Investigation 2: Linear Inequalities in Two Variables

Yoga is a very old science, and a disciplined way of life.

When practicing yoga, on the mat or in every day life, one

brings together mind, body, and spirit.

While yoga students can leave their worries at the door,

studio owners like Autumn are on their toes.

Growing their businesses, and dealing with finances.

Autumn just opened a brand new yoga studio.

And she's trying to figure out the minimum number of students

required each day to break even the first month.

She's offering two options.

$12 per class if you buy a pass of ten classes, and $18

dollars per class if you just drop in.

Open seven days a week, the total cost of running the

studio works out to an average of $360 per day.

So here's the question.

How many students does she need each

day to make a profit?

When income exceeds expenses, you have a profit.

You know the daily expenses.

Now you need to figure out an algebraic expression for the

daily income.

Since daily attendance varies, let x be the average number of

pass holders on a given day.

And y the average number of drop-ins.

The income expression is $12 for every x, and

$18 for every y.

Or 3 ~ 8 plus 18y.

To make a profit, 3 ~ 8 plus 18y must be greater than 360.

This is a linear inequality in two variables, x and y.

Once again, it's linear, because the degree of the

polynomial is one.

Namely, its highest power is one.

To solve such linear inequalities we use the

algebraic properties of inequality and the geometry of

linear function graphs.

Let's begin by converting this inequality from standard form

to slope-intercept form.

Subtracting 3 ~ 8 from both sides, using the subtraction

property, gives 18y greater than 360 minus 3 ~ 8.

Dividing both sides by 18, using the division property,

leaves the direction of the inequality unchanged because

18 is positive.

You obtain y greater than 20 minus 2/3x.

Or y greater than negative 2/3x plus 20.

Let's use what we know about linear equations to understand

what the meaning of this inequality.

The related equation, y equals negative 2x plus 20, is a

linear relationship between two variables.

So its graph is a line in a two dimensional plane.

We call y f of x, or a function of x, since y changes

as x changes.

Let's begin by graphing this line, using the TI Nspire.

Turn on the TI Nspire.

Press the Home key to open a new document.

You may be prompted to save an open document.

After you decide, select two to create a

graph in geometry page.

The blinking cursor is on the function entry

line, by f 1 of x.

Type in negative 2/3x plus 20.

For a fraction placeholder, press Control and

the Division key.

Enter negative 2 in the numerator.

Press the down arrow, and enter 3.

Then press the right arrow.

Press Enter to graph.

Press Escape to move the cursor from the entry line to

the work area.

To change the window, press Menu.

And under Windows, select Windows Setting.

Enter the following values, using tab to move from one

entry to another.

Negative 3, 35, negative 2, 25, then click OK.

Grab and drag the equation to the top of the screen.

To do so, hover over the equation with the pointer.

When it becomes an open pen, click and hold until the open

pen becomes a closed pen.

Use the mat pad to drag an equation upwards.

Press escape to exit grab and drag mode.

Now let's place a movable point on the line.

Press Menu, and other points and lines, select Point On.

Move the pointer to any point on the line.

When you see the words Point On, press enter

to label the quarter.

Press Escape to exit Point On mode.

Notice the blinking point and open pen.

Click and hold to grab the point.

Now use the mat pad to slide the point and you travel up

and down the line.

Notice how the coordinate value, or x and y value,

change accordingly.

This dynamic technology is truly amazing.

You just saw many xy pairs that lie on the line.

For those xy values, 3 ~ 8 plus 18y equals 360, or

equivalently negative 2/3x plus 20.

These are the break even points where

expenses equal income.

For all other xy pairs y is either greater than or less

than negative 2/3x plus 20, representing either

a profit or a loss.

Therefore the lines divides the xy plane into three

distinct regions.

The region of all points above the line, on the line, and

below the line.

Regions one and three are called half planes.

This is true for all straight lines with equation y equals

mx plus b or y equals ax plus b.

I prefer the latter notation.

So, let's generalize.

Let y equal ax plus b for any linear equation in

slope-intercept form.

Its straight line graph divides the

plane into three regions.

Y greater than ax plus b, y equals ax plus b, and y less

than ax plus b.

Let's pursue our investigation to see which of the two half

planes is the solution to our yoga problem.

Suppose you had tried to guess some solutions.

Trial and error is a good exploratory approach.

Access the Tools menu.

Under Page Layout, select Custom Split.

Press the right arrow a few times until you have about a

2/3 1/3 ratio in screen split.

Then click.

Now press Control-tab to toggle to the right.

Press Menu, and select Lists and Spreadsheet.

Move up to the top of column one, beside letter A. Type p

for path holder, then press the down arrow twice.

Enter the following values, pressing Enter or the down

arrow after each one

4, 9, 15, 25.

In the same way, use the mat pad to move to the right of

letter B in column two.

Type d, for drop-in, then press the down arrow twice.

Enter the following values

20, 13, 16, and 5.

Next, toggle back to the graph by pressing Control-Tab.

Press Menu, and under Graph Type, select Scatter Plot.

The entry line is ready to specify the x and

y-coordinates of the points.

Select p, click, and select d.

The scatter plot is displayed.

Of your four guesses, three points are above the

line, and one below.

To see their coordinates, press Menu.

Then select Point On under Points and Lines, as before.

Hover over each point one by one, and click to display the

coordinates.

I trust you're suspecting that the upper half plane is our

solution to making a profit.

To confirm, let's do one last thing.

Press Menu, and under Graph type, Select Function.

In the Function Entry line we can type in an

inequality as follows.

Press the backspace key to erase the equals sign.

Then press the greater than symbol, then

negative 2/3x plus 20.

Press Enter and watch the top half plane become shaded.

This shaded region is the infinite set of all ordered

pairs that are theoretical solutions to our inequality.

We used a lot of different TI Nspire features, so you may

want to go back and review them.

Above the line, you have a profit.

On the line, you break even.

And below the line, there's a loss.

Even though the theoretical solution set is infinite, the

realistic solutions lie closer to the line.

And lie in quadrants one, since negative numbers are

nonsensical in this context.

(4,20), (15,16) and (25,5) are some reasonable solutions.

The first, for instance, means that an

average of 24 students

4 pass holders, and 20 drop-ins

would yield a profit.

The third means that 30 students

25 and 5

would also work.

In the final exercise, let's graph this system of two

inequalities in two variables.

A system simply means we must solve the inequalities

simultaneously.

We must find all points in the plane that satisfy both

inequalities.

Back to our TI Nspire.

The first inequality is already graphed.

And the cursor is on the entry line, ready for the second

inequality.

To graph y less than or equal to x plus 5, first press the

backspace to delete the equal sign.

Add less than or equal.

Then type in x plus 5.

Press Enter and you have a second half plane shaded.

So your solution set is the intersection of the two half

plane solutions.

So solving a system of two inequalities by graphing will

yield the intersection of two half planes.

Depending on the solutions of each one, and on their

relative positions, the solution sets look different.

You can have this intersection, or this one, or

even this one, which would give the empty set.

It's easy to extend the concept to more than two

inequalities and visualize possible solution sets.

You're in for such a challenge at the end of this lesson.

Inequalities are the fundamental building blocks of

linear programming, a mathematical model developed

in 1947 by American mathematician George Dantzig.

It has endless applications.

In business and economics, for example, it's used to either

maximize income or minimize costs for

certain production schemes.

Beyond that, just about every industry, including clothing,

airline, forestry, communication, and

advertisement uses linear programming to solve large

scale problems.

I hope you'll practice with other problems, and become

proficient in solving inequalities.

They'll come in handy not only in future math courses but in

future life situations, as well.

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