A4.e How Do I Graph The Solution Set of A Linear Inequality in Two Variables? Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation

A4.e How Do I Graph The Solution Set of A Linear Inequality in Two Variables?

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Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Warm UpFind each equation of direct variation, given that y varies directly with x.

1. y is 18 when x is 3.

2. x is 60 when y is 12.

3. y is 126 when x is 18.

4. x is 4 when y is 20.

y = 6x

y = 7x

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12-6 Graphing Inequalities in Two Variables

y = 5x

y = x15

Problem of the Day

The circumference of a pizza varies directly with its diameter. If you graph that direct variation, what will the slope be?

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Learn to graph inequalities on the coordinate plane.

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Vocabulary

boundary linelinear inequality

Insert Lesson Title Here

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A graph of a linear equation separates the coordinate plane into three parts: the points on one side of the line, the points on the boundary line, and the points on the other side of the line.

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When the equality symbol is replaced in a linear equation by an inequality symbol, the statement is a linear inequality. Any ordered pair that makes the linear inequality true is a solution.

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Graph each inequality.

y < x – 1

Example 1A: Graphing Inequalities

First graph the boundary line y = x – 1. Since no points that are on the line are solutions of y < x – 1, make the line dashed (creates 2 open half planes). Then determine on which side of the line the solutions lie.

(0, 0)

y < x – 1

Test a point not on the line.

Substitute 0 for x and 0 for y.0 < 0 – 1?

0 < –1?

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Any point on the line y = x 1 is not a solution of y < x 1 because the inequality symbol < means only “less than” and does not include “equal to.”

Helpful Hint

Example 1A Continued

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(0, 0)

Since 0 < –1 is not true, (0, 0) is not a solution of y < x – 1. Shade the side of the line that does not include (0, 0). When a “<“ is used, shade below the boundary line.

y 2x + 1

Additional Example 1B: Graphing Inequalities

1. Graph the boundary line: y = 2x + 1. 2. Make the line solid (creates 2 closed half

planes) because the points that are on the line are solutions of y 2x + 1.

3. Shade above the line (because the is used). This is where the rest of the solutions of y 2x + 1 lie.

(0, 4) Choose any point not on the line.

Substitute 0 for x and 4 for y.

y ≥ 2x + 1

4 ≥ 0 + 1?

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Any point on the line y = 2x 1 is a solution of y ≥ 2x 1 because the inequality symbol ≥ means “greater than or equal to.”

Helpful Hint

Additional Example 1B Continued

Since 4 1 is true, (0, 4) is a solution of y 2x + 1. Shade the side of the line that includes (0, 4). When a “>” is used, shade above the boundary line.

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(0, 4)

2y + 5x < 6

Additional Example 1C: Graphing Inequalities

First write the equation in slope-intercept form.

2y < –5x + 6

2y + 5x < 6

y < – x + 352

Then graph the line y = – x + 3. Since points that

are on the line are not solutions of y < – x + 3,

make the line dashed. Then determine on which

side of the line the solutions lie.

52 5

2

Subtract 5x from both sides.

Divide both sides by 2.

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Additional Example 1C Continued

Since 0 < 3 is true, (0, 0) is a

solution of y < – x + 3.

Shade the side of the line

that includes (0, 0).

52


y < – x + 352

0 < 0 + 3?

0 < 3?

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(0, 0)


y < x – 4

Check It Out: Example 1A

First graph the boundary line y = x – 4. Since no points that are on the line are solutions of y < x – 4, make the line dashed. Then determine on which side of the line the solutions lie.

(0, 0)

y < x – 4

Test a point not on the line.

Substitute 0 for x and 0 for y.0 < 0 – 4?

0 < –4?

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Check It Out: Example 1A Continued

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(0, 0)

Since 0 < –4 is not true, (0, 0) is not a solution of y < x – 4. Shade the side of the line that does not include (0, 0).

y > 4x + 4

Check It Out: Example 1B

First graph the boundary line y = 4x + 4. Since points that are on the line are solutions of y 4x + 4, make the line solid. Then shade the part of the coordinate plane in which the rest of the solutions of y 4x + 4 lie.



y ≥ 4x + 4

3 ≥ 8 + 4?

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Check It Out: Example 1B Continued

Since 3 12 is not true, (2, 3) is not a solution of y 4x + 4. Shade the side of the line that does not include (2, 3).

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(2, 3)

3y + 4x 9

Check It Out: Example 1C

First write the equation in slope-intercept form.

3y –4x + 9

3y + 4x 9

y – x + 343

Subtract 4x from both sides.

Divide both sides by 3.

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43Then graph the line y = – x + 3. Since points that

are on the line are solutions of y – x + 3, make

the line solid. Then determine on which side of the

line the solutions lie.

43

Check It Out: Example 1C Continued

Since 0 3 is not true, (0, 0) is

not a solution of y – x + 3.

Shade the side of the line that

does not include (0, 0).

43


y – x + 343

0 0 + 3?

0 3?

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(0, 0)

A successful screenwriter can write no more than seven and a half pages of dialogue each day. Graph the relationship between the number of pages the writer can write and the number of days. At this rate, would the writer be able to write a 200-page screenplay in 30 days?

Additional Example 2: Career Application

First find the equation of the line that corresponds to the inequality.

In 0 days the writer writes 0 pages.

point (0, 0)

point (1, 7.5)In 1 day the writer writes no more than 7 pages.1

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The phrase “no more” can be translated as less than or equal to.

Helpful Hint

Additional Example 2 Continued

With two known points, find the slope.

y 7.5 x + 0 The y-intercept is 0.

Graph the boundary line y = 7.5x. Since points on

the line are solutions of y 7.5x make the line solid.

Shade the part of the coordinate plane in which the

rest of the solutions of y 7.5x lie.

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m = 7.5 – 01 – 0

7.5 1

= = 7.5


y 7.5x


Since 2 15 is true, (2, 2) is a solution of y 7.5x. Shade the side of the line that includes point (2, 2).


2 7.5 2?

2 15 ?

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The point (30, 200) is included in the shaded area, so the writer should be able to complete the 200 page screenplay in 30 days.


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A certain author can write no more than 20 pages every 5 days. Graph the relationship between the number of pages the writer can write and the number of days. At this rate, would the writer be able to write 140 pages in 20 days?

Check It Out: Example 2

First find the equation of the line that corresponds to the inequality.

In 0 days the writer writes 0 pages. point (0, 0)

point (5, 20)In 5 days the writer writes no more than 20 pages.

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Check It Out: Example 2 Continued

20 - 0 5 - 0m = = 20

5 = 4 With two known points, find the slope.

y 4x + 0 The y-intercept is 0.

Graph the boundary line y = 4x. Since points on the line are solutions of y 4x make the line solid. Shade the part of the coordinate plane in which the rest of the solutions of y 4x lie.

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y 4x


Since 60 20 is not true, (5, 60) is not a solution of y 4x. Shade the side of the line that does not include (5, 60).


60 4 5?

60 20 ?

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The point (20, 140) is not included in the shaded area, so the writer will not be able to write 140 pages in 20 days.


x

y200

180

160

140120

100

80

60

40\

20

Pag

es

5 10 15 20 25 30 35 40 45 50

Days

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1. y < – x + 4

13

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Lesson Quiz Part I

2. 4y + 2x > 12

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Lesson Quiz Part II

Tell whether the given ordered pair is a

solution of each inequality.

3. y < x + 15 (–2, 8)

4. y 3x – 1 (7, –1)

yes

no

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Lesson Quiz: Part III

Documents

A4.e How Do I Graph The Solution Set of A Linear Inequality in Two Variables? Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation