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Determine the absolute value inequality representing π₯ is the set of all real numbers minus the closed interval negative 21 to 27.
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First, letβs consider the information we know about π₯.
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If we think about a number line showing all real numbers, and then we add negative 21 and 27, these are all the excluded values of π₯, the values that π₯ cannot be.
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Letβs add the values that π₯ can be to the same number line.
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π₯ cannot be equal to 27, but it can be greater than 27.
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And then π₯ cannot be equal to negative 21, but it can be less than that, which means π₯ is less than negative 21.
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But now letβs think about what we know about absolute value.
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The definition of absolute value tells us that it is the magnitude of a number without regard to its sign.
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If we have the values of negative π₯ and π₯ on the number line, we say that the absolute value of π₯ is π₯.
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And both values, negative π₯ and π₯, are π₯ units from zero; they have a magnitude of π₯.
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And to write an inequality to represent this value of π₯, what weβll want to do is find the middle of negative 21 and 27.
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If we find the midpoint, the distance between negative 21 and π will be the same as the distance from π to 27.
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The distance from negative 21 to 27 is 48 units.
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Half of that is 24.
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24 units to the right of negative 21 on the number line is three.
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And three plus 24 equals 27.
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And what weβre saying here is that π₯ must be more than 24 units from three.
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If we start at three, we need to go more than 24 units to get into the range of what π₯ can be in the right direction.
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And if we start at three, we need to go more than 24 units in the left direction to find a value that is acceptable for π₯.
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And now we need to translate that into an absolute value inequality.
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We can write that as the absolute value of π₯ minus three must be greater than 24.
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To check that this is true, we break it up into two separate equations, which is π₯ minus three is greater than 24 and the negative of π₯ minus three is greater than 24.
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On the left, we add three to both sides to get π₯ is greater than 27.
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On the right, we multiply both sides of the inequality by negative one so that π₯ minus three is less than negative 24.
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And after we add three to both sides, we get that π₯ is less than negative 21 which confirms the absolute value of π₯ minus three must be greater than 24.
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And that will produce π₯-values such that π₯ can be all reals with the exception of the closed interval negative 21 to 27.