Three algebra inequality questions?

Question

I've got the answers to these, but I just HAVE to know why the answer is like that. You know, that's me. Also, I'd appreciate a demonstration. The most thorough answer gets 10 pts!

1. -5(x - 1) < 5(x + 1)
The question I have here is, why does the left side, -5(x-1),simplify to -5x+5, with that addition sign? In all the rules I learned I don't remember learning that.


2. 4x + 2 - 4x > 5 - x - 4
(Note that the > originally has a line under it, meaning equal to or greater than)
How does the left side simplify completely to 2, which it does? Then it simplifies to 2>1-x and from there you would do 2-1>1-1-x. Why use 1 to subtract? That brings the right side to -x, doesn't it?


3. 1/3x - 2 < 2/3x - 6
(Note that the < has a line under it, meaning ...see above) Here's one I'm stumped with. How would you go about doing this? Start with subtracting/adding the fractions from each side? Or go +6 starting with right side? Or add +2 to the left side and right?

Thanks!

Answer 1

1 expand both sides left side expands to (-5 * x) - (5 * -1)
or -5x - (-5) a minus minus translates to a +

-5x+5 < 5x + 5

combine

0 < 10x

divide both sides by 10
0 < x or also can be read as x > 0

2 combine x's on left 4x - 4x = 0, what is left is the 2
then completing both sides yields

2 >= 1 - x >= means greater than or equal
subt 1 from both sides
1 >= -x
divide both sides by -1 (remember this changes the > sign to a < sign - this only occurs when you multiply or divide both sides by a negative number
-1 <= x or also can be read as x>= -1

3. subtract 1/3x from both sides, add six to both sides (this will put variables on one side and integers on other)

4 <= 1/3x
mult both sides by 3
12 <= x or can be also read as x>=12


hope this wins the 10 points

Answer 2

1) Because that is actually a multiplication problem and there is a thing called the distributive property of multiplication. You multiply everything in the parenthesis by -5. When it comes to turning the minus sign, just turn (x – 1) into (x + -1). Subtracting any positive number is the same as adding a negative number. Also, you should know that a negative times a negative is a positive.

So:

-5 (x – 1) = -5(x + -1) = -5x + 5


2) Just play with each equation a little bit. You can rewrite it, but if it is a addition/subtraction problem, you have to move the previous operator with the number. I will use brackets [ ]to show you: THE BRACKETS DO NOT INDICATE A MULTIPLICATION PROBLEM. I am just using them to show you how to group and rewrite a problem so it becomes clear.

4x +2 - 4x = [4x] [+2] [-4x]
= [4x] [-4x] [+2]
= 4x - 4x + 2.............I removed the brackets
= 0 + 2......................4x - 4x = 0, so just write a 0
= 2

5 - x - 4 = 5 [-x] [-4]
= 5 [-4] [-x]
= 5 - 4 - x......................I removed the brackets.
= 1 - x............................5-4 = 1, so just write a 1

3) Once again take each side and rewrite the problem. With this thing, you have to remember a few things:

a) There are two equations.
b) Fractions suck, so it is best to get rid of the fraction by
"flipping" it. If you see 2/3x, just multiply it by 3/2 and
then it becomes 6/6x or 1x or just x.
c) When you "flip" a fraction by multiplying, you have to
flip the other parts of the equation. For example, in the
right hand side, you have to multiply both 2/3x AND 6
by 3/2. So, if you take 3/2 x 6, it would be 18/2 or just
9.

These rules will allow you to figure out the left side as well

Answer 3

Question 1
Noting that (- 5 ) x (- 1) = 5:-
- 5x + 5 < 5x + 5
5 < 10x + 5
0 < 10x
x > 0
Question 2
2 ≥ 1 - x
x + 2 ≥ 1
x ≥ - 1
Question 3
Hope that question is meant to be:-
(1/3).x - 2 ≤ (2/3).x - 6
You can now do anything you like so long as basic rules are followed.
eg multiply both sides by 3:-
x - 6 ≤ 2x - 18
subtract x from both sides:-
- 6 ≤ x - 18
Add 18 to both sides:-
18 - 6 ≤ x
x ≥ 12

Answer 4

1.
-5*-1 = +5 so -5*(x-1) = -5x+5

2. 4x+2-4x = 4x-4x+2 = 2

5-x-4 = 5-4-x = 1-x

3. multiply through by 3

3*1/3x - 3*2 < 3* 2/3x - 3*6
x-6<2x-18

add 18 to both sides

x-6+18 < 2x-18+18

x-12 < 2x

subtract x from both sides

x-x-12 < 2x-x

12