Question 116727
You can work these problems pretty much following the same rules as you would for solving an
equation. The inequality sign is similar to the equal sign in the equation in separating
the two sides from each other. There is an exception, however. If you multiply or divide
both sides of the inequality by a negative quantity, you reverse the direction of the inequality
sign.
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So let's begin using the processes we would use for solving if the given problem were an equation.
Start with the given:
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17 -(4k - 2) > 2*(k + 3) 
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On the right side, do the distributed multiplication to get:
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17 -(4k - 2) > 2k + 6
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On the left side remove the parentheses. However, since the parentheses are preceded by
a minus sign, when you remove the parentheses you change the signs of the two terms inside
the parentheses to get:
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17 - 4k + 2 > 2k + 6
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Now, as in solving an equation, we try to collect the terms with the variable k on one side
of the > sign and all the numerical terms on the other side. Begin by subtracting 2k
from both sides of the inequality to get rid of the 2k on the right side. This subtraction 
reduces the inequality to:
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17 - 6k + 2 > 6
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On the left side combine the 17 and the +2 to get 19:
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19 - 6k > 6
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Get rid of the 19 on the left side by subtracting 19 from both sides:
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-6k > -13
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Solve for k by dividing both sides by -6. However, recall the rule that if you divide both
sides by a negative quantity, you reverse the direction of the inequality sign. So dividing
both sides by -6 results in:
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k < 13/6
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That is the answer to this problem. The original inequality will be true if k is less than 13/6.
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Let's try it to help verify. If we let k = 2 (which is slightly less than 13/6) then the
original inequality becomes:
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17 - (8 - 2) > 2(2 + 3)
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Combine the terms in the parentheses on both sides to get:
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17 - (6) > 2(5)
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Do the math. On the left side 17 - 6 = 11 and on the right side 2*5 = 10. So when k = 2
the inequality is:
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11 > 10
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That is true. To further check the answer you might want to try letting k = 3 (which is
not less than 13/6) and see what the original inequality becomes when k is that value.
You should find that the resulting inequality is not true ... showing that if k is bigger than
13/6 the inequality will not work.
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Hope this helps you to understand how to work problems such as these. Don't forget the
rule about reversing the direction of the inequality sign if you multiply or divide both
sides by a negative quantity.
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