Question 610682
Given to solve:
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7(x-9)+3x<-3(-4x-9)-3x
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You can generally solve these by following the same rules as you would for solving an equation EXCEPT that for solving inequalities, if you multiply or divide both sides by a negative quantity, you must also reverse the direction of the inequality sign. This being the case, let's begin:
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A good place to start is to do the distributed multiplications on both sides of the inequality sign. On the left side multiply the 7 times each of the two terms in the parentheses. And on the right side multiply the -3 times each of the two terms in the parentheses. After these multiplications the inequality becomes:
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7x - 63 + 3x < 12x + 27 - 3x
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On the left side add the 7x and +3x to get 10x. And on the right side add the 12x and the -3x to get 9x. As a result, the inequality is simplified to:
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10x - 63 < 9x + 27
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Get rid of the 9x on the right side by subtracting 9x from both sides to get:
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x - 63 < 27
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Finally, get rid of the -63 on the left side by adding 63 to both sides and you have the answer:
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x < 90
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This says that the original inequality will be true if you substitute for x any value that is less than 90. Let's just run a sample check. In the original inequality let's substitute 89 for x. Since x is then less than 90, the inequality should be correct. By substituting 89 for x the original inequality becomes:
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7(89 - 9) + 3*89 < -3(-4*89 - 9) - 3*89
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On the left side, in the parentheses the 89 - 9 becomes 80. And on the right side, within the parentheses the -4 times 89 is -356 and then subtracting 9 from that results in -365. So these two changes make the inequality become:
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7(80) + 3(89) <-3(-365) - 3*89
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On the left side the 7 times 80 is 560 and the 3 times 89 is 267. On the right side the -3 times - 365 is + 1095 and the -3 times 89 is - 267. Substituting these changes into the inequality simplifies it to:
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560 + 267 < 1095 - 267
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Now adding the 560  + 267 on the left side results in +827. And on the right side subtracting 267 from 1095 results in +828. This makes the inequality simplify to:
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+827 < +828
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This certainly is true. 827 is less than 828. So if x is 89, the inequality holds true.
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But what happens if we let x equal 90? This means that x equal to, but NOT LESS than 90 as our answer said it should be. When x is equal to 90, the original inequality becomes:
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7(90 - 9) + 3*90 < -3(-4*90 - 9) -3*90
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Let's combine the numbers inside the parentheses. On the left side they combine to 81 and on the right side they combine to -369. This makes the inequality:
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7*81 + 3*90 <-3(-369) - 3*90
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Doing the two multiplications on the left side gives 567 + 270. Doing the two multiplications on the right side results in 1107 - 270. So the inequality reduces to:
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567 + 270 < 1107 - 270
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Adding 567 + 270 on the left side results in 837. And on the right side subtracting 270 from 1107 also results in 837. So the inequality becomes:
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837 < 837
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But this inequality is NOT TRUE. 837 is not less than 837 ... the two sides are equal. Therefore, when x equals 90, the inequality DOES NOT WORK. By these two tests we know that when x equals 89 the inequality is OK, but when x equals 90 it is not. This suggests that x must be less than 90 as our answer told us. These two trials should give us confidence in our answer of x < 90.
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Hope this helps you to understand the problem a little better. Note that we did not use the rule that if you multiply or divide both sides of an inequality by a minus (or negative) quantity, you must also change the direction of the inequality sign. Remember that rule for another problem in which you may need to do such a multiplication or division.