12 - Solving & Graphing Inequalities w/ One Variable in Algebra, Part 1

Summary

This algebra lesson focuses on solving one-variable inequalities, starting with simple problems and progressing to more complex ones. The instructor explains that solving inequalities follows the same rules as equations but with a crucial difference: when multiplying or dividing by a negative number, the direction of the inequality sign must be reversed. The lesson covers various inequality types, demonstrates solving steps, and emphasizes graphing solutions on a number line with open and closed circles to represent inclusive or exclusive endpoints. The goal is to find a range of values that satisfy the inequality, verifying solutions by substituting back into the original inequality.

Takeaways

• 📚 Start by understanding the basics of solving inequalities with one variable, progressing from simple to complex problems for thorough practice.
• 🔍 The rules for solving inequalities are similar to solving equations, involving moving terms to one side and isolating the variable.
• 📉 Inequalities represent a range of values rather than a single solution, which is a key difference from equations.
• 📌 When graphing inequalities, use open circles for 'greater than' or 'less than' and closed circles for 'greater than or equal to' or 'less than or equal to' to indicate inclusion or exclusion of the boundary value.
• 📈 To graph an inequality, shade the appropriate side of the number line based on whether the inequality is 'greater than' or 'less than'.
• ⚠️ A crucial rule when solving inequalities is to flip the inequality sign when multiplying or dividing by a negative number.
• 🔢 Practice checking solutions by substituting values back into the inequality to ensure they satisfy the original condition.
• 🔄 Inequalities can involve various variables, but the process of solving them remains consistent regardless of the variable used.
• 📝 When solving compound inequalities, collect like terms on one side and constants on the other, then isolate the variable.
• 📉 After isolating the variable, the solution to an inequality is a range of values that can be represented on a number line.
• 🚀 These foundational skills in solving one-variable inequalities are essential for tackling more complex inequalities with multiple variables in the future.

Q & A

What is the main focus of the algebra lesson in the provided transcript?

The main focus of the lesson is solving inequalities that involve one variable, starting with simple problems and gradually increasing in complexity.

How are inequalities different from equations in terms of solutions?

Inequalities have a range of solutions, whereas equations typically have a single solution. Inequalities use less than, greater than, less than or equal to, or greater than or equal to signs instead of an equal sign.

What is the first step in solving an inequality like 'X - 7 > -5'?

The first step is to isolate the variable, which involves adding 7 to both sides of the inequality to get 'X > 2'.

How do you represent the solution of the inequality 'X > 3' on a number line?

You place an open circle at 3 on the number line and shade everything to the right, indicating that X can be any number greater than 3 but not including 3 itself.

What is the rule for dividing or multiplying both sides of an inequality by a negative number?

When dividing or multiplying both sides of an inequality by a negative number, you must flip the direction of the inequality sign.

What does the inequality 'X ≥ 2' mean in terms of the range of values for X?

'X ≥ 2' means that X can be any number greater than or equal to 2, including 2 itself.

How do you graph the solution for 'X ≤ -2' on a number line?

You place a closed circle at -2 on the number line and shade everything to the left, indicating that X can be any number less than or equal to -2.

What is the difference between using an open circle and a closed circle when graphing inequalities?

An open circle indicates that the number at that point on the number line is not included in the solution, while a closed circle indicates that the number is included.

In the transcript, what is an example of an inequality that involves dividing by a negative number?

An example given is '-5x < 10', which after dividing both sides by -5, becomes 'x > -2', and the direction of the inequality sign is flipped.

How do you check if a particular number is a solution to an inequality?

You substitute the number into the original inequality and check if the resulting statement is true, indicating that the number is part of the solution set.

What is the final step in solving the inequality '3x - 1 ≥ -4' as per the transcript?

The final step is to divide both sides by 3, which gives 'x ≥ -1' after flipping the inequality sign due to division by a positive number.

Outlines

00:00

📘 Introduction to Solving Inequalities

This paragraph introduces the concept of solving inequalities with one variable. The instructor emphasizes the similarity between solving equations and inequalities, highlighting that the process involves isolating the variable on one side. The key difference is the use of inequality signs (less than, greater than, etc.) instead of an equal sign. The instructor also hints at a crucial point to remember when solving inequalities, which will be explained with an example later. The paragraph sets the stage for a gradual increase in complexity, starting with simple problems to build up to more complicated ones.

05:01

📐 Understanding Inequality Symbols and Graphing

The paragraph delves into the meaning behind different inequality symbols, such as 'greater than' and 'less than,' using the variable X as an example. It explains how these symbols represent a range of values rather than a single solution. The instructor provides examples of inequalities like X > 3 and X ≥ 2, illustrating how to graph them on a number line with open and closed circles to denote inclusion or exclusion of the boundary values. The paragraph also covers how to graph inequalities like X < 1 and X ≤ -2, reinforcing the importance of understanding the difference between open and closed circles in graphing and the implications for the range of solutions.

10:05

🔢 Solving Simple Inequalities Step by Step

This section walks through the process of solving simple inequalities like X - 7 > -5 and 2T > 6. The instructor demonstrates how to treat inequalities like equations by isolating the variable, emphasizing the need to reverse the inequality sign when multiplying or dividing by a negative number. Examples are provided to show how to graph the solutions on a number line, using open circles to indicate values that are not included in the solution set. The paragraph reinforces the concept that inequalities represent a range of values that satisfy the inequality, contrasting this with equations that have a single solution.

15:06

📉 Advanced Inequality Problem Solving

The paragraph tackles more complex inequality problems, such as -5x < 10 and (-T/2) > 3.5, showing how to manipulate the inequality to isolate the variable. It reiterates the rule of reversing the inequality sign when dividing or multiplying by a negative number. The solutions are then graphed on a number line, with the instructor explaining the significance of open and closed circles in representing the solution set. The paragraph also covers scenarios where the inequality involves both multiplication and addition or subtraction, like 3x - 1 ≥ -4, and demonstrates how to solve and graph these inequalities.

20:07

🚀 Wrapping Up Inequality Solutions

The final paragraph summarizes the process of solving one-variable inequalities and sets the stage for future lessons on multi-variable inequalities. It reiterates the importance of understanding the range of values that satisfy an inequality and the methods used to solve them. The instructor also emphasizes the foundational nature of these skills for tackling more complex mathematical problems. The paragraph concludes with a preview of upcoming lessons that will increase in complexity while building on the techniques learned in this session.

Keywords

💡Inequality

An inequality is a mathematical expression that shows a relationship of not being equal between two values or expressions, using symbols such as 'greater than', 'less than', 'greater than or equal to', or 'less than or equal to'. In the context of the video, inequalities are the main focus, where the host teaches how to solve and graph them, starting with simple cases like 'X greater than 3' and moving to more complex scenarios.

💡Variable

A variable is a symbol, often a letter, that represents an unknown value in mathematics. In the video, variables like 'X', 'T', and 'Y' are used to represent these unknowns in inequalities. The process of solving inequalities involves isolating the variable to determine the range of values that satisfy the inequality.

💡Number Line

The number line is a visual representation of numbers laid out in a straight line, with an origin, direction, and unit length. In the script, the number line is used to graph the solutions of inequalities, showing which numbers are included or excluded based on the inequality's direction and type.

💡Graphing

Graphing in the context of the video refers to the visual representation of the solution set of an inequality on a number line. For example, when an inequality states 'X is greater than 3', graphing involves placing an open circle at 3 on the number line and shading all numbers to the right, indicating all values greater than 3 are part of the solution.

💡Solving Equations

Solving equations is the process of finding the values that make the equation true. The video mentions this concept as a foundation for solving inequalities, emphasizing that the process is similar except for the handling of the inequality sign and the range of solutions.

💡Open Circle

An open circle on a number line represents a point that is not included in the solution set. In the script, it is used to indicate that a particular value, while significant, is not part of the solution for an inequality, such as in 'X greater than 3' where 3 itself is not a solution.

💡Closed Circle

A closed circle on a number line signifies that the point is included in the solution set. For instance, in the inequality 'X is greater than or equal to 2', a closed circle at 2 indicates that 2 is a valid solution, along with all numbers greater than 2.

💡Shading

Shading on a number line is used to indicate the range of numbers that are part of the solution to an inequality. In the video, shading to the right of an open or closed circle signifies all numbers greater than the circle's value, while shading to the left indicates all numbers less than the value.

💡Dividing by a Negative Number

When solving inequalities, dividing or multiplying both sides by a negative number requires flipping the direction of the inequality sign. The video script explains this rule, emphasizing its importance to avoid incorrect solutions, using examples like dividing an inequality by -5 and flipping 'less than' to 'greater than'.

💡Isolating the Variable

Isolating the variable is the process of getting the variable alone on one side of the inequality through mathematical operations like addition, subtraction, multiplication, or division. The video demonstrates this by moving terms around and using inverse operations to simplify the inequality until the variable is isolated, allowing for the determination of the solution set.

Highlights

Introduction to solving inequalities with one variable, starting with simple problems and gradually increasing in complexity.

Explanation of the similarity between solving equations and inequalities, with the main difference being the handling of inequality signs.

Illustration of how to interpret and graph inequalities such as 'X > 3', emphasizing the range of values X can take.

Demonstration of using an open circle on a number line to represent values not included in the solution set, such as in 'X > 3'.

Clarification on the difference between open and closed circles when graphing inequalities, with closed circles including the number itself.

Example of solving the inequality 'X - 7 > -5' by treating it like an equation and then adjusting for the inequality sign.

Explanation of the importance of understanding the meaning of solutions in inequalities, as they represent a range of values rather than a single solution.

Process of solving the inequality '2t > 6' by dividing both sides by 2, and the subsequent graphing of the solution.

Rule for flipping the direction of inequality signs when multiplying or dividing by a negative number, a critical step in solving inequalities.

Application of the rule in solving '−5x < 10', showing the transformation of the inequality and the resulting graph.

Approach to solving inequalities with fractions, such as '−T/2 > 3/2', including multiplying by a form of 1 to eliminate fractions.

Solving the compound inequality '3x - 1 ≥ -4' by isolating the variable and adjusting for the 'greater than or equal to' sign.

Verification of solutions by plugging values back into the original inequality to ensure they satisfy the condition.

Solution to the inequality 'y ≤ 7y - 24' by collecting like terms and dividing by a negative number, flipping the inequality sign.

Final example of solving '2x^2 - x < 4 + x', demonstrating the steps to isolate the variable and graph the solution.

Emphasis on the importance of practicing solving inequalities and understanding the range of values they represent.

Anticipation of future lessons involving more complex inequalities with multiple variables, building on the skills learned.

Transcripts

hello welcome back to algebra here in

this lesson we're going to start solving

inequalities that involve one variable

this is the first of a few lessons where

we'll start with very easy problems and

then we'll gradually increase to more

and more complicated problems so that

you have lots and lots of practice

alright so the main thing is up until

now we've solved equations we've done a

great review of getting ourselves

familiar with how to solve equations

with multiple steps we move the things

over to one side of the equal sign we

get the variable to one side and we

divide or multiply to get that variable

by itself on one side of the equal sign

and then we know what it's equal to on

the other now we're going to be dealing

with inequalities so we're going to be

replacing the equal sign with a less

than or a greater than or a greater than

equal to or a less than equal to the

rules of solving these inequalities are

exactly the same as solving equations we

move things over we get the variable to

one side of the inequality we divide or

multiply and that's what we're trying to

do there's one main difference one thing

that you have to keep in mind as we

solve inequalities and I'm gonna wait to

tell you what that is until we can get

to an example so I can show you rather

than just tell you so let's start with

really easy inequality just to show you

what I mean first we're going to recall

what an inequality is so we're going to

talk about for instance the inequality X

greater than 3 what does this mean this

means that the variable X is not equal

to a number it's not just 5 or 6 X is a

whole range of numbers it can be really

infinity numbers right because what

we're saying is since it's greater than

3 we're saying that X can be 4 or 5 or 6

or 7 or 8 but also the numbers in

between like X can be 4 point 5 X can be

3 point 0 1 2 as long as it's bigger

than 3 notice there's no equal sign if

there's an equal sign under the

inequality then you would have to

include 3 any number bigger than 3 3.01

5.7 9 and so on that is what X can be

equal it's a whole range of values so in

order to graph that X is greater than 3

what we do is we go to number 3 on the

number line up here and we put an open

circle the open circle means

that you know obviously 3 is the number

we care about but we're not including

the number 3 in the solution and what we

do is we shade everything to the right

of this on the number line so if you can

see that what it means is everything

bigger than 3 4 5 6 on to infinity and

all of the numbers in between like

here's 3.5 here's 4.5 and so on the open

circle means we are not counting the

number 3 in our solution all right let's

take a look at another really simple one

let's say we have X is greater than or

equal to 2 so this is the exact same

thing except we're also including the

number 2 in the range of X because it's

greater than 2 or it's equal to 2 as

well so we represent that by finding the

number 2 in the number line and we put a

solid dot in place instead of an open

circle and then we shade everything to

the right you could put a little arrow

at the end if you like showing that

you're going on and on to infinity so

this open circle means we're not

including the number 3 everything to the

right of it though those closed circle

means we're including everything to the

right of 2 and also including the number

2 so if we have the inequality X is less

than 1 for instance that would mean

everything smaller than 1 so it would be

0 negative 1 negative 2 and so on to the

left of 1 so we go to the graph to our

number line we find the number 1 which

is right here

it's less than but it's not equal to 1

so we put an open circle here and we

shade everything to the left all of

these numbers to all of these negative

numbers and also the numbers between 0 &

1 here but of course not including the

number 1 itself because it's an open

circle like this so open circle versus

closed circles very important there now

we have one more just to kind of wrap it

up and just kind of give you one of the

really basic examples what if we had X

is less than or equal to negative 2 less

than or equal to negative 2 so first you

find negative 2 in your graph you see

that it's less than or equal to so we

find negative 2 it's going to be a

closed circle because it's also equal to

negative 2 and we're saying the number

if the variable X can be less than or

equal to that so it means we shade to

the left so basically what we're gonna

do for all of these inequality problems

is I'm

this was just kind of a basic review of

what an inequality is and now what we're

gonna do is we're gonna start solving

inequalities where you have to move

things to the left until the right hand

side but the goal is at the end you want

to end up and get an equation you want

to end up with x equals 5 or x equals

negative 2 here you want to end up with

x is less than or equal to negative 2 or

you want to end up with X is less than 1

so you want to move everything to the

left everything to the right so you have

a variable by itself and then once you

get the answer you graph it so you

basically the answers to all of these

things are gonna be ranges of values so

that's the difference between an

inequality and an equation an equation

means they're equal there's one solution

generally for a simple linear equation

like this but for inequalities there's a

whole range of solutions so just to give

you an example of one of the very simple

ones let's say we had the problem X

minus 7 is greater than negative 5 so if

X minus 7 is greater than negative 5 so

what you do first of all is you just

pretend that this is an equal sign right

here you pretend it's an equation so if

this was X minus 7 equals negative 5

what would you do you want X by itself

so you have to get rid of the 7 how do

you do it because you the way you do it

is you do the opposite of the negative

you add 7 to both sides so what you

would get when you add 7 to the left is

just X will be by itself because 7 and

negative 7 will add to 0 on the right

you'll have negative 5 and you'll have

to add the 7 to it so what do you get on

the right hand side X is greater than

what do you do what you get when you add

these you subtract them and the sign

goes with this one you'll get a 2 so

this is not saying that X is equal to 2

this is saying that X is a range of

values bigger than 2 so you go up here

to represent the solution as a graph you

go and you find the 2 on the number line

you put an open circle because it's not

greater than or equal it's just greater

than 2 and then you shade every number

larger than this so what this means it's

very important in math to understand

what you're doing and what the solutions

mean what it means is just like for an

equation you had one solution you can

take that solution and stick it back in

the equation and show that that solution

is correct

that that one's that that makes the

equation equal right with the inequality

what we're saying is

any number bigger than two actually

makes this inequality work so you can

check it let's take the number 3 because

that's bigger than 2 right what we're

saying is any number bigger than 2

should work so let's put it in here what

is 3 minus 7

3 minus 7 if you think about it 3 minus

7 is negative 4

and my question to you is negative 4

greater than negative 5 now it might be

a little bit weird to say that but if

you should look at negative 4 and here's

a negative 5 negative 4 actually is

larger than negative 5 they're both

negative so you have to think about it a

little bit but negative 4 is actually

larger now if we pick a number even

bigger let's say let's pick 10 because

that's also bigger than - let's put 10

minus 7 what is 10 minus 7 10 minus 7 is

3 is 3 larger than negative 5 of course

3 is over here negative 5s over here so

that would satisfy as well so you see as

you keep plugging numbers larger larger

and larger n it's gonna get more and

more and more greater on this side so

it's always gonna work now what happens

when you put the number 0 in here

because that's not gonna work it's not

greater than - let's put 0 in 0 - 7 is

negative 7 is negative 7 greater than

negative 5 whoops we'll put a question

mine no it's not if you look over here

negative 7 will be over here that's

definitely not bigger than negative 5 so

you see when you get the inequality down

to the end you're getting a range of

values and and those values are what

would work when you put them back into

the inequality to begin with and so on

that's what the idea is so let's do one

more simple one before we move on to a

little more complicated ones what if we

had 2 times the variable T is greater

than 6 so you're basically just in your

mind envision or pretend that this is an

equal sign what would you do well I

would divide this I would have 2t less

than 6 I would divide the left by 2 and

if I do that I have to divide the right

by 2 and I do that so that this cancels

so I would get T on the Left 6 divided

by 2 is 3 so what I would do if I wanted

to graph this is I'd have to find the

number 3 which is over here I have to

put an open circle because it's not less

than or equal to it's just less than and

then all numbers to the left of 3

so we go to my graph here and I would

shape everything to the left so any of

these numbers to the left of three but

not including three would work so just

pick one let's take a 0 in 0 is less

than three right so two times zero is

zero that is less than six that works

what if you put in negative one that's

less than negative one times two that's

gonna give you negative 2 that's also

less than 6 and so on but if you go the

other way too far let's put 10 in 10 is

definitely not less than 3 so it

shouldn't work 10 times 2 is 20 that is

not less than 6 now let's look at the

special point what happens when you set

it equal to 3 2 times 3 is 6 right so

you say 6 & 6 but notice it's an

inequality what is on this side has to

be less than what's on the right but 6

is not less than 6 if 6 is equal to 6 so

the number 3 itself is not part of the

solution because when you put 3 in here

it doesn't work because it 6 is equal to

6 that's not less than that's why we

have an open circle here because the

number 3 doesn't work for the solution

so as we go through here we're gonna

solve a few more problems getting

practice and graphing every one of these

solutions on the number line all right

for our next problem let's say we have

negative 5x is less than 10 negative 5x

is less than 10 so you treat it like an

equation now what you need to do is you

need to divide both sides by what by

negative 5 to get it by itself so what

we'll have is just to make it 100% clear

let me rewrite this so I don't kind of

kind of mess up the first thing that I

wrote down here the problem statement

let me take this away for right now

we'll take that away from right now so

what we're gonna do then is we're going

to divide the left side by negative 5

and when we do that also we have to

divide the right side by negative 5 now

if you remember back at the beginning of

the lesson I told you that solving

inequalities was exactly the same as

solving equations I mean you use the

same rules except for one thing you have

to remember and this is that thing when

you divide both sides of this inequality

or if you multiply both sides of this

inequality by a negative number any

negative number then what you have to do

is this inequality sign you have to flip

directions

all right that's a that's a general rule

I could go into why you have to do that

but honestly it's not worth it's not

worth doing because ultimately there's a

there's a reason and it has to do with

the number line and what happens when

you divide by a negative number but the

bottom line is every time you solve an

inequality if you divide by negative

five you have to flip the direction of

the arrow if it's a less than or equal

to then you would flip it to greater

than or equal to if you divide by

negative two you're gonna flip the sign

of that arrow if you divide by negative

17 you'll flip the sign of that arrow if

you divide by negative 0.5 you're gonna

flip the sign of that arrow right now

here's the thing because division and

multiplication are related right the

same thing happens when you multiply by

a negative number so if I have to

multiply this by negative 10 I'm gonna

flip that arrow if I'm gonna multiply

this by negative 1/2 I'm gonna flip that

arrow so it's a very simple rule to

remember anytime you multiply or divide

an inequality by a negative number you

flip the sign of this arrow if you don't

do it you would get the wrong answer so

what do we have we have the negative 5

canceling with the negative 5 so on the

left you have X greater than now what's

10 divided by negative 5 that's negative

2 so this is the final answer X is

greater than negative 2 so the way you

graph is you go up and find negative 2

it's not greater than equal to it's just

greater than so you put an open circle

because we'd not including the number

negative 2 in our solution and we shade

everything to the right that's the graph

of this inequality all right what if we

had the inequality negative T over 2

greater than 3 halves there's a lot of

different ways to do this I can think of

2 ways right now but what we're gonna do

is we're gonna try to get rid of this

let's do it like this three halves we're

gonna do it like this we're gonna take

the left-hand side of this guy since

it's we're multiplying by negative 1 we

have a negative 1/2 here essentially

what we're gonna do is we're gonna

multiply by negative 2 over 1 right here

and when we do it to the left-hand side

we also have to multiply the right by a

negative 2 over 1 why are we multiplying

by negative 2 over 1 well first of all

we're multiplying by negative so we can

kill the negative sign we don't want any

negative signs on the left

and two over one will cancel with the

two because the two will cancel with the

two so the only thing you'll have left

when the negatives cancel is a T on the

left hand side but when we do this

multiplication we must flip the sign of

this inequality so we'll have less than

and what do we have here the two

cancer's with the two but now we have a

negative times three means negative

three this is the final answer it works

exactly the same this is exactly what

you would do if you had an equal sign

here to get this by itself but of course

with an equal sign you'd have to flip

the direction of anything here we had to

flip this direction so to graph it we go

find negative three and we put an open

circle because it's not equal to

negative three it's just less than

negative 3 and we shade everything to

the left all right

that's basically it let's do one more

since these are so small I think I can

fit it on this board what if we had 3x

minus 1 is greater than or equal to

negative 4 now the only difference

between this and the other equations

obviously there's a there's a negative

one term here is now we have greater

than or equal to so it doesn't change

how you do it if it's greater than or

equal to you just have to carry that

sign down throughout and of course if

you have to flip the direction because

if you divide by negative or multiply by

negative then you'll flip it to the

other direction with an equal sign under

it so what we do now is we say what do

we have to do first if this were an

equal sign we would get rid of the 1 so

we would add 1 so it'd mean 3 X greater

than or equal to we add one to the left

we add one to the right what is negative

4 plus 1 negative 4 plus 1 negative 3 on

the right make sure you understand add 1

add 1 alright and then what did we do to

get rid of the X well we get 2/3 so

it'll be X greater than or equal to

negative 3 on the right we'll have to

divide by that 3 we divide the left by

it by 3 killing it we divide the right

by 3 and what you get at the end of the

day is X greater than equal to what do

we have negative 1 here this is the

final answer greater X greater than or

equal to negative 1 so now to plot this

we find negative 1 and we put a solid

dot because it's greater than or equal

to negative 1 and then we shade every

to the right all of these numbers

including the number one negative one

will be correct if you stick them into

this value of x it will satisfy this

inequality in fact if you put negative

one in here because we're saying it's

greater than or equal to negative one if

you put negative one in here what will

you get three times negative one is

negative three negative three minus one

is negative four negative four is that

greater than or equal to negative four

yes because it's equal to which is

allowed in this inequality so the

answers that you get you should be able

to put them back in and verify that they

are correct all right what if we had the

inequality y greater I'm sorry less than

or equal to seven times y minus 24 now

first of all just like with equations

and inequalities you'll see all kinds of

variables running around sometimes

you'll see X sometimes you'll see why

sometimes you'll see T sometimes you'll

see a or B or W doesn't matter you treat

it all the same

it's exactly the same you're trying to

find out what values of Y work with this

inequality so what you do first of all

is you have to collect all the Y terms

on one side all the other stuff on the

other side and that's what you need so

in order to to get this done how do you

get this 7y over here well this is a

positive 7y so to move it over you have

to subtract it so on the left it would

be y minus 7y on the right hand side

it'll be zero here because you

subtracted 7y and you'll have the

negative 24 that's still there so we

subtract 7y from the left we subtract

seven from y from the right that makes

it zero and what do we have when we do

this subtraction one minus seven you

should know now is negative 6y and it's

gonna be negative twenty-four right like

this so what's the final answer to get Y

by itself what do we do we have to

divide by a negative remember divided by

negative means we have to flip that sign

so when we divide by the negative six

we'll have Y by itself this sign at this

point flips around notice we have an

inequality with an equal sign so it

flips the other direction and what is

going to be over here negative 24

divided by negative six so we divide by

negative six divided by negative six and

the final answer will be positive

because negative divided by negative is

positive six times four is 24 so it's y

greater than or equal to four now to

plot this guy

we go find the number four we put a

solid circle because it's greater than

or equal to and all of the values to the

right of that is what we have so four

five six seven and so on if you stick

them in here you will find that this

inequality is satisfied all right now we

have one more that we're going to do in

this lesson what if we have 2x whoops

not to x2 minus X that's what I was

trying to write less than four plus X so

it's the same sort of deal with

equations right you have some X's on the

left some X's on the right you got to

move the X's over here you got to move

the numbers over here so what do we do

we're gonna subtract X to move it over

here so what do we have to minus x from

here but we're gonna subtract X so we'll

have another minus X then we'll have a

less than sign then we'll have a 4 when

we subtract X from the right then this

becomes 0 because X minus x is 0 we

subtract X from the left we have written

it out here and then the next step we'll

say it's minus 2x less than 4 just like

this now we have to take the numbers the

number 2 and move it to the right by

what this is a positive 2 so we'll have

to subtract 2 like this what's not equal

to in this case we'll have 4 minus 2 on

the right so we subtract 2 from the left

it disappears subtract 2 from the right

and what we have is negative 2 x less

than 4 minus 2 is 2 right now what do we

do in order to get X by itself we divide

by negative 2 so you remember we divide

by negative that means we flip the sign

of this arrow and it'll be on the right

2 divided by negative 2 on the left we

divide by negative 2 it disappears on

the right we divide by negative 2 or we

get is X greater than negative 1 2

divided by 2 is 1 positive divided by

negative is negative so what we have is

X greater than negative 1 but not equal

to negative 1 so we put an open circle

here and then we shade everything to the

right open circle at negative 1 shade

everything to the right so just to pick

an example if you want notice we're

saying X is greater than negative 1 so

we're saying 0 should work right just as

this single example to put in here 2

minus 0 that gives us 2 4 plus 0 gives

you 4 to

less than four it works and so you can

pick numbers on the on the left and on

the right and just kind of make sure but

essentially this defines a range of

values that work when you use them and

plug them back into your initial

inequality so that's a really good

overview of solving inequalities and

what we call one variable right because

later on we'll have inequalities that

have two variables and we'll get to that

later on down the road but these skills

that you're learning here are absolutely

essential for you to understand how to

do more complicated types of problems so

follow me onto the next lesson we'll

continue solving inequalities and we'll

make the con the problems a slightly

more complicated along the way but

essentially we'll be doing the same

things as we're doing here with just a

few more steps when the problems get a

little bit more challenging