Inequalities and GraphingInequalities and Graphing

Chapter 7.4Pages 345-349

GraphGraph

-6 -4 -2 0 +2 +4 +6

2 7x

Graph’s SolutionGraph’s Solution

-6 -4 -2 0 +2 +4 +6

2 7x

What’s the Deal?What’s the Deal?

• In this lesson We will review: – graphing from y-intercept form and

standard form.– balancing inequality equations.

• We will find that inequalities have an infinite number of solutions.

• We will use dashed and solid lines for our equations.

Find the equation in y-intercept Find the equation in y-intercept form then graph.form then graph.

2x + 3y = 6

-2x -2x

3y= 6 - 2x

Then divide all terms by 3

3y = 6 - 2x

3 3 3

y= 2 – 2/3x

Rewrite: y = -2/3x +2

Graph.

Find the same equation in y-Find the same equation in y-intercept form with an intercept form with an

inequality.inequality.2x + 3y < 6

-2x -2x

3y< 6 - 2x

Then divide all terms by 3

3y < 6 - 2x

3 3 3

y< 2 – 2/3x

Rewrite: y < -2/3x +2

Graph with a dashed line (<).

This is similar to the open circle.

equation solutions.equation solutions.

Now is where the fun begins.

The x and y values of MANY points will make our equation correct.

Actually, an infinite number of points will make our inequality correct.

Do the origin values make the Do the origin values make the equation correct?equation correct?

We use 2x + 3y < 6

Next: fill in (0,0) for x and y into the original equation.

Is 2(0) + 3(0) < 6?

0+0 < 6

The origin is ONE coordinate The origin is ONE coordinate from the many that work.from the many that work.

If 2x + 3y < 6

Next: fill in another value on the same side as (0,0)

Let’s pick (-1, -2)

Is 2(-1) + 3(-2) < 6?

-2 + -6 < 6

Many coordinates work.Many coordinates work.So we can graph them.So we can graph them.

If 2x + 3y < 6

All these values will work.

We can show this by using shading or slanted lines.

The dotted lines show that any point on the line makes the equation equal. And NOT part of this equation.

NOT a solution.NOT a solution.

We found 2x + 3y < 6

Use (3,0)

2(3) + 3(0) < 6

6+0 is NOT < 6

Graph everything down and to the left of the dashed line instead of graphing every point.

-2x + 3y < -15+2x +2x 3y < -15 + 2xDivide all terms by

3 3y < -15 + 2x 3 3 3 y < -5 + 2/3xRewrite: y < +2/3x

- 5

Graph the inequality -2x + 3y Graph the inequality -2x + 3y < -15< -15

Now try the values of the origin in the equation.

-2x + 3y < -15-2(0) + 3(0) < -

15 Is 0+0 < -15?

Graph the inequality -2x + 3y Graph the inequality -2x + 3y < -15< -15

Since (0,0) does NOT work, graph the opposite side of where the origin is found.

-2(0) + 3(0) < -15 0+0 isn’t < -15?

Graph the inequality -2x + 3y Graph the inequality -2x + 3y < -15< -15

Graph with lines or shading.

Show 0 + 0 is not < -15Show 0 + 0 is not < -15

AssignmentAssignmentpg. 351: 17 - 31 odds pg. 351: 17 - 31 odds

(Stay Tuned for part 2)(Stay Tuned for part 2)

Graphing InequalitiesGraphing InequalitiesPart TwoPart Two

Warm-upWarm-up

• Graph 3x - y > -2

• Solution on the next slide

Warm-up equationWarm-up equation

Graph 3x - y > -2 -3x -3x -y > -2 – 3xDivide all terms by -

1 and switch the sign.

-y < -2 - 3x-1 -1 -1y < +1 + 3x ORy < +3x + 1

Warm-up answerWarm-up answer

Check 3x - y > -2with the origin

(0,0)3(0) – (0) > -20>2 so the origin

works. Shade right.

Let’s add another equationLet’s add another equation

GraphSlope is 0. (0x +

1)Shade up.

1y

What’s the solution?What’s the solution?

GraphWhere do the

graphed lines from both equations meet?

1y

Graph the Graph the inequalities inequalities

Subtract x from both sides.

z < -x + 3Add x to both sides.

z < +x + 3

3z x 3z x

Graph the Graph the inequalities inequalities

Graph as if z is y.

z < -1x + 3Add x to both sides.

z < +1x + 3Fill in the values of the origin

into both equations and shade.

0 + 0 < + 30 - 0 > + 3

3z x 3z x

Graph the Graph the inequalities inequalities

0 + 0 < + 3? yes0 - 0 > + 3? No

The solutions are found where the shading overlaps.

3z x 3z x

extrasextras

GraphingGraphing