Question 461824: solve the compound inequalities
6>-3x+5 or 9<= -5x+3
I dont understand this system
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! You can work these just like equations, with one exception. That exception is that whenever you multiply or divide both sides by a negative number, you have to reverse the direction of the inequality sign.
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Just like solving an equation, the goal is to solve for x.
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Let's give it a try with the first part of the problem:
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If you were solving this like an equation, one of the first steps might be to subtract 5 from both sides so that you get rid of the +5 on the right side. Let's do that:
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and by subtracting this simplifies to:
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Now to solve for x, divide both sides by -3, the multiplier of the x to get:
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Doing the division results in:
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But don't forget the important rule ... if you divide or multiply both sides by a negative number, then you have to reverse the direction of the inequality sign. We divided both sides by negative 3. Therefore, we need to reverse the direction of the inequality sign. When you do that you get the answer:
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And that is the answer. Since the arrow always points toward the smaller value, you interpret this answer as either  is smaller than  or that is greater than . Both of those interpretations mean the same thing.
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If you think in terms of the number line and you put a dot on the line at , then can be any number on the number line that is to the right of that dot.
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A quick easy check of this answer is to return to the original problem that we started with and set x equal to zero. If x does equal zero, it will lie to the right of on the number line and, therefore, it should be in the set of answers that makes the original inequality true. Let's try it:
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If x equals zero, this becomes;
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This is true because 6 is greater than 5. So x=0 is part of the answer set. You may want to try other positive and negative numbers to see if they make the original inequality true. Any number to the left of on the number line should make the inequality untrue. Any number to the right should make it be OK.
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Now let's work the second part of the problem by following the same method. Start with:
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Subtract 3 from both sides:
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Divide by -5:
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Reverse the inequality sign because of dividing by negative 5. This makes the answer become:
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This tells you that x is smaller than or equal to  . You interpret this on a number line as follows: put a dot at on the number line. Then x can have any value to the left of that dot and it can also have a value equal to that dot, that is equal to .
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Since this is the second part of a compound inequality, you combine the two answers. Visualize this on the number line. From the second part of this problem x can be any value left of or equal to and from the first part, x can be any value to the right of . But x cannot be a number between these two values (including that x cannot take the value .
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Check my work to make sure I didn't make some dumb mistake.
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Hope this gives you a little better insight into solving inequalities.
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