statements which include an equal sign are equations...some are identities.

Slide 7 / 138 Equations and Identities

Here are some identities: 2 + 3 = 5 9 - 2 = 7 2x = 2x 9/3 = 3 These are always true...there are no values that can be assigned to the variables for which these would be untrue.

Slide 8 / 138 Equations and Identities

Here are some equations: x = 5t v = 7 + x a = x - 8 In all these cases, the variables are interdependent. They are

• nly true for certain sets of variables.

Changing the value of the variable on the right side of the equation, changes the possible values of the variables on the left side...and vice versa. These are equations (not identities) since knowing the value of

• ne variable changes the possible value(s) of the other(s).

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Tables

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Slide 10 / 138 Writing Equations

Tim's age is equal to Kathy's plus 8 T = K + 8 Tim is eight years older than Kathy. Write an equation for Kathy's age. click for mathematical equation click for equation in words Write an equation in words. Then translate that into a mathematical equation.

Slide 11 / 138 Writing Equations

Tim's age is equal to Kathy's plus 8 T = K + 8 Tim is eight years older than Kathy. Write an equation for Kathy's age. click for mathematical equation click for equation in words Write an equation in words. Then translate that into a mathematical equation.

Slide 12 / 138 Writing Equations

Bob's height equals double Fred's height less 6 B = 2F - 6 click for mathematical equation click for equation in words Bob is 6 inches less than twice the height of Fred. Write an equation for Bob's height. Write an equation in words. Then translate that into a mathematical equation.

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Write an equation in words. Then translate that into a mathematical equation.

Writing Equations

Speed equals distance divided by time s = d/t Speed is equal to the distance traveled in an amount of time. click for equation in words click for mathematical equation

Slide 14 / 138 Tables and Expressions

Let's use this table to find some solutions to the equation s = d/t; where s represents speed (in meters/second), d represents distance (in meters) and t represents time (in seconds). d (m) t (s) s (m/s) 30 2 60 4 90 6 120 2 240 4 360 6 We've entered the distance traveled and the time it took to travel that distance in two

• f the columns.

Use the equation (s = d/t) to find the speeds and fill in the blank column. s = d/t

Math Practice

Slide 15 / 138 Tables and Expressions

Note that in the first three sets of answers, the object was moving at a speed of 15 m/s. The final three sets of answers are for an object traveling four times faster, at 60 m/s. d (m) t (s) s (m/s) 30 2 15 60 4 15 90 6 15 120 2 60 240 4 60 360 6 60 But, in all cases, knowing the value of two of the three variables determines the values of the third. s = d/t

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Determining Solutions of Equations

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A solution to an equation is a number that makes the equation true. In order to determine if a number is a solution, replace the variable with the number and evaluate the equation. If the number makes the equation true, it is a solution. If the number makes the equation false, it is not a solution.

Determining the Solutions of Equations Slide 18 / 138

Which of the following is a solution of the equation from the solution set? x + 7 = 9 {2, 3, 4, 5} Write the equation four times. Each time replace x with one of the possible solutions and simplify to see if it is true. 2 + 7 = 9 3 + 7 = 9 4 + 7 = 9 5 + 7 = 9 9 = 9 10 = 9 11 = 9 12 = 9 Yes No No No Answer: 2 is the solution to x + 7 = 9

Determining the Solutions of Equations

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Which of the following is a solution of the equation? y - 12 = 8 {17, 18, 19, 20} Write the equation four times. Each time replace y with one of the possible solutions and simplify to see if it is true. 17 - 12 = 8 18 - 12 = 8 19 - 12 = 8 20 - 12 = 8 5 = 8 6 = 8 7 = 8 8 = 8 No No No Yes Answer: 20 is the solution to y - 12 = 8

Determining the Solutions of Equations Slide 20 / 138

1 Which of the following is a solution to the equation? x + 17 = 21 {2, 3, 4, 5}

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2 Which of the following is a solution to the equation? m - 13 = 28 {39, 40, 41, 42}

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3 Which of the following is a solution to the equation? 12b = 132 {9, 10, 11, 12}

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4 Which of the following is a solution to the equation? 3.5 + d = 7.5 {2.5, 3, 3.5, 4}

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Solving an Equation for a Variable

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Like in any game there are a few rules. There are three rules which will allow you to solve any one-step equation.

The Rules for Solving Equations Slide 26 / 138

Here are the three rules. Let's examine them, one at a time.

• 1. To "undo" a mathematical operation, you must perform the

inverse operation.

• 2. You can do anything you want (except divide by zero) to one

side of an equation, as long as you do the same thing to the

• ther.
• 3. You can always switch the left and right sides of an equation.

The Rules Slide 27 / 138

• 1. To "undo" a mathematical operation, you must do the opposite.

We learned earlier that for every mathematics operation, there is an inverse operation which undoes it: when you do both

• perations, you get back to where you started.

When the variable for which we are solving is connected to something else by a mathematical operation, we can eliminate that connection by using the inverse of that operation.

The Rules Slide 28 / 138

• 2. You can do anything you want (except divide by zero) to one

side of an equation, as long as you do the same to the other side. If the two expressions on the opposite sides of the equal sign are equal to begin with, they will continue to be equal if you do the same mathematical operation to both of them. This allows you to use an inverse operation on one side, to undo an operation, as long as you also do it on the other side. You can just never divide by zero (or by something which turns out to be zero) since the result of that is always undefined.

The Rules Slide 29 / 138

• 3. You can always switch the left and right sides of an equation.

Once an equation has been solved for a variable, it'll be a lot easier to use if that variable is moved to the left side. Mathematically, this has no effect since the both sides are equal. But, it's easier to use the equation if the side for which you are solving is on the left and values are substituted on the right.

The Rules Slide 30 / 138

When solving equations, the goal is to isolate the variable on one side of the equation in order to determine its value (the value that makes the equation true). x + 7 = 32 The goal: get "x" by itself on one side of the equal sign.

Solving Equations

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For each equation, write the inverse operation needed to solve for the variable. a.) y + 7 = 14 subtract 7 b.) a - 21 = 10 add 21 c.) 5s = 25 divide by 5 d.) x = 5 multiply by 12 12

click

Inverse Operations

click click click

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5 What is the inverse operation needed to solve this equation? 7x = 49 A Addition B Subtraction C Multiplication D Division

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6 What is the inverse operation needed to solve this equation? x - 3 = 12 A Addition B Subtraction C Multiplication D Division

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7 What is the inverse operation needed to solve this equation? x 5 A Addition B Subtraction C Multiplication D Division

= 8

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8 What is the inverse operation needed to solve this equation? 25 + x = 30 A Addition B Subtraction C Multiplication D Division

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Solving One Step Addition & Subtraction Equations

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HINTS: To solve equations, you must work backwards through the order of

• perations to find the value of the variable.

Remember to use inverse operations in order to isolate the variable

• n one side of the equation.

Whatever you do to one side of an equation, you MUST do to the

• ther side!

One Step Equations Slide 38 / 138

Example: y + 9 = 16

• 9 -9 The inverse of adding 9

y = 7 is subtracting 9 Remember - whatever you do to one side of an equation, you MUST do to the

• ther!!!

One Step Equations Slide 39 / 138 One Step Equations

Try These! Solve each equation. x - 8 = -2 +8 +8 x = 6 2 = x - 6 +6 +6 8 = x

click click

Slide 40 / 138 One Step Equations

Try These! Solve each equation. x + 2 = -14

• 2 -2

x = -16 7 = x + 3

• 3 -3

4 = x

click click

Slide 41 / 138 One Step Equations

Try These! Solve each equation. 15 = x + 17

• 17 -17
• 2 = x

x + 5 = 3

• 5 -5

x = -2

click click

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9 Solve. x - 6 = 11

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10 Solve. j + 15 = 34

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11 Solve. 23 + t = 100

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12 Solve. w - 225 = 300

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13 Solve. 37 + x = 57

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14 Solve. y - 17 = 51

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15 Solve. n - 15 = 23

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Solving One Step Multiplication & Division Equations

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Example: 6m = 72 6 6 The inverse of multiplying by 6 m = 12 is dividing by 6 Remember - whatever you do to one side of an equation, you MUST do to the

• ther!!!

One Step Equations Slide 51 / 138 One Step Equations

Try These! Solve each equation. 3x = 15 3 3 x = 5

• 4x = -12
• 4 -4

x = 3

• 25 = 5x

5 5

• 5 = x

click click click

Slide 52 / 138 One Step Equations

Try These! Solve each equation. x 2 x = 20 = 10 (2) (2) x

• 6

x = -216 = 36 (-6) (-6)

click click

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16 Solve. 115 = 5x

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17 Solve. = 12 x 9

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18 Solve. 51 = 17y

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19 Solve. 3 = x 7

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20 Solve. 108 = 12r

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21 Solve. 33 = 11m

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22 Solve. 23 = x 5

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Writing Equations

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We already know how to translate words into expressions. We just need to learn what the equal sign would be represented by. Think of situations where you would use an equals sign.

Writing Equations Slide 62 / 138 Equals

List words that indicate equals.

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23 Which equation represents seven minus five is six less than a number. A 7 = 5 + n -6 B 75 = 6 - n C 7 - 5 = n - 6 D 7 - 5 = 6 - n

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24 Which equation represents six less than a number comes to the sum of three and seven. A n - 6 = 3 + 7 B 6 - n = 3 + 7 C 7 - 3 = n + 6 D 3 + 6 = n - 7

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25 Which equation represents ten times a number totals sixty plus twenty. A 10n + 60 = 20 B n10 = 6 + 20 C 60 + 20 + 10 = n D 10n = 60 + 20

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26 Which equation represents twenty plus four is the same as the product of fourteen and a number. A 24 = 14n B 14n = 2 + 4 C 20 + 4 = 14n D 20 + 4 x 14 = n

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You will now use your knowledge of writing equations to write an equation for real-life scenarios. George is buying video games online. The cost of the video is $30.00 per game. He spent a total of $120.00. How many games did he buy in all? Lets pull out the pieces of information, and put them in place.

Writing Equations Slide 68 / 138

George is buying video games online. The cost of the video is $30.00 per game. He spent a total of $120.00. How many games did he buy in all? Notice that the video games are "per game". We are never told how many games he bought. So we use a variable to represent the number of games. Lets use "g". · $30.00 per game translates to 30g · He spent a total of $120.00 translates to = 120 · How many games did he buy in all? means that we are solving for "g". We know that total means equals. This is the question we need to answer.

click

click

Writing Equations Slide 69 / 138

30

cost of

• ne video

game

number

• f games

120

totals

amount he spent

Lets put it all together and solve the equation.

Writing Equations

=

g

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27 Alice has the 5 newest DVDs, which is 4 less than the amount Jon has. Which equation represents the number

• f DVDs Jon has.

A n + 5 = 4 B 5 = n - 4 C 5 - 4 = n D 4 - n = 9

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28 Now solve the equation... Alice has the 5 newest DVDs, which is 4 less than the amount Jon has. Which equation represents the number

• f DVDs Jon has.

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29 Jasmine, who bought $5 worth of candy, spent $3 more than Leah spent. Which equation represents the amount that Leah spent? A x - 3 = 5 B 5 = x + 3 C 5 + x = 3 D 3x = 5

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30 Now solve the equation... Jasmine, who bought $5 worth of candy, spent $3 more than Leah spent. Which equation represents the amount that Leah spent?

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31 Kate got a 93 on her quiz retake. That is 14 points higher than her original grade. Part A Select an answer from each group to create an equation that can be used to determine g, the

• riginal grade.

g = A + B - C x D / E 14 F 93

G 14 H 93

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32 Kate got a 93 on her quiz retake. That is 14 points higher than her original grade. Part B What is the value of g, the original grade?

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33 Two brothers put their money together to buy a $19 video game. One contributed $8. Part A Select an answer from each group to create an equation that can be used to determine d, the number of dollars the other brother contributed. d = A + B - C x D / E 19 F 8

G 19 H 8

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34 Two brothers put their money together to buy a $19 video game. One contributed $8. Part B What is the value of d, the number of dollars the

• ther brother contributed?

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35 James is 3 times as old as Thomas, who is 8 years

• ld.

Part A Select an answer from each group to create an equation that can be used to determine j, James' age. = A 3 B 8 C j D + E x F 8 G j

H 3 I 8 J j

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36 James is 3 times as old as Thomas, who is 8 years

• ld.

Part B What is the the age of James, j?

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37 Kellie bought 8 towels and spend $39.60. Each towel costs the same amount. Part A Select an answer from each group to create an equations that can be used to determine t, the price, in dollars, of 1 towel. t = A + B - C x D / E 8 F 39.60

G 8 H 39.60

From PARCC EOY sample test calculator #1

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38 Kellie bought 8 towels and spend $39.60. Each towel costs the same amount. Part B What is the price, in dollars, of 1 towel?

From PARCC EOY sample test calculator #1

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Writing Simple Inequalities

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Slide 83 / 138 Symbols

What do these symbols mean? Less Than Less Than

• r Equal To

Greater Than Greater Than

• r Equal To

click to reveal

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An inequality is a statement that two quantities are not equal. The quantities are compared by using one of the following signs: Symbol Expression Words < A < B A is less than B > A > B A is greater than B < A < B A is less than or equal to B > A > B A is greater than or equal to B

Inequality

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When am I ever going to use it? Your parents and grandparents want you to start eating a healthy

• breakfast. The table shows the nutritional requirements for a healthy

breakfast cereal with milk. Healthy Breakfast Cereals (per serving) Fat Less than 3 grams Protein More than 5 grams Fiber At least 3 grams Sugar At most 5 grams Suppose your favorite cereal has 2 grams of fat, 7 grams of protein, 3 grams of fiber and 4 grams of sugar. Is it a healthy cereal?

Inequality

Answer & Math Practice

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Is a cereal with 3 grams of fiber considered healthy? Healthy Breakfast Cereals (per serving) Fat Less than 3 grams Protein More than 5 grams Fiber At least 3 grams Sugar At most 5 grams

Inequality

Answer

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Is a cereal with 5 grams of sugar considered healthy? Healthy Breakfast Cereals (per serving) Fat Less than 3 grams Protein More than 5 grams Fiber At least 3 grams Sugar At most 5 grams

Inequality

Answer

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When you need to use an inequality to solve a word problem, you may encounter one of the phrases below. Important Words Sample Sentence Equivalent Translation is more than Trenton is more than 10 miles away. d > 10 is greater than A is greater than B. A > B must exceed The speed must exceed 25 mph. The speed is greater than 25 mph. s > 25

Inequality Slide 89 / 138

Here are some more expressions you may encounter: Important Words Sample Sentence Equivalent Translation cannot exceed Time cannot exceed 60 minutes. Time must be less than or equal to 60 minutes. t < 60 is at most At most, 7 students were late for class. Seven or fewer students were late for class. n < 7 is at least Bob is at least 14 years

• ld.

Bob's age is greater than or equal to 14. B > 14

Inequality Slide 90 / 138

How are these inequalities read? 2 + 2 > 3 Two plus two is greater than 3 2 + 2 ≥ 4 Two plus two is greater than or equal to 4 2 + 2 < 5 Two plus two is less than 5 2 + 2 ≤ 5 Two plus two is less than or equal to 5 2 + 2 ≤ 4 Two plus two is less than or equal to 4 2 + 2 > 3 Two plus two is greater than or equal to 3

Read Inequalities

Slide 91 / 138 Writing Inequalities

Let's translate each statement into an inequality. x is less than 10 20 is greater than or equal to y x < 10 words inequality statement translate to 20 > y

Slide 92 / 138 Try These

• 1. 14 is greater than a
• 2. b is less than or equal to 8
• 3. 6 is less than the product of f and 20
• 4. The sum of t and 9 is greater than or equal to 36

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• 5. 7 more than w is less than or equal to 10
• 6. 19 decreased by p is greater than or equal to 2
• 7. Fewer than 12 items
• 8. No more than 50 students
• 9. At least 275 people attended the play

Try These Slide 94 / 138

$7.50 7.5 at least > An employee earns e A store's employees earn at least $7.50 per hour. Define a variable and write an inequality for the amount the employees may earn per hour. Let e represent an employee's wages.

Writing Inequalities Slide 95 / 138

39 Write an inequality for the sentence: m is greater than 9 A m < 9 B m < 9 C m > 9 D m > 9

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40 Write an inequality for the sentence: 12 is less than or equal to y A 12 < y B 12 < y C 12 > y D 12 > y

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41 Write an inequality for the sentence: The grade, g, on your test must exceed 80% A g < 80 B g < 80 C g > 80 D g > 80

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42 Write an inequality for the sentence: y is not more than 25 A y < 25 B y < 25 C y > 25 D y > 25

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43 Write an inequality for the sentence: The total, t, is fewer than 15 items. A t < 15 B t < 15 C t > 15 D t > 15

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44 Write an inequality for the sentence: k is greater than or equal to twenty A k < 20 B k < 20 C k > 20 D k > 20

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45 Cirrus clouds form more than 6,000 meters above

• Earth. Choose an inequality to represent h, the

height, in meters, of cirrus clouds. A h < 6000 B h < 6000 C h > 6000 D h > 6000

From PARCC EOY sample test non-calculator #20

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46 Let x represent any number in the set of even integers greater than 1. Which inequality is true for all values of x? A x < 0 B x > 0 C x < 4 D x > 4

From PARCC PBA sample test calculator #3

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Solutions to Simple Inequalities

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Remember: Equations have one solution. Solutions to inequalities are NOT single numbers. Instead, inequalities will have more than one value for a solution. This would be read as, "The solution set is all numbers greater than or equal to negative 5."

Solution Sets

1 2 3 4 5 6 7 8 9 10

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10

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Let's name the numbers that are solutions to the given inequality. r > 10 Which of the following are solutions? {5, 10, 15, 20} 5 > 10 is not true So, 5 is not a solution 10 > 10 is not true So, 10 is not a solution 15 > 10 is true So, 15 is a solution 20 > 10 is true So, 20 is a solution Answer: {15, 20} are solutions to the inequality r > 10

Solution Sets Slide 106 / 138

30 ≥ 5d; {4,5,6,7,8} 30 ≥ 5d 30 ≥ 5(4) 30 ≥ 20 30 ≥ 5d 30 ≥ 5(5) 30 ≥ 25 30 ≥ 5d 30 ≥ 5(6) 30 ≥ 30 30 ≥ 5d 30 ≥ 5(7) 30 ≥ 35 30 ≥ 5d 30 ≥ 5(8) 30 ≥ 40

Solution Sets

Which of the following numbers are solutions to the given inequality.

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47 Which of the following are solutions to the inequality: x > 11 {9, 10, 11, 12} Select all that apply. A 9 B 10 C 11 D 12

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48 Which of the following are solutions to the inequality: m < 15 {13, 14, 15, 16} Select all that apply. A 13 B 14 C 15 D 16

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49 Which of the following are solutions to the inequality: x > 34 {32, 33, 34, 35} Select all that apply. A 32 B 33 C 34 D 35

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50 Which of the following are solutions to the inequality: 3x > 15 {4, 5, 6, 7} Select all that apply. A 4 B 5 C 6 D 7

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51 Which of the following are solutions to the inequality: 6y < 42 {6, 7, 8, 9} Select all that apply A 6 B 7 C 8 D 9

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Graphing Solution Sets to Simple Inequalities

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Since inequalities have more than one solution, we show the solution two ways. The first is to write the inequality. The second is to graph the inequality on a number line. In order to graph an inequality, you need to do two things:

• 1. Draw a circle (open or closed) on the number that is your

boundary.

• 2. Extend the line in the proper direction.

Graphing Inequalities Slide 114 / 138

Determining Whether to Use an Open or Closed Circle An open circle on a number shows that the number is not part of the solution. It serves as a boundary only. It is used with "greater than" and "less than". The word equal is not included. < > A closed circle on a number shows that the number is part of the solution. It is used with "greater than or equal to" and "less than or equal to". < >

Graphing Inequalities The Circle

Slide 115 / 138 Determining Which Direction to Extend the Line

Extend Line to the Left: If the variable is smaller than the number, then you extend your line to the left (since smaller numbers are on the left). Extend the line to the left in these situations: # > variable variable < # Extend Line to the Right: If the variable is larger than the number, then you extend your line to the right (since bigger numbers are

• n the right).

Extend the line to the right in these situations: # < variable variable > #

Slide 116 / 138

When graphing inequalities, ask yourselves each question below. What is the number in the inequality? What kind of circle should be used? In what direction does the line go?

Graphing Inequalities Slide 117 / 138

Step 1: Rewrite this as x < 1. Step 2: What kind of circle? Because it is less than, it does not include the number 1 and so it is an open circle.

• 1
• 2
• 3
• 4
• 5

1 2 3 4 5

Graphing Inequalities

x is less than 1

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Step 4: Draw a line, thicker than the horizontal line, from the dot to the

• arrow. This represents all of the numbers that fulfill the inequality.

Step 3: Draw an arrow on the number line showing all possible

• solutions. Numbers greater than the variable, go to the right.

Numbers less than the variable, go to the left.

• 1
• 2
• 3
• 4
• 5

1 2 3 4 5 x < 1

Graphing Inequalities

• 1
• 2
• 3
• 4
• 5

1 2 3 4 5

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1 2 3 4 5 6 7 8 9 10

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10

You try Graph the inequality x > 2 Graph the inequality

• 3 > x

click 2 on the number line for answer click -3 on the number line for answer

Graphing Inequalities

1 2 3 4 5 6 7 8 9 10

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10

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Try these. Graph the inequalities.

• 1. x > -3
• 2. x < 4
• 1
• 2
• 3
• 4
• 5

1 2 3 4 5

• 1
• 2
• 3
• 4
• 5

1 2 3 4 5

Graphing Inequalities

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Try these. State the inequality shown. 1. 2.

Graphing Inequalities

• 1
• 2
• 3
• 4
• 5

1 2 3 4 5

• 1
• 2
• 3
• 4
• 5

1 2 3 4 5

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52 This solution set would be x ≥ -4. True

False

1 2 3 4 5 6 7 8 9 10

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10

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• 1
• 2
• 3
• 4
• 5

1 2 3 4 5

53 A x > 3 B x < 3 C x < 3 D x > 3 State the inequality shown.

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5 6 7 8 9 10 11 12 13 14 15

54 A 11 < x B 11 > x C 11 > x D 11 < x State the inequality shown.

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55 A x > -1 B x < -1 C x ≤ -1 D x ≥ -1 State the inequality shown.

• 1
• 2
• 3
• 4
• 5

1 2 3 4 5

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56 A

• 4 < x

B

• 4 > x

C

• 4 ≤ x

D

• 4 ≥ x

State the inequality shown.

• 1
• 2
• 3
• 4
• 5

1 2 3 4 5

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57 A x > 0 B x < 0 C x ≤ 0 D x ≥ 0 State the inequality shown.

• 1
• 2
• 3
• 4
• 5

1 2 3 4 5

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A store's employees earn at least $7.50 per hour. Define a variable and write an inequality for the amount the employees may earn per hour. Graph the solutions. Let e represent an employee's wages.

1 2 3 4 5 6 7 8 9 10

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10

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58 The sign shown below is posted in front of a roller coaster ride at the Wadsworth County Fairgrounds. If h represents the height of a rider in inches, what is a correct translation

• f the statement on this

sign?

From the New York State Education Department. Office of Assessment Policy, Development and

• Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011

All riders MUST be at least 48 inches tall.

A h < 48 B h > 48 C h ≤ 48 D h ≥ 48

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Glossary & Standards

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Back to Instruction

Equation

A mathematical statement, in symbols, that two things are exactly the same (or equivalent).

4x+2 = 14 3y + 2 = 11 11 - 1 = 3z + 1 7x = 21

(where x = 3) (where z = 3) (where y = 3)

a.k.a. function d = rt

Slide 132 / 138

Back to Instruction

Identity

An equation that has infinitely many solutions.

3(x - 1) = 3x - 3

3x - 3 = 3x - 3

• 3x -3x
• 3 = -3

7(2x + 1) = 14x + 7 14x + 7 = 14x + 7

• 14x -14x

7 = 7

3x - 1 = 3x + 1

• 3x -3x
• 1 = +1

Slide 133 / 138

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Inequality

A comparison of two numbers that are not, or may not, be equal.

larger

smaller

larger

smaller

Greater than Less than

Greater than or equal to Less than

• r equal

to

Slide 134 / 138

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Inverse Operation

The operation that reverses the effect of another operation.

Addition

Subtraction Multiplication Division + _ x ÷ 11 = 3y + 2

• 2
• 2

9 = 3y ÷ 3 ÷ 3 3 = y

• 5 + x = 5

x = 10 + 5 + 5

Slide 135 / 138

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Solution

A value you can put in place of a variable that would make the statement true. x + 4 = 9 Solution: x = 5 The answer to a math problem. 3y = 6 Solution: y = 2 Slide 136 / 138

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Solution Set

A set of values that can make a statement true.

The #s in a solution set are written in curly brackets.

{ } 2y = 16 y = {8} 3 < y < 7 {4,5,6,7} y= Slide 137 / 138

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Variable

A letter or symbol that represents a changeable or unknown value.

4x + 2

variable

x = ?

2x = 6

x x

Slide 138 / 138

Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of

• thers.

MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.