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### Course: Pre-algebra > Unit 7

Lesson 7: Intro to inequalities with variables- Testing solutions to inequalities
- Testing solutions to inequalities (basic)
- Plotting inequalities
- Plotting an inequality example
- Graphing basic inequalities
- Inequality from graph
- Plotting inequalities
- Inequalities word problems
- Inequalities word problems
- Graphing inequalities review

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# Plotting an inequality example

Drawing a number line helps visualize 'x is less than 4'. We mark 4 with a circle, not a dot, because 4 isn't included. Then, we color the line below 4, showing all values less than 4. Easy peasy, lemon squeezy. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- Hello! Can anyone help me? How do you remember the difference between: ≤ ≥ < > and the open dot and the closed dot. Also, for example, if there's the problem: x ≤ 5 . Here's my questions (P.S.: This all applies to a number line): Which direction should the line go? Should the dot be opened or closed? Is there a way to remember it? Thanks to y'all for answering! Plz try and answer soon! Bye!(19 votes)
- Not sure if you know the signs or not, but one way of thinking about greater than and less than is to make signs with your thumb and pointing finger. If the sign looks like your left hand (<), left is less than. If the sign looks like your right hand (>), Tony the Tiger says right is Grrrrrrrrrrrreater. If you have a line underneath (≥ or ≤) you have to add the phrase or equal to, so ≥ is greater than or eqaul to and ≤ is less than or equal to.

As far as the open and closed circle, the best way is to understand what it really means. If you were just going to show x=3, you would put a closed dot on 3. So a closed dot means the point counts and you need the equal sign below the line (≥ or ≤). If you have an open circle, the point does not count, thus no equal line. If you have a positive variable on the left, the sign points toward the direction that you draw the line (so x< and x≤ both point toward the left, so start at your point (either open or closed), draw left and end with an arrow <------. If the sign points to the right (> or ≥), then start at point (either open or closed) and draw to the right --------->.

Does this help, or do you need more?(18 votes)

- when do you close the circle?1!(17 votes)
- graph x is less than 4.0 lets draw yourself a number line(8 votes)

- what does it mean when the dot is open on the 2 and the line is going both negative and positive ways?(14 votes)
- It means that All Real Numbers except 2 is the solution.(8 votes)

- For some reason, I cannot remember when to use an open circle, and when to use a solid dot on these number lines.(9 votes)
- I always remember: an open circle is around the number, so it doesn't actually touch the number, meaning it does not include the number itself. A filled in dot is really on the number itself, so that does include the number.(17 votes)

- Is 6/8 greater than 6/10(3 votes)
- Yes. To double check, you can convert
`6 / 8`

to`30 / 40`

and`6 / 10`

to`24 / 40`

.

Clearly,`6 / 8 = 30 / 40`

is greater than`6 / 10 = 24 / 40`

.(9 votes)

- How would you graph x=4?(4 votes)
- Vertical line going through 4 on x axis.(3 votes)

- How i can solve this:

16< |6-3x| < 19 ?(3 votes)- Split it into a compound inequality:

16< |6-3x| and |6-3x| < 19

Solve each individually, then find the intersection of the two results.(5 votes)

- I don't get the concept of the closed and open circle??(3 votes)
- Think of the open and closed circles as your hand. If your hand is open, you can't grasp anything or "contain" any object. This is kind of like 5 < x. The circle will be open because it does not contain the 5 because x is "greater than" 5. x has to be greater than 5, so 5 is not an answer to the inequality. Now if your hand is closed around an item you can contain it within your hand. For an inequality example, let's use 2 ≥ x. the circle is closed because 2 is contained and agrees with the inequality given. 2 is a solution because 2 is greater than or EQUAL TO x.(5 votes)

- Would it still be x < 4 if you did not put a circle on 4?(2 votes)
- The dot or circle is always used so there is no ambiguity as to where the inequality starts. An open dot tells you that the inequality is "<" or ">" with the arrow's direction telling you which applies. A solid dot tells you that the inequality is ">=" or "<=".(6 votes)

- solve problem 4 is greater than z less than 3(4 votes)
- 4>3-z

At first we subtract 3 from both sides

1>-z

Then we divide by -1. Remember that when we divide (or multiply by -1, the signs flip like from < to >)

1/-1>-z/-1

And we get ...

-1<z

If you (or anybody) needs help, please comment.(0 votes)

## Video transcript

Graph x is less than 4. So let's draw ourselves
a number line over here. So let me draw a number line. I'll start here at 0,
so 0, 1, 2, 3, 4, 5. And we could go below 0. We'd have negative 1, negative
2, negative 3, negative 4. I could keep going. Now, we want to graph all of
the x's that are less than 4, but we're not including 4. It's not less than
or equal to 4. It's just less than 4. And to show that we're not
going to include 4, what we're going to do is we're going
to draw a circle around 4. So this shows us that
we're not including 4. If we were including 4, I
would make that a solid dot. And to show that we're going to
do all the values less than 4, we want to shade
in the number line below 4, going down
from 4, just like that. And then we can just shade
in the arrow just like that. So this right here is all
of the values less than 4. And you could test it out. Take any value
where there's blue. So there's blue over
here, negative 2. Negative 2 is
definitely less than 4. If you take this value
right here, this 2, it's definitely less than 4. 4 is not included because
4 is not less than 4. It's equal to 4. 5 is not included because
5 is not less than 4.