cc6+ unit 7 equation and inequality · web viewequation ‐ equations can be solved and usually...

CC6+ Unit 7 Equation and Inequality

Notes 7.1 Expressions, Equations and Inequalities I Expression : _ a mathematical phrase containing numbers, operations ( ), and/or variables‐ Expressions can be evaluated or simplified but they cannot be "solved"Examples:9 ÷ 3+42 12

4+4 (3 ) 5 x−2 9m−12 3+x+2

3−2+4 5 w−3 w 4+ y+ y+ y 5 y+ y

Numerical expressions can be simplified to one number.9 ÷ 3+42 12

4+4 (3 ) 3−2+4

Algebraic expressions can be simplified5 y+ y 4+ y+ y+ y 5 w−3 w

Algebraic expressions can be evaluated for given variables5 x−2 when x=3 9m−12 when m=4

II Equation‐ Equations can be solved and usually have one solution‐ Like a balanced scale‐ a mathematical sentence that states two expressions are equivalent‐ has an equal sign in between Examples: 3+x+2=9 5 w−3 w=9 ÷ 3+22−1 3+5=5 x−2 12

4+4 (3 )=9 m−12

III. Inequality‐ a mathematical sentence that states one expressions is greater than or less thananother expression-has inequality signs‐ Inequalities can be solved but have many solutions‐ Like an unbalanced scaleExamples :3+x+2≤9 5w−3 w ≥ 9 ÷ 3+22−1 3+5<5 x−2 12

4+4 (3 )>9 m−12

Video: http://www.youtube.com/watch?v=jQAxpFAKQ6M&feature=share&list=PLNDkuWRw1gGTgaYk6dQhGp10UT41lPFTM

CC6+ Unit 7

http://www.youtube.com/watch?v=jQAxpFAKQ6M&feature=share&list=PLNDkuWRw1gGTgaYk6dQhGp10UT41lPFTM



Grade 6 Name: ______________________ Date: _______ Period: __

Notes 7.1 Expressions, Equations and Inequalities Guided Notes

3y2 + 2y – 3 = 30 5n – (8 ÷ 2) 7m – m3 ¿ 6

____________________

List characteristics of each “item” above and then create a Venn diagram to compare and contrast similarities and differences.

Video: http://www.youtube.com/watch?v=jQAxpFAKQ6M&feature=share&list=PLNDkuWRw1gGTgaYk6dQhGp10UT41lPFTM

Practice 7.1 C

I. Categorize each of the following mathematical “items” into the proper columns:

CC6+ Unit 7



5x + 3x + 2x 45 45x – 4 = 86 4 + 9y > 36 5 – 3 + 12 x 2 9 < 23b

t – 7 = 7 2w + 3 = 23 98 ÷ 2x ≤ 14 u – 5 ≥ 12 5d – d + b 7g = 102 g

Expressions Equations Inequalities

II. Write each “statement” mathematically and identify it as an expression, equation, or inequality:

1. Each fountainhead had two nozzles each and there were 5 faucets.

__________________________________________________________________________.

2. All three of the fountainheads sprayed out the same amount of water, which totaled twelve

thousand gallons of water.

__________________________________________________________________________.

3. The two filters processed at least five-hundred-thousand gallons of water a week.

Practice 7.1H Identify Expressions, Equations, and Inequality

Name___________________________ Date_____________ Pd____

CC6+ Unit 7


Determine if each statement/item applies to Expressions, Equations, and/or Inequalities. Some statements may apply to more than one. Put a check in the correct column(s).

Expression

Equation Inequality

1) Contains an equal sign

2) Cannot be ‘solved’

3) 3 x+4

4) 10<21

5) Has multiple solutions

6) 12=3+ x

7) Can contain numbers

8) A mathematical sentence

9) 7 w

10) 4=12 ÷ 3

11) Can contain variables

12) Can be compared to a scale

13) A mathematical phrase

14) x≥ 9

15) x+8<37−2

16) Like a balanced scale

17) 7+1

18) 7+1=8

19) 7+1≥ 8

20) Can contain operation symbols (+,-,x,÷)

21)34+7 (2 )−12+30

22) 5−x=12

23) 5−x

24) Usually has one solution

25) Like an unbalanced scale

Practice 7.1 C Equations VS inequality Sorting Cards

CC6+ Unit 7


3+5=5 x−2 5 w−3 w=9 ÷ 3+22−1

124

+4 (3 )>9 m−12 3+x+2<9

5 x−2 7+9

3+5 ≤ 5x−2 124

+4 (3 )

3+x+2=9 5w−3w

9 ÷ 3+22−1 9

Expressions VS Equations and Inequalities Sorting cards

3+5=5 x−2 5 w−3 w=9 ÷ 3+22−1

124

+4 (3 )>9 m−12 3+x+2<9

5 x−2 7+9

3+5 ≤ 5x−2 124

+4 (3 )

3+x+2=9 5w−3w

9 ÷ 3+22−1 9

IDENTIFYING EXPRESSIONS, EQUATIONS, & INEQUALITIES

CC6+ Unit 7


Sample Quiz Unit 7 Part 1

Problems AnswersIdentify each item as an expression, equation, or inequality. Write your answer in the answer column.

1. 7 w+8+6 w−3

2.20 w

4>10

3. 4 x=8

4. 14+9

5. 18 g÷ 2 ∙3=5 g−6

6. x ≤ 4

7. 7

8. 8=8

9. 9<10

10. a+b−cd>cd+efg

Notes 7.2A Solutions to Equations and Inequalities Guided Notes

CC6+ Unit 7


SOLUTION? It’s the value or values that make an equation or inequality true.EXAMPLE: Is t = ____ a solution to the following ________________________?6t + 13 = 6t – 13 14 – 2t = 2t – 14 9t ÷ 9 = 4t ÷ 4STEP 1: _____________________ – “plug-in” or replace the variable with the possible solution.____________________ __________________ ________________

STEP 2: _____________________ – evaluate the __________________________ on either side of the equation/inequality.

_____________________ ________________ ________________

____________ ____________ ____________

STEP 3: ___________________ – determine if the mathematical statement is _________.

__________________ ______________________ __________________

Is t=____ a solution for the following ____________________________?

(* Follow the same steps as above for inequalities as equations.)

9t ÷ 9 < 4t ÷ 4 9t ÷ 9 > 4t ÷ 4 9t ÷ 9 ≤ 4t ÷ 4 9t ÷ 9 ≥ 4t ÷ 4

_____________ _____________ _______________ ______________

______________ _______________ ______________ ______________

______________ _______________ ______________ ______________

______________ ________________ ____________ ______________

Video : http://www.youtube.com/watch?v=0EMUtIV13H8&feature=share&list=PLNDkuWRw1gGTgaYk6dQhGp10UT41lPFTM

Notes 7.2 B Solutions to Equations and Inequalities Solution the value or values that make an equation or inequality true

Is m=4a solution to 5m+10≤ 7m−2?

To determine if a given value is a solution:

CC6+ Unit 7

http://www.youtube.com/watch?v=0EMUtIV13H8&feature=share&list=PLNDkuWRw1gGTgaYk6dQhGp10UT41lPFTM

http://www.youtube.com/watch?v=0EMUtIV13H8&feature=share&list=PLNDkuWRw1gGTgaYk6dQhGp10UT41lPFTM


1. Substitute the given value into the equation or inequality2. Simplify the expression on either side of the equation or inequality3. Determine if the simplified expressions satisfy the equal sign or inequality symbol 5 m+10>7m−2

5 (4 )+10>7 (4 )−2

Simplify the expression on either side of the equation or inequalityNOTE: the >,<, or = sign seperates the 2 sides (2 expressions

5 m+10>7 m−2

5 (4 )+10>7 (4 )−2

20+10>28−2

30>26

Determine if the simplified expressions satisfy the equal sign or inequality symbol.

http://www.youtube.com/watch?v=MG4DvlWQaGE&feature=share&list=PLNDkuWRw1gGTgaYk6dQhGp10UT41lPFTM

Practice 7-2H State whether the given value is a solution to the equation or inequality. Write YES or NO. Show work to prove your answer.

1) 2 x+7=17 ; x=5 8) 5 w<3w+6 ; w=2

2) 35=7h−8 ;h=6 9) 63<3+6 r ;r=10

CC6+ Unit 7





3) 63≤ 3+6 r ;r=10 10) 50−3 x=42; x=12

4) 9+6 a=57 ; a=8 11) 9 k+3<8 ∙3 ;k=2

5) 10+4>20−3 v ;v=4 12) 5 g−15≥ 60 ;g=15

6) 9 h=20+6 h;h=7 13) 79−8 p=34+ p ; p=5

7) 8+6 c≤ 9 c−30 ;c=4 14) 8 r+1=5r+10;r=3

Practice 7-2 H2 State whether the given value is a solution to the equation or inequality. Write YES or NO.

CHALLENGE: If the value is not a solution, can you determine which value(s) would be a solution?

1 . 5 x – 8=18+4, for x=6 2. 4 x2 – 5 (5)=12, for x=3

3. ¿ , forn=3 4. 17 x−8 (2 x−4 )>32, for x=3

5. 2 x+12+8 x>32, for x=2 6. 6(3 x−2)+5<50, for x=3

CC6+ Unit 7


Test each value in the ‘Replacement Set’ column to determine if the values are solution(s) to the given equation/inequality. Be sure to list all numbers that work to make the statement true. There may be 1, more than one, or no solutions to each. ** Do your work in the space below the chart. Equation Replacement Set Solution(s)

7. 5 x+2=17 {1, 2, 3, 4}

8. 3 x−2>4 {2, 3, 4, 5}

9. 2 x2+4=54 {1, 3, 5, 7}

10. 7 x−7<30 {2, 4, 6, 8}

11. 2 (2 x+4 )>20 {3, 5, 6, 9}

12. 5 x – 6=24 {1, 2, 3, 4}

Solving 1-Step Equations and Inequalities Guided NotesHow do we find the value of the variables in these equations?______________________, where we just guess different values until the math statement is true?What do equations and a balanced scale have in common?

The ________________ on both sides of the equal/inequality sign must have the __________.Therefore, whatever operation you do to one side of the equation, you must do the same operation to the other side as well.If we can ________________ the variable on side of the equation, we can solve for the variable! _________________ the variable is a strategy, where the goal is to get the variable all by itself on one side of the equal or inequality sign.

CC6+ Unit 7


x – 7 = 5 8y = 24 m3 = 9

a. b. c.

1st : Ask, what is being done to the variable?

a. ________________________ b. ________________________ c. _____________________

2nd : Ask, what is the inverse operation? “How do I undo this operation?”

a. __________ b. _________________ c. ________________________

3rd : Perform this operation to both sides of the equation/inequality to isolate the variable and keep the equation/inequality balanced.

a. x – 7 = 5 b. 8y = 24 c. 3(m3 ) = 3(9)

___ = ____ ____ = ____ ____ = _____

y + 8

>

Practice 7.3 One –Step Equation

Solve each equation. Make sure to show your work by using the inverse property. Circle your answer.

1. 8 + v = 26

2. 3 + p = 8

3. 15 + b = 23

4. m + 4 = 12

5. x – 7 = 13

CC6+ Unit 7

INEQUALITIES are solved just like equations, but remember that your final answer is not single, but rather a range of multiple solutions! 15 > y + 8

-8 -8 ( y must be any number ___ > y less than 7.)


6. m – 9 = 23

7. p – 6 = 5

8. v – 15 = 26

9. n + 9 = 16

10. 8x =104

11. 14b = 56

12. b18

=6

13.10n=40

14. v8=2

15. k11

=14

16.15 x=0

17. 17x = 204

18. 21 = 7n

19. m4=13

20.126=14 k

21. 143 = 11x

22. 16 + x = 19

23.5= a18

24. 17 = x – 15

Practice 7.3 Hands on Equation

Equation: Check:

CC6+ Unit 7


Class Example:

Class Example:

Example 1:

Example 2:

Class Example:

2x + 3x = 15

CC6+ Unit 7


Example 3:

4x + 2x = 15 + 3

Example 4:

x + 7x = 2 + 14

Class Example:

x + 4 = 13

Example 5:

5 + x = 12

Example 6:

17 = 9 + x

Class Example:

5x – 2x = 12

Example 7:

8x – 6x = 20

Example 8:

7x – x = 18

Practice 7. 4 Equations &Word Problems with Decimals

Solve and check the equations.

CC6+ Unit 7


1) 2.23 + x = 6.5

2) x – 4.75 = 9.2

3) 0.06 + y = 3.6

Write an equation for each word problem. Solve and check.

5) A plastic jug holds 5.2l of liquid. A water bottle holds 3.9l less liquid than the jug. What is the volume of liquid in the water bottle?

6) Sammy had $964.21in his checking account at the beginning of the month. The bank posted a balance of $2695.36 on the 15th of the month. How much money had Sammy deposited during this part of the month?

CC6+ Unit 7


CC6+ Unit 7

Practice 7.4 One step Equation


CC6+ Unit 7

Practice 7.5C One Step Equation


CC6+ Unit 7

Practice 7.5H Equation Word Problems


Notes 7.6 Graphing Equations and Inequalities on Number Lines Guided Notes

Graphing an Equation on a Number Line:

(equal) X = 4

The value of x, which is 4, is represented by a darkened _________ at 4 to the ___________ of 0 on the number line.

Graphing Inequalities on a Number Line:

(greater than) Y ¿ 3

The value of y, which is any number _____________ than 4, is represented by an _______________ at 3 to the right of zero with an ______________ pointing to right on the number line. An open circle means it _____________ equal that value, and the arrow (ray) pointing to the right means that it can be any value ______________ than the number that is circled.

(less than) Z ¿ 5

The value of z, which is any number ________ than 5, is represented by an _______________ at 5 to the right of zero with an _____________ pointing to left on the number line. An open circle means it ____________ equal that value, and the arrow (ray) pointing to the left means that it can be any value __________ than the number that is circled.

(greater than or equal) W ≥ -2

The value of w, which is ___________ to -2 and any number ____________ than -2, is represented by a ________ at 2 to the left of zero with an arrow pointing to __________ on the number line. The point means it _____________ that value, and the arrow (ray) pointing to the right means that it can be any value _______________ than the number where the point is located as well.

(less than or equal) V ≤ 1

The value of v, which is ___________ to 1 and any number ___________ than 1, is represented by a ___________ at 1 to the right of zero with an arrow pointing to __________ on the number line. The point means it ____________ that value, and the arrow (ray) pointing to the left means that it can be any value _____________ than the number where the point is located as well.

WCPSS Video: http://www.youtube.com/watch?v=TIdrfYfNmDs&feature=share&list=PLNDkuWRw1gGTgaYk6dQhGp10UT41lPFTM

Practice 7.6 C Graphing Inequalities

CC6+ Unit 7

http://www.youtube.com/watch?v=TIdrfYfNmDs&feature=share&list=PLNDkuWRw1gGTgaYk6dQhGp10UT41lPFTM

http://www.youtube.com/watch?v=TIdrfYfNmDs&feature=share&list=PLNDkuWRw1gGTgaYk6dQhGp10UT41lPFTM


Graph Inequalities Inequalities can be graphed on a number line. This helps you tosee which values make the inequality true. You can also write inequalities for a graph.

Example 1

Graph each inequality on a number line.a. x ≤ 8 b. x > 8

6 7 8 9 10 6 7 8 9 10

The closed dot means 8 does make the sentence true. The line means that numbers less than 8 make the sentence true.The open dot means 8 does not make the sentence true. The line means that numbers greater than 8 make the sentence true.Write an inequality for each graph.

a. b. -5 -4 -3 -2 -1 0 3 4 5 6 7 8

The open dot means -2 is not included in the graph. The arrow points left, so the graph includes all numbers less than -2.The inequality is x < -2.

The closed dot means 5 is included in the graph. The arrow points right, so the graph includes all numbers greater than 5.The inequality is x ≥ 5.

ExercisesGraph each inequality on a number line.

1. x > 7 2. a ≤ -2 3. d < -4

5 6 7 8 9 -4 -3 -2 -1 0 -6 -5 -4 - 3 -2

4. w > -9 5. t ≥ -5 6. n < -11

-11 -10 -9 -8 -7 -7 -6 -5 -4 -3 -13 -12 -11 -10 -9

Write the inequality for each graph.

7. 8. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -5 -4 -3 -2 -1 0 1 2 3 4 5

9. . 10. -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 0 1 2 3 4 5 6 7 8 9 10

Video Graphing Inequality http://www.virtualnerd.com/middle-math/equations-functions/inequalities/number-line-graph-inequality

Notes 7.6 Inequalities

CC6+ Unit 7

An open dot indicates that the number marked does not make the sentence true. A closed dot indicates that the number marked does make the sentence true.

The direction of the line indicates whether numbers greater than or less than the number marked make the sentence true.

http://www.virtualnerd.com/middle-math/equations-functions/inequalities/number-line-graph-inequality


A mathematical sentence that contains any of the symbols listed below is called an inequality.

< > ≤ ≥

• is less than

• is fewer than

• is greater than

• is more than

• is less than or equal to at most no more than

• is greater than or equal to at least no less than

Example 1 Write an inequality for the sentence.Fewer than 70 students attended the last dance.Words Fewer than 70 students attended the last dance.Symbols Let s = the number of students.Inequality s < 70You can substitute a value for a variable in an inequality and determine whether the value makes the inequality true or false.

Example 2For the given value, state whether each inequality is true or false.a. 5y - 6 < 14; y = 5 b. r - 16 ≥ -12; r = 45y - 6 < 14 Write the inequality. r - 16 ≥ -125(5) - 6 < 14 Replace the variable with the given value. 4 - 16 ≥ -1219 < 14 Simplify. -12 ≥ -12This sentence is false. Although -12 > -12 is false, -12 = -12 is true. So, this sentence is true.ExercisesWrite an inequality for each sentence

1. The maximum diving depth is no more than 45 feet below sea level. _________________

2. Adult male elephants can weigh over 12,000 pounds. _____________________________________

3. The maximum fee for any student is $15.________________________________________

4. You must be at least 38 inches tall to ride the roller coaster._________________________________

For the given value, state whether the inequality is true or false.

5. m + 8 ≥ 5; m = -3 6. 4+ p < -2; p = 6

7. b + 12 ≤ 15; b = 1 8. j + 7 < -8; j = 0

Video : Definition of Inequality http://www.virtualnerd.com/middle-math/equations-functions/inequalities/inequality-definition

Notes 7.7 Solve Inequalities by Adding or Subtracting

CC6+ Unit 7

http://www.virtualnerd.com/middle-math/equations-functions/inequalities/inequality-definition


Use the Addition and Subtraction Properties of Inequalities to solve inequalities. When you add or subtract a number from each side of an inequality, the inequality remains true.

Example

Solve 12 + y > 20. Check your solution.

12 + y > 20 Write the inequality.

12 - 12 + y > 20 - 12 Subtraction Property of Inequality

y > 8 Simplify.

To check your solution, try any number greater than 8.

CHECK 12 + y ¿?

20 Write the inequality.

12 + 9 ?

20 Replace y with 9.

21 > 20 ✔ This statement is true.

Any number greater than 8 will make the statement true. Therefore, the solution is y > 8.

ExercisesSolve each inequality. Check your solution.

1. 12 < 8 + b 2. t - 5 > 4 3. p + 5 < 13

4. 5 > y - 6 5. 21 < n -18 6. s - 4 ≤ 3

7. 14 > w + (-2) 8. j + 6 ≥ -4 9. z + (-4) < -2.5

10. b+3.89≤ 4.25 11. g - 214 ≥ 3 1

4

Video : Learnzillion http://learnzillion.com/lessons/2614-write-and-graph-inequalities-shopping http://learnzillion.com/lessons/2615-write-and-graph-inequalities-temperature

CC6+ Unit 7


CC6+ Unit 7

Practice 7.6 One-Step Inequality

cc6+ unit 7 equation and inequality · web viewequation ‐ equations can be solved and usually...

Documents