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Find all values of 𝑥 that satisfy negative 18 is less than or equal to negative 𝑥 plus four, which is less than or equal to negative one.
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Write your answer as an interval.
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This is a compound linear inequality.
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It has more than one inequality symbol in the problem.
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This means that when we solve this, we’ll get a range of what 𝑥 can be.
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Solving a compound inequality is basically the same as solving a simple inequality, except that whatever we do to one side, we have to do to all three pieces of our equation.
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Let’s take a look.
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To get 𝑥 by itself, I’ll want to subtract four, but that means that I’ll need to subtract four from negative 18, negative one, and from the middle.
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Negative 18 minus four is negative 22.
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We don’t change our inequality symbol; we just bring it down.
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We haven’t changed our negative 𝑥, so we bring that down as well.
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Positive four minus four equals zero; they cancel each other out; we have nothing to bring down here.
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We bring down our second inequality symbol, and then we say negative one minus four equals negative five.
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Now it looks like 𝑥 is isolated, but our problem is asking to find all the values of 𝑥, and what we’re currently showing is the values of negative 𝑥.
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In order to solve this problem, we’ll need to multiply our entire compound inequality by negative one.
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Starting here, we’ll say negative one times negative 22 equals 22.
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Now here’s something very important to remember.
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When we’re dealing with inequalities and we’re multiplying or dividing by a negative, we have to flip the sign.
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We’ve just turned this less than symbol into a greater than symbol.
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We can then move on to multiply negative one times negative 𝑥, which would equal 𝑥.
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We have to distribute this negative one to our second inequality symbol and flip that sign as well.
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And finally we can distribute negative one to the negative five, which leaves us with five.
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If I want to read this inequality, I would read it like this: 22 is greater than or equal to 𝑥, which is greater than or equal to five.
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On a graph, it would look something like this: 𝑥 has to be greater than or equal to five but less than or equal to 22.
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That in interval notation looks like this: bracket five comma 22 bracket.