WEBVTT
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Let's talk about interval notation.
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Here's an example of an interval from 3 to 5, including the endpoints.
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The first thing I'd like to do is draw a graph on a number line.
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Here are my endpoints: 3 and 5.
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This interval includes all the real numbers x
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between 3 and 5 and also includes the endpoints.
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I'm going to shade all those numbers between 3 and 5.
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Because the endpoints are included, I want to use square brackets.
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Now let's think about the inequality.
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The real number x in this interval
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has to be less than 5 or equal to 5.
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It also has to be 3 or higher,
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so it's greater than or equal to 3.
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For the second example, we have the inequality given.
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I like to go ahead and look at the graph
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on the number line again here. There are my endpoints.
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This tells me that x is a number between negative 2 and 1.
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This time, I'm not going to include the endpoints.
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We shade the numbers, and we use parentheses
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to show that the endpoints are not included.
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Now I want to write the interval. When you look at the graph,
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you can sort of see the intervals staring right at you.
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It's all the numbers from negative 2 to 1, not including those endpoints.
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Notice again I used parentheses because the endpoints are not included.
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Our third example has the symbol infinity in it.
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The symbol is used to indicate unboundedness in the positive direction.
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Again, I want to look at the graph.
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I only have one number for an endpoint: negative 3.
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That infinity again tells me that
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I want an interval that's unbounded in the positive direction.
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Any number that's bigger than negative 3 should be included, so I want to shade all of this.
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Now what about negative 3? The parentheses tell me it's not included,
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so I'm going to use parentheses over here as well because it's not included.
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Now for the inequality, x is any real number
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that is bigger than negative 3, so x is greater than negative 3.
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Again, the endpoint is not included.
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Here again, we have x is less than 2.
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Let's look at the graph.
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Where on the number line will we find numbers that are less than 2?
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They are to the left.
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Anything smaller than 2 will be on this side.
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Again, it was not included, so I'm going to use parentheses.
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Now what would the interval look like?
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Since it's going to the left without bound,
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this is unbounded in the negative direction, which is negative infinity.
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It goes to the endpoint 2, which is not included.
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Lastly, this interval is actually all real numbers.
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It goes from negative infinity to positive infinity.
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If we look at the graph on the number line, that's everything.
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How would we write an inequality?
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We actually could just write in words: all real numbers x.
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The inequality would be negative infinity is less than x,
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which is less than infinity.
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Given an inequality, you should be able
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to write the interval or draw the graph.
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Given the graph, you should be able to come up with the inequality and the interval.
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Given the interval, you should be able to get the other two.
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This is an important skill that you need to practice until you understand it.
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Now let's look at solving inequalities.
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Here are two inequalities to solve.
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The first one is a linear inequality.
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The goal is to isolate the x
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on one side of the inequality symbol.
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My first step: subtract 6 from both sides,
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so 7 minus 6 will leave me with 1.
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Now I want to distribute the 5.
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Subtract 5 from both sides.
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1 minus 5 will give us negative 4.
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Here's where you want to be careful.
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You're going to divide both sides by a negative number,
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so remember to flip the inequality symbol: reverse it.
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That's going to leave me with x is
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greater than or equal to positive 4/5.
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This is in inequality form, and I want my answer to be in interval notation.
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This is all real numbers x that are greater than or equal to 4/5.
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4/5 is my endpoint.
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I want anything bigger than 4/5, so that's towards positive infinity.
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I want to include the 4/5, so I use a square bracket.
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Now our second example is actually a compound inequality.
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Notice it has two inequality symbols.
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The goal here is to isolate the x in the middle.
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Remember that whatever you do to the middle
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you also need to do on the left and on the right.
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My first step: multiply everything by positive 5.
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Negative 1 times 5 is negative 5.
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I'm going to have the 2x minus 4 in the middle.
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5 times 0 is of course still 0.
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Add 4 to all the parts of the inequality.
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That's going to give me negative 5 plus 4, which is negative 1.
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There's 2x. If I add 4 to 0, I get 4.
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Now I'm going to divide by positive 2.
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Inequality symbols stay the same.
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4 divided by 2 is 2.
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Again, I want my answer to be in interval notation.
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Notice that the solutions to the inequality
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are all the numbers x between negative 1/2 and 2, not including the endpoints.
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Use parentheses because the endpoints are not included, and there's your interval.
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Be sure to practice solving these inequalities
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and using interval notation until you're comfortable with the process.