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Question 554427: How to solve the inequality -4(2x+7)<16?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Given to solve:
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In general, you can work inequalities such as this one using the same rules as you would for equations with the exception that if you divide or multiply both sides by a negative quantity, the sign of the inequality will reverse direction.
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So, let's begin by doing the distributed multiplication on the left side. Multiplying both terms inside the parentheses by -4 results in:
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Next, let's get rid of the -28 on the left side by adding +28 to both sides. The -28 and the +28 will cancel each other on the left side and on the right side the +16 and the +28 total to +44. So the inequality is now:
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Just as we would do for an equation, we can solve for x by dividing both sides by -8. But remember the rule that if we divide or multiply both sides by a minus quantity, the direction of the inequality sign changes.
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Dividing the -8x by -8 results in just x and dividing 44 by -8 gives  . And reversing the direction of the inequality sign due to the division by the MINUS 8 makes it change direction so it points to the right as shown:
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This tells us that the original inequality that we were given will only be true for values of x that are greater than . On the number line, such values lie to the right of .
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Let's raise our confidence in that answer by taking the original inequality and substituting a value for x that is greater than to see if it works. Let's try letting x equal -5:
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Start with:
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Replace x by -5 to get:
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Inside the parentheses the 2 multiplies the -5 to give -10 and the +7 adds to the -10 to result in -3. This leads to:
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and then:
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Multiplying the -4 times the -3 gives +12 so when x equals -5, the inequality becomes:
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That certainly is true ... 12 is less than 16. So for a value of x that is slightly greater than the original inequality that we were given is true.
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For practice, why don't you see what happens when x is replaced by something slightly less than meaning that it lies slightly to the left of on the number line. If you try letting x equal -6 in the original inequality, you should get:
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But that is not true. +20 is greater than +16. So x cannot be -6 and still have the original inequality be true. Somewhere between x equals -5 and x equals -6, the value of x will make the inequality go from being true to being untrue. That helps to give us confidence that our answer is correct. x must be greater than some value between -5 and -6.
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For more practice, why don't you try letting x be exactly and see what happens when you substitute that value for x into the original inequality? You should get:
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That is not true. +16 is equal to, not less than +16. This tells you that, on the number line you must go a little to the right of to make the original inequality be true. This establishes that our answer is correct. Namely, it must be that:
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Hope this helps you to understand the problem a little better.
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