Inequalities--Segments01.mp4

Algebra Nspirations

Inequalities: Solving Linear Inequalities in One and Two Variables

What do the following have in common?

Estimating your semester grade in a course before the final

exam, expressing a range of numerical values for target

heart rate when exercising, or pH levels in a healthy body,

and figuring out the break even point in an up and coming

small business venture.

One answer, and the topic of this program, is inequalities.

Hello, my name is Monica Neagoy.

And I will explore with you the multiple uses of algebraic

inequalities in describing and solving these and other

problem situations.

In this program, we'll investigate linear

inequalities in one variable, linear inequalities in two

variables, and systems of linear

inequalities in two variables.

Investigation 1 will lay the foundation.

The concept, properties, graphing solutions, interval

and set notations, and will highlight important

differences between equations and inequalities.

In Investigations 2 and 3, the TI Nspire will help us

visualize and better understand the meaning of

linear inequalities in two variables and learn how to

solve them graphically.

It may come as a surprise to you that inequalities are used

in mathematics almost as much as equations.

The only difference in appearance between an equation

and an inequality is that the equal sign is replaced by one

of four inequality signs, less than, greater than, less than

or equal, or greater than or equal.

But we'll see that there are some major differences.

Let's begin with an investigation I trust

you can relate to.

Suppose you set a goal for this school year and promised

yourself to score a B or higher in all your courses.

Imagine your four math grades earned thus far are 90, 78,

85, and 67.

You're trying to figure out what grade you need to score

on the final exam in order to average 85 or

better in the course.

The final is a two hour exam, so it's worth double.

Let the variable x denote the unknown grade.

Since the final exam carries twice the weight of a one hour

test, we use 2x when calculating the average.

Next, let's express the average of

all semester grades.

Even though there are five exams in all, we divide by 6

because the final exam counts twice.

You want this average to be 85 or better.

Mathematically this means greater than or equal to 85.

Combining these two expressions yields our first

inequality.

It has one variable, x, and is linear because

the degree is one.

Recall that in the case of one variable, the degree is the

variable's highest exponent.

x equals x to the first power.

Let's solve this inequality algebraically.

Multiplying both sides by six yields a denominator

of one on the left.

For the fraction, this equals the numerator.

And on the right, six times 85.

Working out the arithmetic gives 320 plus x, greater than

or equal to 510.

Subtracting 320 from both sides yields 2x greater than

or equal to 190.

Finally, dividing both sides by 2, we get x greater than or

equal to 95.

So you'd have to score 95 or better on your final exam to

get a minimum of 85 for the course.

That's the cutoff point for a B. I trust you can do it.

We'll use this example to examine three aspects of

inequalities.

The algebraic properties, the ways of representing

solutions, and the nature of these solutions.

Beginning with the algebraic properties, notice that we

multiplied both sides of the

inequality by the same quantity.

Subtracted from both sides the same quantity, and divided

both sides by the same quantity.

And we found our solution, x greater than or equal to 95.

We applied the same properties of equations, assuming they'd

work for inequality.

And they did.

Let's find out why.

For one, the addition and subtraction properties are the

same for equations and inequalities.

Namely, adding or subtracting the same number, call it c, to

both sides of an inequality yields an equivalent

inequality.

These properties also hold for less than or equal and greater

than or equal.

The direction of an inequality symbol also remains unchanged

if both sides are multiplied or divided by the same number

c, provided c is positive.

In our average problem, we first multiplied by 6 and

later divided by 2.

Since both numbers are positive, the inequality

symbol remained unchanged.

And now for the first fundamental difference between

equations and inequalities.

If we multiply or divide both sides of a true inequality by

a negative number, we must reverse the direction of the

inequality symbol to obtain a true equivalent inequality.

Again, the multiplication and division properties also hold

for less than or equal and greater than or equal.

To make sense of this, take the true

statement 2 less than 5.

Multiplying both sides by negative 2 gives negative 4 on

the left, and negative 10 on the right.

Negative 4, being closer to 0, is clearly greater than

negative 10.

So indeed, switching the original less than symbol to a

greater than symbol yields another true inequality.

Rules hold for all inequalities, whether

numerical or algebraic.

The second fundamental difference between linear

equations and inequalities in one variable is the nature of

their solution set.

Whether you begin with a simple equation, or a

complicated one, you'll always find a final solution set.

Barring absolute values, the solution is unique.

In contrast, the solution of an inequality is an infinite

set of numbers, denoted by expressions such as x greater

than or equal to 95 or x less than negative 2.

Let's use these examples to examine different ways of

representing inequalities.

x equals 95 is positive.

That's represented by this point.

All values greater than 95 lie to the right

of 95 on the x-axis.

Combining x equals 95 and x greater than 95 gives x

greater than or equal to 95.

That is, all points lying on the colored ray from 95 to

positive infinity.

Another way of graphing this inequality is to change the

dot for 95 to a square bracket, which also denotes

the inclusion of 95.

These two are graphical representations.

This is the corresponding interval location.

An interval closed on the left and open on the right.

Finally, the set notations look like this.

The set of all x values n r, or real numbers, such that x

is greater than or equal to 95.

Or simply the set of all x such that x is greater than or

equal to 95.

This simplified set notation assumes

that x is a real number.

In the case of a strict inequality, such as x less

than negative 2, the solid dot at negative 2 is replaced by

an open dot to denote that the boundary

point is not included.

Likewise, the bracket is replaced by a parenthesis,

which also denotes the exclusion of negative two.

This interval notation, therefore, denotes an open

interval at both ends.

Lastly, the set notation would look similar, except with a

different inequality sign.

So there you have it, for the two main differences.

Symbol reversal when multiplying or dividing by a

negative number, and infinite solution sets.

One last concept

compound inequalities.

We found x greater than or equal to 95 for

the final exam score.

In reality, you can't score more than 100.

So x must also be less than or equal to 100.

Meeting both conditions simultaneously gives us this

compound or double inequality.

You encounter double inequality in many contexts

that are bound by real world constraints.

Things For instance, normal human body temperatures, t,

are usually between 36 and 37.2 degrees Celsius.

Another example, a severe tornado's wind speed, w, is

somewhere between 136 and 165 miles per hour.

Or 218 and 266 kilometers per hour.

Before our next investigation, try this exploration.