Solving linear inequality problems?

Question

Okay, I think I am good with how to solve once I figure out where to move my like terms. For example

-4x+8<1/2x +3 (sorry I don't know how to make the fraction look right here).

So I move add 4x to both sides. By moving my "x" like terms to the right.

Now if I have 2x-7>-2/3x-2 , apparently I move my "x" like terms to the left by adding 2/3x to each side.

Here's my question, how do I know where to move my "x" like terms. Is there a standard? Sometimes to the right, sometimes to the left and I don't understand why. I am pretty confused.
Thanks.

The first inequality problem is not a 2x, but rather a one half..I couldn't get the 1 over the 2 looking right.
Also, someone mentioned always move your "x" variables to the left, but this doesn't seem to be the case and this is why I am confused. On my program for these problems, it has the "x" like terms moving right and left, either - or . This is why I am confused.

Okay so if there is no rule about where to move your like terms i.e. "x" etc, then how do I know which way to go? I can't get out of this set of problems until I learn them, and I can't figure out the trick behind knowing where to send the each set of like terms.

Answer 1

Hi Heidi Ann,

There is no standard that you have to move variables to a specific side when solving inequalities. You can solve a problem to x > 3 or to 3 < x.- Both of them mean the same thing, so either is a correct solution.

However, as a teacher, I have a preference. I tell my students to always move their variable to the left-hand side when solving inequalities. The reason for this is that if they are to draw a number line graph to show the inequality, if the variable is on the left side, then the inequality sign points the direction they should shade their arrow on the inequality.

x>3 means make a hollow dot at x = 3 and then shade
O==|==|=>
3....4...5

3< x means the same thing, but too many students see the less than side and shade the wrong way.

So, while you can solve the inequality with variables on either side, I prefer variables on the left.

I hope that helps!!

Answer 2

No there is no set rule. THe trick is to always move the variables to one side and the constants to the other. In inequalities remember to change the sense of the inequality if you divide by a negative number.

example 3X - 4 > 5X - 10
if you move the 3X you get 6 > 2X
or 3 > X meaning X < 3

but if you moved the 5X you get -2X > -6
dividing by -2 reverses the sign X < 3 (same thing)

Answer 3

>how do I know where to move my "x" like terms. Is there a standard?
Generally you move the x-terms so that the coefficient of x is positive (because if not, you would need an extra step to multiply by a negative number so that x-coefft becomes 1 or positive.)

Here's one example of yours:
-4x+8<1/2x +3

Multiply the inequality by 2. When you multiply by a positive number, the inequality sign does not change:

-8x+16 < x +6
Add 8x-6 to each side
10 < 9x
Divide by 9 (again positive, no sign change)
10/9 < x
x > 1.1111...

Answer 4

If you always move them to the left, sometimes you get a negative times x. If you want to avoid that, pick out where the x term has the lowest number in front (coefficient), and add what it takes to eliminate it, but now you'll sometimes end up with x on the right. It's a choice, no standard.

Example:

-4x - 7 ≥ 6 + 2x
-6x - 7 ≥ 6
-6x ≥ 13
x ≤ -13/6.................or

-4x - 7 ≥ 6 + 2x
-7 ≥ 6 + 6x
-13 ≥ 6x
-13/6 ≥ x

Answer 5

Multiply both sides of the inequality by 2 first,
-8x + 16 < x + 6
Solve for x by collecting like terms,
16 - 6 < x + 8x
10 < 9x
x > 10/9

Answer 6

Is your right hand side of the inequality
(1/2)x+3
or
1/(2x+3)
or
[1/(2x)]+3

The answer will depend on which one is the correct one.
However, if you are used to solving equalities, you apply the same rules/techniques for inequalities, i.e. add/subtract same entity on both sides, multiply/divide both side with the same entity, raise to same power etc. Other solutions are well illustrated.

Answer 7

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