You can put this solution on YOUR website! Let x be the unknown number.
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Then 2 times the unknown number is 2*x.
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Increase that by 28 and you have 2*x + 28
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This is to be less than or equal to 6 times the unknown number ... or less than or equal to 6*x.
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In inequality form this is:
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We need to solve this for x. So we need to collect the terms containing x on one side of the
inequality and all the numbers on the other side. Start by getting rid of the +28 on the
left side of the inequality. Do this by subtracting 28 from the left side. But when you
do that subtraction you also have to subtract 28 from the right side. This makes the inequality
become:
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Now get rid of the 6*x term on the right side by subtracting 6*x. But when you subtract 6*x
from the right side, you must also subtract 6*x from the left side. This subtraction
from both sides simplifies the equation to:
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Finally, to solve for +x divide both sides of the inequality by -4 which is the multiplier
of x. Now an important rule comes into play. When you divide or multiply both sides of
an inequality BY A NEGATIVE QUANTITY the inequality arrow must CHANGE DIRECTION.
So this division by -4 causes several major changes. The -4x becomes just x and the -28
becomes +7. And also the inequality arrow switches direction. Substituting all these
changes into the equation results in:
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So on the number line, x must be +7 or be to the right of +7. Let's try a value just to
convince ourselves that this answer is correct. Let's set x equal to +8 which is to
the right of +7 on the number line. If you return to the original problem and set x equal
to +8 the original problem statement becomes:
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2 times 8 is 16 and increased by 28 is 44. That must be less than or equal to 6 times 8 which is 48.
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That works. How about we try a value of x that is less than 7 ... say x = 6.
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2 times 6 is 12 and increased by 28 is 40. That must be less than or equal to 6 times 6 which is 36.
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This does not work. These couple of spot checks serve to convince us that our answer
looks good ... x must be greater than or equal to +7.
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Hope this helps you to understand inequality problems a little more. And don't forget
the rule that if you multiply or divide both sides of an inequality by a negative number
you must reverse the direction of the inequality arrow.
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