SOLUTION: ''Solve for x and show work.''
2|x - 5| - 7| x - 5| > |x - 5|
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Question 628276: ''Solve for x and show work.''
2|x - 5| - 7| x - 5| > |x - 5|
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
It might help to look at |x - 5| as another variable.
Let q = |x - 5|. Then your inequality would be:
2q - 7q > q
Looking at the inequality this way might make it easier to see what to do. First, simplify:
-5q > q
Next, get the variable on just one side. Because of the special rule regarding multiplying or dividing an inequality by a negative, I am going to get the variable on the side with the higher coefficient. Since 1 > -5 I'm going to get the variable on the right side by adding 5q:
0 > 6q
Next we divide both sides by 6:
0 > q
But a solution for q is not what we wanted. We want a solution for x. So we replace q with |x - 5|:
0 > |x - 5|
and solve for x. Note: At some point you will not need a temporary variable like q. You will be able to see how to go directly from
2|x - 5| - 7| x - 5| > |x - 5|
to
-5|x - 5} > |x - 5|
to
0 > 6|x - 5|
to
0 > |x - 5|
This inequality says "The absolute value of x-5 is less than zero." (Note: Always read an inequality from where the variable is. In this case the variable is on the right side. So we read it from right to left. (Remember, Math is not English. We don't do everything left to right in Math like we do in English. This is a case that calls for right-to-left reading!) Notice how ">" means greater than when read left-to-right but it means "less than" when read right-to-left.
But how can an absolute value be less than zero? Answer: It can't! This is why there is no solution to this inequality.
But if you don't notice that there is no solution you could go ahead and try to solve it anyway. The next step would be to rewrite the inequality without an absolute value. This results, as I hope you have learned, in two inequalities:
0 > x - 5 and -0 < x - 5
Note that we use "and" and not "or". "And" is used for "less than" inequalities of absolute values and, as we discussed earlier, this inequality, when read properly, is a "less than"! TO solve these we just add 5 to each side:
5 > x and 5 < x
Now it should be clear that there is no solution. One inequality says x must be less than 5, the other one says x must be greater than 5 and the "and" says that both of these must be true. But it is impossible for any number to be both less than and greater than 5 at the same time!
P.S. Some things I hope you will remember from this:- Read inequalities from where the variable is. Sometimes this will mean left-to-right reading and sometimes it will mean right-to-left reading.
- When solving inequalities it is to your advantage to have the variable end up with a positive coefficient. This is true because:
- The last step in solving is to divide both sides by the coefficient of the variable. If this coefficient is negative then we will divide by a negative number.
- There is a special rule that applies when multiplying or dividing an inequality by a negative number. It is very easy to forget this rule so avoid it if you can.
Let's look at some examples. At one point in this problem we had:
-5q > q
If we subtract q from each side (as many people would do because they like the variable on the left side) we would get:
-6q > 0
Next we divide both sides by -6. If we forget the rule we would get:
q > 0
which would lead to a solution in a problem where there is no solution. If we remember the rule then we know that when we multiply or divide an inequality by a negative number we should, at the same time, reverse the inequality symbol. So
-6q > 0
becomes
q < 0
after we divide by 6. This special rule can always be avoided. Say you had this inequality:
-3q + 4 > 10
The variable is already on just one side of the equation. But it is on the "wrong" side because its coefficient is negative and the last step would be to divide by -3 (which would require the special rule). If we add 3q to each side we get:
4 > 3q + 10
Then we can solve this with the last step being a division by positive 3. Yes, this was an extra step. But unless you are extremely good with the special rule, this extra step is probably worth it. - Using a temporary variable is not required but it can be helpful in figuring out how to solve problems. Your inequality has the same absolute value in three places. By replacing it with a temporary variable it was easier to see what we could do and how we do it. Just remember that is it temporary. Eventually you have to substitute back in for it.
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