GRAPHING SYSTEMS OF LINEAR INEQUALITIES -...

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FOM 11 T14 – SYSTEMS OF LINEAR INEQUALITIES 1 © R. Ashby 2017. Duplication by permission only. MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) A SYSTEM OF LINEAR INEQUALITIES = a problem where 2 or more inequalities are graphed on the same grid, the solution of which is the overlapping shaded area. GRAPHING SYSTEMS OF LINEAR INEQUALITIES I) Systems of linear inequalities are questions that require you to graph more than one inequality on the same grid. This results in an area on the grid where the shaded area of each inequality overlap. This is the solution to the problem. A) USE THESE STEPS TO SOLVE REAL LIFE SITUATIONS INVOLVING LINEAR INEQUALITIES 1: Graph the boundary and shaded area for each inequality. 2: Stipple the overlapping shaded area if the MATHEMATICAL restrictions on one or more of the variables is anything other than real numbers. B) SAMPLE PROBLEMS 1: Study these examples carefully. Be sure you understand and memorize the process used to complete them. INSTRUCTIONS: Graph each system of linear inequalities. 1) x, y ( ) y 2 x 1 , x !, y ! { } x, y ( ) y 3 , x !, y ! { } 1: Graph the boundary and shaded area for each inequality. 1a: Graph the first inequality. 1b: Graph the second inequality. y 2 x 1 y 3 y-intercept = b = 1 This inequality has a horizontal boundary slope = m = 2 1 passing through the y-axis at y = 3 2: Stipple the overlapping shaded area if the MATHEMATICAL restrictions on one or more of the variables is/are anything other than real numbers. Because the MATHEMATICAL restrictions are real numbers x !, y ! the overlapping area is not stippled. The solution to the system is the graph created in step 1 (shown above).

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FOM 11 T14 – SYSTEMS OF LINEAR INEQUALITIES 1

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MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) A SYSTEM OF LINEAR INEQUALITIES = a problem where 2 or more inequalities are graphed on the same

grid, the solution of which is the overlapping shaded area.

GRAPHING SYSTEMS OF LINEAR INEQUALITIES I) Systems of linear inequalities are questions that require you to graph more than one inequality on the same grid.

This results in an area on the grid where the shaded area of each inequality overlap. This is the solution to the problem. A) USE THESE STEPS TO SOLVE REAL LIFE SITUATIONS INVOLVING LINEAR INEQUALITIES

1: Graph the boundary and shaded area for each inequality. 2: Stipple the overlapping shaded area if the MATHEMATICAL restrictions on one or more of the

variables is anything other than real numbers.

B) SAMPLE PROBLEMS 1: Study these examples carefully. Be sure you understand and memorize the process used to complete them. INSTRUCTIONS: Graph each system of linear inequalities.

1)

x, y( ) ⎮ y≥ 2x−1 , x∈!, y∈!{ }x, y( ) ⎮ y≥ −3 , x∈!, y∈!{ }

1: Graph the boundary and shaded area for each inequality. 1a: Graph the first inequality. 1b: Graph the second inequality.

y≥ 2x−1 y≥−3

y-intercept = b = −1 This inequality has a horizontal boundary slope =

m = 2

1 passing through the y-axis at y = −3

2: Stipple the overlapping shaded area if the MATHEMATICAL restrictions on one or more of the variables is/are anything other than real numbers.

Because the MATHEMATICAL restrictions are real numbers x∈!, y∈! the overlapping area is not stippled. The solution to the system is the graph created in step 1 (shown above).

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2)

x, y( ) ⎮ y > x , x∈ I, y∈ I{ }x, y( ) ⎮ y <5 , x∈ I, y∈ I{ }

1: Graph the boundary and shaded area for each inequality. 1a: Graph the first inequality. 1b: Graph the second inequality.

y > x → y >1x + 0 y <5 y-intercept = b = 0 This inequality has a horizontal boundary slope =

m = 1

1 passing through the y-axis at y = 5

2: Stipple the overlapping shaded area if the MATHEMATICAL restrictions on one or more of the variables is/are anything other than real numbers.

Because the MATHEMATICAL restrictions are Integers x∈ I, y∈ I the overlapping area is stippled

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FOM 11 T14 – SYSTEMS OF LINEAR INEQUALITIES 3

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3)

P, M( ) ⎮ M < 3 , P∈W, M ∈W{ }P, M( ) ⎮ M ≤3P−4 , P∈W, M ∈W{ }

1: Graph the boundary and shaded area for each inequality. 1a: Graph the first inequality. 1b: Graph the second inequality.

M < 3 M ≤3P−4

This inequality has a horizontal boundary M-intercept = b = −4 passing through the M-axis at y = 3 slope =

m = 3

1

2: Stipple the overlapping shaded area if the MATHEMATICAL restrictions on one or more of the variables is/are anything other than real numbers.

Because the MATHEMATICAL restrictions are Whole numbers, x∈W, y∈W , the points in the overlapping shaded area that have non-negative entire number coordinates are stippled as shown.

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FOM 11 T14 – SYSTEMS OF LINEAR INEQUALITIES 4

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4)

x, y( ) ⎮ 3x−7 y≥ −45 , x∈ I, y∈ I{ }x, y( ) ⎮ 9y +5x≥32 , x∈!, y∈W{ }

1: Graph the boundary and shaded area for each inequality. 1a: Graph the first inequality. 1b: Graph the second inequality.

3x−7 y≤ −45

3x−3x−7 y≤−3x + −45−7 y≤−3x−45−7−7

y≤−3−7

x− 45−7

y≤ 37 x + 45

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b = 457 ! 6.4 m = 3

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9y +5x≥32

9y +5x−5x−≥−5x + 32

9y≥−5x + 32

99

y≥−59 x + 32

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y≥−59 x + 32

9

b = 329 ! 3.6 m =

−59

Because the y-intercepts are fractions, not entire numbers, calculate the x-intercepts for each inequality then graph them using the x & y-intercepts. Because you are calculating specific points on the grid, change the inequality signs to a = signs.

3x−7 y = −45

3x−7 0( )= −45

3x−0 = −45

3x = −45

33

x =−453

x -int = −15

9y−5x = 32

9 0( )+5x = 32

0+5x = 32

5x = 32

55

x = 325

x -int = 325 ! 6.4

Continued on the next page.

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FOM 11 T14 – SYSTEMS OF LINEAR INEQUALITIES 5

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2: Stipple the overlapping shaded area if the MATHEMATICAL restrictions on one or more of the variables is/are anything other than real numbers.

Because the most restrictive MATHEMATICAL restriction is Whole numbers, y∈W , the points in the overlapping shaded area that have non-negative entire number coordinates are stippled as shown.

5)

x, y( ) ⎮ y≤ −76 x +800 , x∈!, y∈!{ }

x, y( ) ⎮ y≤ −57 x + 600 , x∈!, y∈!{ }

1: Graph the boundary and shaded area for each inequality. 1a: Graph the first inequality. 1b: Graph the second inequality.

y≤

−76 x +800

y≤

−57 x + 600

y-intercept = b = 800 y-intercept = b = 600

slope = m = −7

6 slope = m = −5

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Because the y-intercepts are very large, calculate the x-intercepts for each inequality then graph them using the x & y-intercepts. Because you are calculating specific points on the grid, change the inequality signs to a = signs.

y = −76 x +800

0( )=−76 x +800

0 + 76 x =

−76 x + 7

6 x +800

76 x = 800

6( ) 76 x( )= 800 6( )

7x = 480077 x( )= 4800

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x -int = 48007 ! 685.7

y = −57 x + 600

0( )=−57 x + 600

0 + 57 x =

−57 x + 5

7 x + 600

57 x = 600

7( ) 57 x( )= 600 7( )

5x = 420055 x( )= 4200

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x -int = 840 Continued on the next page.

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2: Stipple the overlapping shaded area if the MATHEMATICAL restrictions on one or more of the

variables is/are anything other than real numbers. Because the most restrictive MATHEMATICAL restrictions is Natural numbers, y∈! , the points in the overlapping shaded area that have positive entire number coordinates are stippled as shown.

C) REQUIRED PRACTICE 1: Complete these problems in the order listed. SHOW THE PROCESS!! 1) Page 307: Question 2. {Ans. Page 557} 2) Page 317-319: Questions 1, 2, 4, 7, 9, 3 & 12. {Ans. Page 558-560}

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ASSIGNMENT: PRINT THIS INFORMATION ON YOUR OWN GRID PAPER

LAST then FIRST Name T14 – SYSTEMS OF LINEAR INEQUALITIES Block:

Show the process required to complete each problem to avoid receiving a zero grade. Neatness Counts!!! (Marks indicated in italicized brackets.)

REMEMBER TO USE GRID PAPER FOR ALL ASSIGNMENTS!!! GRAPH EACH INEQUALITY ON ITS OWN GRID!!!

Show the process required to complete each problem to avoid receiving a zero grade. Neatness Counts!!! (Marks indicated in italicized brackets.)

REMEMBER TO USE GRID PAPER FOR ALL ASSIGNMENTS!!!

∞ Graph these systems of inequalities. BE SURE YOU INCLUDE ALL RELEVENT INORMATION.

1)

x, y( ) ⎮ y≥ −2x−4 , x∈!, y∈!{ }x, y( ) ⎮ y > 2 , x∈!, y∈!{ }

(5) 2)

x, y( ) ⎮ 2x + y >8 , x∈W, y∈W{ }x, y( ) ⎮ 4y + x≥8 , x∈W, y∈W{ }

(9)

3)

x, y( ) ⎮ 5x−6y≤ 40 , x∈!, y∈!{ }x, y( ) ⎮ 8x + 7 y≤59 , x∈!, y∈!{ }

(15)

∞ Following the instructions. (1) /30