Do you use distributive property in absolute value?

Question

I have the problem 2|x-7|+3>5 The answer I got was 83 is this correct? I don't think you distribute with aboslute value. Is this correct?

Thanks!

Answer 1

2|x - 7| + 3 > 5

Subtract 3 from both sides:

2|x - 7| > 2

Divide both sides by 2:

|x - 7| > 1

Split up into two cases. The problem involves > (greater than), so use "OR"

x - 7 > 1 OR x - 7 < -1

x > 8 OR x < 6

This is the final answer.

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To answer your other question:

It is correct to replace 2|x - 7| with |2x - 14|. This is a perfectly good way to do the problem, but it (arguably) involves slightly more work than the solution above.

However, you can only distribute like this if the number outside the bars is *postive*. Be careful:

-2|x + 5| is NOT the same as |-2x - 10|.

As a previous answerer stated, if you feel unsure about when you are allowed to distribute, it may be better to avoid distributing when possible.

Answer 2

Not really; no; sorta.

You can use a sort of "distributive property", but only if the number being distributed (or "factored") is not negative. In your case, 2 is not negative, so 2*|x-7| = |2*x - 2*7|. But you don't need to do this at all with your problem. First, add -3 to both sides of the inequality to get:
2*|x-7| > 2
Then multiply both sides by 1/2 to get:
|x-7| > 1
This is true if either of the following inequalities is true:
x - 7 > 1
-(x - 7) > 1
The first easily converts to x>8, so that part of your answer is right. The second converts to:
-x + 7 > 1
-x > -6
x < 6
So the other side of your answer was off.

Doing this the "distributive" way, we noted that:
|2*x - 2*7| + 3 > 5
Again, add -3 to both sides:
|2*x - 2*7| > 2
|2*x - 14| > 2
This is true if either of the following is true:
2*x - 14 > 2
-(2*x - 14) > 2
Solve these and again get either (x>8) or (x<6). I'll do the second one, the part that you missed:
-2x + 14 > 2
-2x > 2 - 14
-2x > -12
x < (-12) / (-2)
x < 6

Answer 3

Properties Of Absolute Values

Answer 4

remember that an absolute value says that distinctive merchandise can not in any respect be destructive. the entire equation can visit the negatives yet not the section interior the brackets. Therfore you remedy 4x-one million>= 0. at one million/4 it equals 0. the final graphing approach for the time of that's to come across the cost for which the ingredient pivots. that ought to be the place actual the cost = 0. subsequently one million/4. The graph the equation the place x=one million/4 you may desire to get y=sixteen (one million/4,sixteen). Then p.c.. values for xone million/4, a pair of them. Graph all those factors. then you definately might have a sharp style of graph with the vertex being at (one million/4,sixteen)

Answer 5

My teacher told me not to use the distributive property in absolute value as it might mess up.......its always good to be on the safe side :-)
2 Ix-7I>2
Ix-7I>1
x<6 ; x>8

Answer 6

you do use the distributive property and you also have to get rid of the +3