Question 95838: Can you please help with this one? I don't understand.
Solve the Inequality
2(x-4)+7<5-x Found 2 solutions by edjones, bucky:Answer by edjones(8007) (Show Source):
You can put this solution on YOUR website! You can work problems of this sort using the same procedures that you use for solving an
equation. There is one exception ... if you multiply or divide both sides by a negative
quantity, then you must reverse the direction of the inequality sign.
.
Let's do the problem. You were given:
.
2(x - 4) + 7 < 5 - x
.
Do the distributed multiplication on the left side by multiplying 2 times each of the two
terms inside the parentheses. When you multiply 2 times the terms in parentheses the inequality
becomes:
.
2x - 8 + 7 < 5 - x
.
On the left side you can combine the -8 and the +7 and get -1. This simplifies the inequality
to:
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2x - 1 < 5 - x
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Now you can get rid of the -x on the right side by adding x to both sides. On the left
side when you add x it combines with the 2x to give you 3x. On the right side when you
add x it combines with the -x and they cancel each other out. The inequality then is:
.
3x - 1 < 5
.
Now get rid of the -1 on the left side by adding +1 to both sides. On the left side the
+1 combines with the -1 and they cancel each other. On the right side the +1 combines with
the +5 and the result is +6. This makes the inequality:
.
3x < 6
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Now solve for +x by dividing both sides by +3 because +3 is the multiplier of x. When you
do this division to both sides the inequality becomes:
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x < 6/3
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which simplifies to;
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x < 2.
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This means that the original inequality will be true if x is less than +2, and it will not
be true if x is greater than +2. Let's try it. Suppose we let x be +1. That is less than
+2 so the original inequality should be true. The original inequality is:
.
2(x - 4) + 7 < 5 - x
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If we substitute 1 for x it becomes:
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2(1 - 4) + 7 < 5 - (+1)
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On the left side the numbers in the parentheses combine to -3 and that gets multiplied
by 2. So the inequality becomes
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-6 + 7 < 5 - 1
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Now the numbers on the left side (-6 + 7) combine to +1 and the numbers on the right (5 and -1)
combine to +4. So when x is +1 the inequality becomes:
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+1 < +4
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That certainly is true because +1 is less than +4.
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Now what if we let x = +3. That is greater than x = +2 so the inequality should not work.
Start with the original inequality:
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2(x - 4) + 7 < 5 - x
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and substitute +3 for x to make the inequality become:
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2(3 - 4) + 7 < 5 - (+3)
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In the parentheses the +3 and the -4 combine to give -1 and that gets multiplied by the
2 to become -2. So the inequality becomes:
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-2 + 7 < 5 - 3
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On the left side the -2 and the +7 combine to give +5. On the right side the 5 and the
-3 combine to give +2. So when x = +3 the inequality becomes:
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+5 < +2
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That's not true! +5 is not less than +2. So when x is +3 the inequality does not work.
.
When we started this check we said that if x was less than +2 the inequality would work
and if x was not less than +2 the inequality would not work. We let x be +1 which is
less than +2 and found that the inequality did work. Then we let x be +3 which is greater
than +2 and we found the inequality did not work. It looks as if our solution of:
.
x < 2
.
is a good solution.
.
Hope that this helps you to understand the problem and how we went about finding a limit
on the values that x could be to make the inequality work.
.
Note that we work this using the same procedures that you would use to solve an equation.
Since we did not multiply or divide both sides by a minus number we did not need to reverse
the direction of the inequality sign.
.
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