Expert: Josh - 12/9/2005

Question
I am haveing a lot of trouble with my algerbra could you please help me? What I don't get is problems like 29x-13>54 you know stuff like that. I don't understand how to work them out. please help

Answer
Hi Chris,

The type of problem you have described is called "inequalities". An "inequality" is an expression (just like an equation) that involves one of the following signs:
greater than, ">"
less than, "<"
greater than or equal to, ">=" OR
less than or equal to, "<="

Essentially, we have to work out some bound(s) for the size of an unknown quantity, which we may write as "x".

Let us start with a general rule.
GENERAL RULE
"Whatever you do onto one side of the inequality, you apply the same operations to the other side of the inequality"....This ensures that the expression remains logically true.

O.K. with the example you've given me,
29x-13>54

We have to work out some bound on the size of x.
Step 1: As always, the general idea is to separate the unknown variable, x, from the known quantities (any numbers).
So, we add 13 to both sides of the inequality.
29x-13 > 54 becomes [Line 1]
29x-13+13 > 54+13 ...[Line 2]

You see, we haven't changed the nature of the expression. We've only manipulated the expression to get one step closer to the answer, but whatever holds for x in Line 1, must also hold in Line 2. Why?? Because all we have done is add a constant amount (i.e., 13) to both sides of the equation. This operation is REVERSIBLE. We can, as a check, subtract 13 from both sides of Line 2 again, to get back to Line 1.

Note: we are permitted to add, subtract, multiply both sides of an inequality by a non-equal constant, without altering the logic of the expression.
e.g., we can (i) add 13 to both sides,
(ii) subtract any number from both sides,
(iii) multiply non-zero number on both sides.

So, what do we choose. These operations are not performed to make things more complicated, of course. They must serve a purpose. The goal is to simplify the expression, as we have done in Line 1.

29x-13 > 54 becomes [Line 1]
29x-13+13 > 54+13 ...[Line 2]
Anyway, continuing from Line 2, we get
29x > 67
Finally, dividing both sides by 29, we find out what one unit of the unknown is "worth",
x>67.

Here is one more thing to remember,

GOLDEN RULES:
1. We don't touch the inequality sign, when we add or subtract some number from both sides.
2. We invert the inequality sign, whenever we multiply or divide both sides by a negative number.
i.e., From ">" to "<" or vice versa; ">=" to "<=" or vice versa.

e.g., suppose 2x-2 > 4
Step 1: add 2 to both sides to eliminate the minus two
2x -2+2 > 4+2 ...next, simplify to get
2x > 6 ...this is fine
Step 2: dividing both sides by 2, we keep the inequality "as is", since we are dealing with a positive (NOT negative) number
answer: x > 3

e.g., this one involves inverting the inequality sign.
suppose -2x+1 > 4 [**]
Step 1: subtract 1 from both sides, nothing spectial here.
-2x +1 -1 > 4-1
-2x > 3
Step 2: #next, we have to divide by -2 (negative number), so we have to change ">" to "<"
-2x/(-2) < 3/(-2)
we get
x < -3/2.

To convince yourself that this must be the case, plug in different numbers of x, smaller than -3/2 into [**].
e.g., pick x=-2
you see that -2x+1>4 becomes -2(-2)+1>4. This is true, since 5>4. If we did not invert the sign, 5<4 would be false. We could not have obtained the right solution.

A good starting point: I suggest that you go over the rules I've given you.

Cheers.