Linear Equations Linear Inequalities Drive on The Education Highway

LinearEquations

Linear Inequalities

Drive on The Education Highway

Linear Equations and Graphing

1. Parts of a Coordinate Plane

2. Slope

3. Slope-intercept Form of a

Linear Equation

4. Graphing by x- & y-intercepts.

Parts of a coordinate plane.

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

Click on the correct quadrant numbers. Correct answer = applause.

Quadrant

I II III IV

Quadrant

I II III IV

Quadrant

I II III IV

Quadrant

I II III IV

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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

Click on the correct axis names. Correct answer = clapping.

x-axis

y-axis

x-axis y-axisLessonStart

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

Click on the point for the origin. Correct answer = clapping.

x

yLessonStart

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

Click on point (-3, 5). Correct point = applause.

x

yLessonStart

You chose a line segment instead of a point.

Go back and try again.

You chose point (5, -3).

Each ordered pair is in the form (x, y) -- it follows the alphabet in its internal order.

You find the x value first, then you find the y value. Where they meet is the point.


-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

Click on the correct ordered pair for the black point.

-5/4

(4, -5)

(-5, 4)

-4/5

x

yLessonStart

You did not choose an ordered pair.


You chose the ordered pair for the pale green point.

Remember: x comes before y in the alphabet and in an ordered pair.


Return to Main Menu.

Return to Prior Problem.

Continue to Next Lesson.

Slopes

1. What is a slope?

2. Slope formula

3. Types of slopes

What is slope?

Slope is the slant of a line.

Slope = rise change in y’s

run change in x’s

Slope is a fraction/integer.

LessonStart

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

How to determine the slope when the line goes up.

1. Count the number of units up from the right point to the left point.

1

2

3

45

6

2. Put that number on top of the fraction line.

Slope = 6

3, Count the number of units to the right.

1 2 3 4 5 6 7 8 9

4. Put that number under the fraction line.

9

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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

How to determine the slope when the line goes down.

1. Count the number of units down from right point to left point.

-1-2

-3-4

-5-6

2. Put that number on top of fraction line.

Slope = -6

3. Count the units to the right.

1 2 3

4. Put that number under the fraction line.

3

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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

Determine the slope of the line shown.

-1/3 3/1

-3/1 1/3LessonStart

The line does not go down.


LessonStart

The line does not rise 3 units,

then run 1 unit to the right.


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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

Determine the slope of the line shown.

-2/3 3/2

-3/2 2/3LessonStart

The line does not go up.


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The line does not rise -2 units,

then run 3 units to the right.


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Slope Formula:

m = (y1 - y2)(x1 - x2)

where m = slope

and (x1, y1), (x2, y2) are

points on the line.

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Example: Find the slope for a line with points (-3, 4) and (7, -2) on it.1. Assign values as follows:

2. Substitute them into the formula and solve.

m = 4 - (-2) = 6 = -3 -3 - 7 -10 5

(x1, y1) = (-3, 4)

(x2, y2) = (7, -2)

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Find the slope of the line with points (5, 6) and (2, 9) on it.

1. (x1, y1) = (5, 6) (2, 9)

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1. (x1, y1) = (5, 6)

2. (x2, y2) =(5, 6) (2, 9)

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1. (x1, y1) = (5, 6)

2. (x2, y2) = (2, 9)3. m =

6 + 95 + 2

6 - 95 - 2

5 - 26 - 9

5 + 26 + 9

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The slope formula is a case of subtraction

on top and bottom.


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You have your x’s and y’s upside down.

Remember:

“Y’s guys are always in the skies!”


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You are also adding when you need to subtract.


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1. (x1, y1) = (5, 6)

2. (x2, y2) = (2, 9)3. m = 6 - 9 = -3 = -1

5 - 2 3

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Find the slope of the line with points (7, 5) and (3, -4) on it.

m = 5 - 4 = 17 - 3 4

7 - 3 = 45 - (-4) 9

5 - (-4) = 9 3 - 7 -4

5 - (-4) = 9 7 - 3 4

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Remember:

“Y’s guys are always in the skies!”


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It is not 5 - 4, it is 5 - (-4).


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You must start with the y and x from the first point and end with the y and x from the second

point.


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Types of Slopes:

1. Positive and Negative Slopes

2. Special Types of Slopes

3. Determining Types of Slopes by

Looking at Graphs of Lines

4. Determining Types of Slopes

Algebraically.LessonStart

Positive and Negative Slopes.

Type Graph AlgebraPositive Up left to right. Positive

Fraction

Negative Down left to right. Negative

FractionLessonStart

2 Special Types of Slopes

Type Graphs AlgebraZero Horizontal Line 0/a, a 0

UndefinedVertical Line a/0

No Slope

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Determining Types of Slopes by

Looking at Graphs of Lines

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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

Is the slope of the line positive, negative, zero, or undefined?

-0 +LessonStart

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y


-0+LessonStart

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y


-0 +LessonStart

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y


-0 +LessonStart

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

Click on the line with the negative slope.

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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

Click on the line with the zero slope.

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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

Click on the line with the no slope.

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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

Click on the line with the positive slope.

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Determining Types of Slopes

Algebraically.

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Is the slope of the line with the 2 points listed positive, negative, zero, or undefined?

Points (-3, 5) and (-9, -4)

+ - 0

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Points (3, 5) and (3, -4)

+ - 0

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Points (-3, -5) and (-9, -4)

+ - 0

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Points (-3, -4) and (-9, -4)

+ - 0

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Slope-intercept Form

of a Linear Equation

1. Slope-intercept equation

2. Graphing by slope-intercept

3. Writing slope-intercept equations

Slope-intercept Form:

y = mx + b

where m = slope

and b = y-intercept.

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Example:

y = -1/2x + 4

-1/2 = m = slope

4 = b = y-intercept

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Graphing by Slope-intercept

1. Graphing lines with

positive/negative slopes.

2. Graphing lines with zero or

undefined/no slopes.

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The Slope-intercept SongYou make the last number first.

It’s either up or down.

Make the slope in 2 numbers,

Or you look like a clown.

Top one’s up or down,

And the bottom’s always right.

You’d better do it well,

Or you’ll get a fright. (Tune: “Hokey-Pokey”)

LessonStart

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

Graph y = -1/2x + 4

1. Last number is 4, so go up 4 on the y-axis from the origin and plot a point.

1

2

3

2. Slope is already 2 numbers. Top one is -1, so go down 1 from the point you just plotted.

3. The bottom number is 2, so go 2 units to the right and plot a point.

4. Draw a line through the 2 points you plotted.

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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

Graph y = 2x - 3

1. Last number is -3, so go down 3 units from the origin and plot a point.

1

2

2. The slope is only 1 number so put a 1 under the 2.

3. Go up 2 from the point you just plotted.

4. Go 1 unit to the right and plot a point.

5. Draw a line through the 2 points you plotted.

1

LessonStart

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

Click on the graph for y = 2/3x - 2

LessonStart

The slope is not negative.


LessonStart

You graphed the last number on the x-axis instead of the y-axis.


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Top number is 2, and the bottom is 3,

so you do not go up 3 and over 2.


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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

Click on the graph for y = -4x + 3

LessonStart

The slope is not positive.


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The -4 is not the y-intercept,

nor is the 3 the x-intercept.


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The -4 is the slope, not the x-intercept.


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Two Special Graphs:Line with a zero slope

And

Line with an undefined slope.Lesson

Start

Line with a zero slope:

y = # (no x)

graphs as a horizontal line.

“Why, o y, do I look upon the horizon?”

LessonStart

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

Graph the equation y = 2.

LessonStart

Line with an undefined/no slope:

x = # (no y)

graphs as a vertical line.

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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

Graph the equation x = -4.

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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

Click on the graph for x = 3.

LessonStart

You chose the graph for x = -3.


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The x = # (no y) line does not graph

as a horizontal line.


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-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

Click on the graph for y = -3½.

LessonStart

The y = # (no x) line does not graph

as a vertical line.


LessonStart

You chose the graph for y = 3½.


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Writing Slope-intercept Equations:

1. When given a slope and the

y-intercept.

2. When given a slope and one point

on the line.

3. When given 2 points on the line.

m = ¾, b = -1

m = -¼, (8, 3)

(3, 7), (5, 12)

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Writing a slope-intercept equation when given a slope and the y-

intercept.

Substitute the slope and the y-intercept

for the m and b in the equation.

Example: m = ¾, b = -1

y = mx + b Slope-int. equation

y = ¾x - 1 The new equation

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1. Click on the correct equation for a line with

slope = 5/3 and y-intercept = 2.y = 5/3x + 2y = 2x + 5/35/3y = 2x y = -5/3x + 2

2. Click on the correct slope and y-intercept

pair for y = 7x - 5.

m = 7, b = -5 m = -5, b = 7 m = -7, b = 5 m = 1/7, b = -5

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Writing a slope-intercept equation when given a slope and a point on the

line.

1. Substitute the x, y, and m in the

slope-intercept form.

2. Solve for b.

3. Substitute the slope and the b in a

clean slope-intercept form.LessonStart

Example: Write the equation of the line with

slope = -¼ and point (8, 3). y = mx + b

3 = -¼(8) + b

1. Substitute the slope, x, and y in the equation.

3 = -2 + b

+2 +2

5 = b

2. Solve for b.

y = -¼x + 5

3. Substitute the slope and b in the equation.

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1. Click on the correct substitution for a line

with slope = 1/3 and point (5, 9).

9 = 1/3(5) + b 5 = 1/3(9) + b9 = 1/3x + 59y = 5x + 1/3

2. Click on the correct equation for a line

with slope = -2/3 and point (-6, 4).

y = -2/3xY = -2/3x + 4y = -2/3y = -2/3x - 6

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Writing a slope-intercept equation for a line with 2 points given:

1. Find the slope of the line.

2. Use that slope and the first point to

find the y-intercept.

3. Substitute the slope and the

y-intercept into the equation.

LessonStart

Example: Write and equation for a

line with points (3, 7) & (5, 12).

1. Find the slope of the line.

m = (y1 - y2)

(x1 - x2)

m = 7 - 12 = -5 = 5

3 - 5 -2 2

Continued on next screen.LessonStart

Write and equation for a line with

points (3, 7) & (5, 12).

m = 5/2 y = mx + b

2. Use the slope and

the first point to

solve for the

y-intercept.

7 = (5/2)(3) + b

2(7) = 2(15/2) + 2b

14 = 15 + 2b

-15 -15

-1 = 2b -1/2 = b

2 2Continued on next screen.Lesson

Start

Write and equation for a line with

points (3, 7) & (5, 12).

m = 5/2, b = -1/2

3. Substitute the slope and the y-intercept for the m and the b in the equation.

y = mx + b

y = 5/2x - 1/2

LessonStart

1. Click on the slope for a line with points

(-2, 8) and (7, -5).13-9

3-9

-913

139

2. Click on the y-intercept for a line with

points (-2, 8) and (7, -5).469

869

-58613

LessonStart

3. Click on the correct equation for a line

with points (3, 7) and (4, 8).

y = x + 4

y = 3x + 7

y = -x + 4

y = 3/4x + 8

LessonStart




Graphing by x- and y-intercepts.

X-intercept: where the line crosses

the x-axis.

Y-intercept: where the line crosses

the y-axis.

x

y

x-intercept

y-intercept

How to graph by x- & y-intercepts:

1. Cover the x term with your index finger and solve the resulting equation for y.

2. Go up or down on the y-axis from the origin that many units and plot a point.

3. Cover the y term with your index finger and solve the resulting equation for x.

4. Go left or right on the x-axis from the origin that many units and plot a point.

5. Draw a line through your points.

LessonStart

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

Graph the line for 3x + 2y = 6.

1. Cover the x term and solve for y.

3x + 2y = 6.

y = 3

2. Go up 3 units on the y-axis.

1

2

3. Cover the y term and solve for x.

3x + 2y = 6.

x = 2

4. Go right 2 units on the x-axis.

1

5. Draw a line through the points plotted.

LessonStart

1. Click on the correct intercepts for

3x - 4y = 24.

x-int: 8y-int: -6

x-int: -6y-int: 8

x-int: 8y-int: 6

x-int: 6y-int: 8

LessonStart

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

2. Click on the graph of 3x - 6y = 12.

LessonStart

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

5

4

3

2

1

-1

-2

-3

-4

x

y

3. Click on the correct equation for the line shown.

-6x - 9y= -36

-6x - 9y= -36

-9y - 6x= -36

-9y - 6x= -36

4x + 6y = 36

4x + 6y = 36

6y + 4x= 36

6y + 4x= 36

LessonStart




Graphing Linear Inequalities

Type of

Line

Where to

Shade

Solving Inequalities

How to Determine

the Type of Line to Draw

Inequality Symbol

Type of Line

> or < Dotted Line

> or < Solid Line

Choose the type of line for the inequality

given.1. y > 3x - 2

a. Solid b. Dotted

2. y > ¼x - 5

a. Solid b. Dotted

LessonStart

Choose the inequality symbol for the line shown.

< or >

< or >

LessonStart

Choose the inequality symbol for the line shown.

< or >

< or >

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For Positive, Negative, & Zero Slopes

For Undefined

or No Slopes

Where to Shade for Positive, Negative, and Zero Slopes:

The inequality must be in

y mx + b

format.

can be:

>, >, <, or <.

LessonStart

If the inequality is:

Shade

y > mx + bor

y > mx + bAbove the

line

y < mx + bor

y < mx + bBelow the

lineLessonStart

x

y

Graph y > x - 2.1. Graph the line

y = x - 2.

2. Since y >, shade above the line.

LessonStart

x

y

Graph y < x - 2.1. Graph the line

y = x - 2.

2. Since y <, shade below the line.

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Do you do anything

different when the line is

dotted rather than solid?

LessonStart

LessonStart

x

y

Graph y > x - 2.

2. Since y >, shade above the line.

1. Graph the line

y = x - 2, but make the line dotted.

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x

y

Graph y < x - 2.1. Graph the line

y = x - 2, but make the line dotted.

2. Since y <, shade below the line.

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x

y

Graph y > -½x + 3Type of

line:

Solid

Dotted

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x

y


line:

Solid

Dotted

Shade ___ the line.

Above Below

LessonStart

x

y


line:

Solid

Dotted

Shade ___ the line.

Above Below

LessonStart

x

y

Choose the correct inequality for the graph shown.

y < 1/3 x + 2

y < 1/3 x + 2

y > 1/3 x + 2

y > 1/3 x + 2

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Where to Shade for Undefined or No Slopes:

The inequality must be in

x # (no y)

format.

can be:

>, >, <, or <.

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If the inequality is:

ShadeTo the

x > #or

x > #Right of the

line

x < #or

x < #Left of the line

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x

y

Graph x > -21. Draw a dotted vertical line at x = -2.

2. Shade to the right of the line.

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x

y

Graph x < -2.1. Graph the line

X = -2.

2. Shade to the left of the line.

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x

y

Graph x > 3.Choose type of

line.

Solid

Dotted

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x

y


line.

Solid

Choose where to shade.

Left Right

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x

y


line.

Solid

Choose where to shade.

Right

LessonStart

Solving Inequalities

You use the same algebraic methods as solving equations except when you multiply/divide both sides by the same negative number.

In that case, you switch the direction of the inequality symbol.

LessonStart

Solve -3x - 2y < 12.

-3x - 2y < 12+3x +3x

-2y < 3x + 12-2 -2 -2 y < -3/2 x - 6>

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Choose the correct inequality.

1. 2x + 5y > -10

y > -2/5 x - 2y < -2/5 x - 2

y > 2/5 x + 2 y < 2/5 x + 2

2. 3x - 2y > 10y > -2/3 x - 5 y < -2/3 x - 5

y > 2/3 x - 5y < 2/3 x - 5

LessonStart

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