Solving for an inequality, help?

Question

There are three problems that follow this, they are what needs to be solved. I am so confused. If you would like a link to the image of the problems instead than reading this, you can find the exact same information at the far bottom of the image here http://palcs.digitalschooling.com/brand/media/images/1193//Algebra_2/New%20Algebra%202/New%20Lessons/homework%20lesson%2012.JPG

Since you don't know whether X is positive or negative, you have to consider two cases: (part 1) X > 0 or (part 2) X < 0.

Part 1.) X > 0
Multiply both sides of the given inequality by 2x, which is positive.
10+x < 6x => 10 < 5x => 2 < x
In this case, the solution is x > 0 and x > 2, or x > 2.

Part 2.) X < 0
Proceeding as before, except that now 2x is negative, you have
10+x > 6x => 10 > 5x => 2 > x
The solution is x < 0 and x < 2, or x < 0.

[[[A / indicates a fraction]]]
Problem a.) 11/3 - 2/x > 5

Problem b.) 8/x - 7 < 9

Problem c.) 3/4 + 12/x-1 > 1




Thanks so much in advance.

Answer 1

a.) 11/3 - 2/x > 5
x > 0:
Multiply by 3x and get 11x - 6 > 15x
Put all terms with x on one side, all other terms on the other:
-6 > 4x
Divide by 4:
-6/4 > x, or x < -3/2
But we assumed x > 0, so there is no solution.

x<0:
Multiply by 3x and get 11x - 6 < 15x
Put all terms with x on one side, all other terms on the other:
-6 < 4x
Divide by 4:
-6/4 < x, or x > -3/2.
We also assumed x < 0, so the solution is x < -3/2.

The final solution is x < -3/2.

b) 8/x - 7 < 9
x>0:
Multiply by x and get 8 - 7x < 9x
Put all terms with x on one side, all other terms on the other:
8 < 16x
Divide by 8:
8/16 < x, or x > 1/2.
We assumed x>0, so x>1/2.

x<0:
Multiply by x and get 8 - 7x > 9x
Put all terms with x on one side, all other terms on the other:
8 > 16x
Divide by 8:
8/16 > x, or x < 1/2.
We assumed x<0, so x<0.

Combining these solutions, we have x>1/2 or x<0.

3) 3/4 + 12/(x-1) > 1
x>1:
Multiply by 4(x-1) and get 3(x-1) + 48 > 4(x-1).
Multiply everything out:
3x-3 + 48 > 4x-4
Put all terms with x on one side, all other terms on the other:
49 > x.
We assumed x>1, so 1 < x < 49.

x<1:
Multiply by 4(x-1) and get 3(x-1) + 48 < 4(x-1).
Multiply everything out:
3x-3 + 48 < 4x-4
Put all terms with x on one side, all other terms on the other:
49 < x.
But we assumed x<1, so there is no solution.

Our final solution is 1

Answer 2

James L looks correct, except for a fumble at the end of problem a). The trouble with rfamilymember's solution is that when switching from
-2/x > 4/3 to
2/x < 4/3, the sign should have been changed on the right side too.

Even so, the next step, inverting both sides, is valid only if x>0, as you can see by considering what happens if x=-2:
2/(-2) < 4/3 is true, but -2/2>3/4 is false.

So after 2/x < -4/3, it should have been:
Clearly x < 0 otherwise the left side would be positive and therefore greater than the right side. Hence -2/x is positive and we can multiply both sides by (3/4)*(-x/2), giving
-3/4 < x/2
hence
x > -3/2

But we've already observed that x < 0, and so the solution is
-3/2 < x < 0

For problems b) and c), James L is correct, so give him the points.

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Answer 3

Problem a.) 11/3 - 2/x > 5
-2/x>15/3-11/3>4/3
2/x<4/3
x/2>3/4
x>3/2

Problem b.) 8/x - 7 < 9
8/x<16
x/8>16
x>2


Problem c.) 3/4 + 12/x-1 > 1
12/x>2-3/4
12/x>5/4
x/12<4/5
x<48/5

Answer 4

you first ought to get X by using itself. to do this subtract the three away. So now you have 2X=-2 The to get the X thoroughly by using itself you divide the two away (for the reason that's the guideline, to divide). So X =-a million (and that's when you consider which you divided the two far off from the X and on the different part it became divided by using the -2. desire that enables.

Answer 5

I've just finished inequalities, but this stuff looks like jibberish to me.

We are probably learning it different ways may possible somewhat I don't know how?