Question 89927: This is from a worksheet: Solve the inequality:
8x - 9 (divided by -4) < - 23 (divided by 8) Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website! Given the inequality:
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In general you can solve these using the same rules and procedures as you would for an equation.
However, there is one exception. That exception is that if you multiply or divide both sides
of the inequality by a NEGATIVE quantity you must change the direction of the inequality
sign.
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So let's slightly re-write the problem to get the negative signs in front of the terms. On
the left side we will take the minus sign from the denominator and put it in front of the term.
On the left side we put the minus sign from the numerator in front of the entire term.
This does not change the problem at all. What we have is:
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So far, so good. Now let's get rid of the minus signs by multiplying both sides of this
inequality by -1. But remember the rule ... if you multiply or divide both sides by a
negative, you need to reverse the direction of the inequality sign. Therefore, multiplying
both sides by -1 results in:
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Notice the new direction of the inequality sign.
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Let's next get rid of the denominators by multiplying both sides by 8:
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Note that on the left side the denominator 4 divides into the 8 in the numerator 2 times.
And on the right side the denominator 8 divides into the 8 in the numerator 1 time. These divisions result in:
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which becomes:
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Multiply out the left side by multiplying 2 times each of the terms in the parentheses
to get:
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Get rid of the 18 on the left side by adding 18 to both sides to get:
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which simplifies to:
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Next solve for x by dividing both sides by +16 to get:
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and if you convert the right side to a decimal by dividing 16 into 41 the answer becomes:
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This means that the original inequality you were given will be true as long as x is greater
than +2.5625.
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Let's try by letting x be +3. That certainly is greater than +2.5625. If we put +3 into the
original inequality in place of x we get:
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On the left side the numerator becomes 24 - 9 which is 15 ... and the inequality then becomes:
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If you divide 15 by -4 you get -3.75 ... and if you divide -23 by 8 the right side
becomes -2.875
so the inequality is:
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Is this correct? Sure is. You can tell that -3.75 is less than -2.875 because it lies to
the left of -2.875 on the number line. (Bigger numbers are to the right on the number line.)
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If you next choose a value for x that is less than 2.5625 the inequality should not work.
For ease you can choose 2. Substitute that value into the original inequality and you get:
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On the left side the numerator becomes 16 - 9 which is 7. So the inequality becomes:
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On both sides divide the denominators into their numerators to get:
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Is this correct? No, it is not because -1.75 is not to the left of -2.875 on the number line.
It is to the right. Therefore, when x is 2, the inequality does not work. This helps to
build confidence that somewhere between x = 2 and x = 3 the inequality goes from "not working"
to "working". Our answer looks as if it is probably correct.
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Hope this helps you to understand the problem a little better.
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