WEBVTT
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Let's look at solving equations and inequalities involving absolute value.
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Our first example is an equation.
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So we have an expression inside the absolute value,
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that is equal to a positive real number.
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There are two possibilities.
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Either that expression, 4x plus 5, equals positive 3 or it must equal negative 3.
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We need to solve each of these linear equations to find our solutions.
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Subtract 5 from both sides.
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So 3 minus 5 will be negative 2. Divide by 4.
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Negative 1/2 is one value of x that satisfies the equation.
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Subtract 5 from both sides of the second one. We get negative 8.
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We are going to divide both sides by 4 again.
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So now we have two solutions to the equation.
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x could be negative 1/2 and negative 2. Those are our solutions.
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We can write this in set notation using braces.
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Now let’s look as some inequalities.
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Here is our first absolute value inequality.
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So we have the expression x minus 3. It's absolute value
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is less than or equal to the real number 1.
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So we know that x minus 3 needs to be something from negative 1 to 1.
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Now I have a compound inequality and I can solve it,
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by getting the x isolated in the middle here.
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Add 3 to each part.
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So now I have the inequality with the x isolated,
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and all I need to do is write the answer in interval notation.
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Notice that my end points are included since it was a less than or equal to inequality.
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So I am going to use square brackets on my interval.
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Let’s look at a second example.
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Here is our third example.
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This is another absolute value inequality.
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We have the absolute value of 2x plus 1 is greater than this positive number 5.
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There are two possibilities.
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2x plus 1 is greater than 5 or
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the other possibility is that 2x plus 1 is less than negative 5.
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Leaving this off is the most common mistake students make in solving these problems.
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So again we have two linear inequalities. We need to solve each one.
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Subtract 1 from both sides here. Divide by 2. This is part of our solution.
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Now let’s solve the other one. Subtract 1 from both sides. Divide by 2.
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Alright, so our solution is all the real numbers x that are bigger than 2
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or less than negative 3.
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And I need to write this in interval notation.
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Now it is customary to write the interval with the smaller numbers first,
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on the left, so let’s start with this one.
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This is all real numbers less than negative 3.
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I do not want to include that negative 3, so use parentheses.
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And I want anything that is smaller than negative 3,
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so that is towards negative infinity, and that is also never included.
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There is the first one.
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The other interval is all real numbers x that are bigger than 2.
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So these will be all the numbers from 2 to infinity.
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Again, parentheses to show the end points are not included.
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So the solution is all the real numbers that are in this interval or in this one.
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And a mathematical symbol we could there is the union symbol.
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Now be sure to practice solving these types of inequalities and equations
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until you are comfortable with the process.