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Find algebraically the solution set of the inequality the absolute value of six minus 𝑥 is less than three.
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We begin by recalling that the absolute value of a number is its distance from zero on a number line.
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In this question, we are told that the absolute value of six minus 𝑥 is less than three, which means that six minus 𝑥 lies between negative three and three.
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Note that the question gives us a strict inequality, so negative three and three are not included.
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We can then express this as a double inequality: six minus 𝑥 is greater than negative three and less than three.
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Subtracting six from each part of the inequality, we have negative 𝑥 is greater than negative nine and less than negative three.
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We can then divide through by negative one, recalling that when multiplying or dividing by a negative number, this changes the direction of the inequality symbol.
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For example, a less than symbol becomes a greater than symbol.
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We can therefore conclude that if the absolute value of six minus 𝑥 is less than three, then 𝑥 is greater than three and less than nine.
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As we are asked to give our answer using set notation, we have the open interval from three to nine.
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In order to check our solution, it is worth substituting a value from in our interval into the original inequality.
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If we let 𝑥 equals four, the left-hand side becomes the absolute value of six minus four.
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This is equal to the absolute value of two, which in turn is equal to two.
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And as this value is less than three, this suggests that our solution set of the open interval from three to nine is correct.