3

Notes

All Khan Due @ end of the dayFinal Exam Grades Posted @ v6 Math

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Notebooks: 1st leave today, 2nd Tuesday, 3rd

Wed., 4th Thursday.

Warm Up:

1. Solve the inequality: -1 < 𝟑 −𝟐𝒙

𝟓< 3

4 > x and x > -6; write as one inequality -6 < x < 4

2. Solve for x: 1 -𝟐

𝒙 −𝟐

1 + 𝟐

𝒙 −𝟐

Eliminate both denominators. Hint: There are 2 terms in both the numerator and the denominator.

𝒙 − 𝟐 1 -𝟐

𝒙 −𝟐𝒙 − 𝟐 − 𝟐

𝒙 − 𝟐 + 𝟐

𝒙 − 𝟒

𝒙

A quick look at the 3rd quarter

Warm Up:

5. Graph the inequality: 2x + 3y > 12

x

y

2-2

(0,4)

(6,0)

3. Write an equation for a horizontal line

6. Write an equation for a line perpendicular to: −𝟏

𝟑x = 2y + 12.

4. Write an equation with an undefined slope

Recall…

Graph n < 3 on a number line.

-3 -2 -1 0 1 2 3 4

a. (4, 5); y < x + 1

Tell whether the ordered pair is a solution of the inequality.

y < x + 1

Substitute (4, 5) for (x, y).Substitute (1, 1) for (x, y).

b. (1, 1); y > x – 7

y > x – 7

5 4 + 1

5 5 <

1 1 – 7

>1 –6

(4, 5) is not a solution. (1, 1) is a solution.

The boundary line also represents the related linear equation....what is the related equation?

What is the Inequality?

y > x + 4

Graphing Linear Inequalities

Step 1Solve the inequality for y (solved EXACTLY like an equation; slope-intercept form).

Step 2Graph the solution (the boundary line). Use a solid line for ≤ or ≥. Use a dashed line for < or >.

Step 3Shade the half-plane above the line for y > or ≥. Shade the half-plane below the line for y < or y ≤. Check your answer.

Graph y > 3 on the coordinate plane.

x

y

x

y

Graph x < -2 on the coordinate plane.

Graph y > -3x + 2 on the coordinate plane.

x

y

Boundary Line

y = -3x + 2m = -3 b = 2

Test a point not on the line

test (0,0)

0 > -3(0) + 2

Not true!

Graph y -3x + 2 on the coordinate plane.

x

y

Instead of testing a point

If in y = mx + b form...

Shadeup

Shadedown

Solidline

Dashedline

> <

Graphing Linear Inequalities

Graph the solution of the linear inequality.5x + 2y > –8

Step 1 Solve the inequality for y.Step 2 Graph the boundary line and Use a dashed line for >.

y = −𝟓

𝟐x – 4

Step 3: Test a point not on the line. Use (0,0) when you can.

5(0) + 2(0) > -8, True or False?

If true, include that point in your shading.

Graph the inequality.

3x - 4y > 12-3x -3x

-4y > -3x + 12

-4 -4

y < x - 3

m = b = -3

Boundary Line

x

y

3

4

𝟑

𝟒

𝟑

𝟒

Problem

You have less than $5.00 in nickels and dimes, find an inequality and sketch a graph to describe how many of each coin you have.

Let n = # of nickels

Let d = # of dimes

0.05 n + 0.10 d < 5.00

or 5 n + 10 d < 500

Graphing Linear Inequalities

5n + 10d < 500

n d

0 50

100 0

0 10 20 30 40 50 60 70 80 90 100n

d

60

50

40

30

20

10

0

When dealing with angled lines,If the inequality is > or > ,then you shade above

If the inequality is < or < ,then you shade below

Finding a point not on the line is still the safest and

surest method to determine where to shade.

Graph y -3x + 2 on the coordinate plane.

x

y

Instead of testing a point

Arrange the equation in y = mx + b form...

Shadeup

Shadedown

Solidline

Dashedline

> <

Graphing Linear Inequalities

Graph on the coordinate plane.

3x - 4y > 12-3x -3x

-4y > -3x + 12

-4 -4

y < x - 33

4

m = b = -33

4

Boundary Line

x

y

Graphing Linear Inequalities

If the point makes the inequality true, shade that side of the line.

If the point does not make the inequality true, shade the opposite side of the line.Use (0,0) as a test if this point is not on the line.

Quadrilateral ABCD has diagonals AC and BD. Determine whether segment AC is perpendicular to BD

AC, m = 7

BD, m = −𝟏

𝟕

The lines are perpendicular.