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Why does, for example, 2y+1 is less than or equal to 7 count as an inequality?
If x is 3 then 2y+1 is equal to 7, not ‘inequal’, yeah?
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Thanks, ganesh.
I've worked my way through those pages already; very helpful. But the question I posted remained afterwards.
It seems to me to be a misnomer ('inequality') given that the two quantities involved are not always necessarily inequal, e.g, the example I gave in my original post, 2y+1 can be equal to 7 when y is 3
(I made a mistake in the original post - I put x when I meant y).
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hi paulb203
Welcome to the forum.
In an equation two expressions (LHS and RHS) are said to be equal in value. There's a whole set of techniques for solving problems using equations. An equation may have more than one solution.
Sometimes a whole range of values may form the solution to a problem. In such cases an algebraic 'thing' is created that is not an equation. This is where the word inequality is used. It's just the name for such a thing so don't get hung up on why it is called that. Let's see an example.
A company charges a fix amount, say $10, to deliver goods plus the cost of the goods. Let's say they cost $5 each.
So an expression for the charge would be 10 + 5n where n is the number of items being bought.
Now I'll create a problem that uses an inequality. Let's say that when the total cost exceeds $100 the company wants payment in advance whereas at $100 and below they'll accept cash on delivery. What's the n value when payment in advance kicks in?
The expression for the total cost was 10 + 5n and the in advance payment happens when this is over $100. So I can write
10 + 5n > 100
You can solve this just like an equation.
Subtract 10 from each side:
5n > 90
divide by 5:
n > 18
So a customer will have to pay in advance if they want to buy more than 18 items (ie. 19, 20 etc)
If I change the sentance to when the total cost is $100 or above the company wants payment in advance, then the inequality becomes
10 + 5n ≥ 100 and the solution is now n ≥ 18 meaning that the rule kicks in at 18, rather than above 18.
There just one special rule for dealing with inequalities that relates to what we mean by >
On the number line ( negatives on the left, positives on the right) A > B means A is further to the right on the line than B. So
6 > 3
-3 > -6
When using algebraic rules to solve an inequality you might have a situation where you multiply or divide by a negative amount. This has the effect of reflecting the LHS and RHS through zero. This means the inequality sign must change.
example. 6 > 3 multiply by -1 and it becomes -6 < -3. If you don't switch the sign like this your algebra will go wrong.
Example.
10 - 3x > 4
If I solve by avoiding multiplying/or dividing by a negative I get this correct solution
add 3x to each side
10 > 4 + 3x
subtract 4
10 - 4 > 3x
6 > 3x
divide by 3
2 > x. We can re-write this as x < 2 ; it means the same thing.
Look what happens if I solve this way:
10 - 3x > 4
subtract 10
-3x > -6
Divide by -3 but forgetting to switch the sign
x > 2
This we alreacy know is not correct. So you can appreciate why the sign must be switched too.
X < 2 is correct.
Hope this helps,
Bob
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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Thanks a lot, Bob. That was very helpful. I had fun working out the 10+5n example before looking at the answer.
I did;
100-10=90
90/5=18
But I couldn't come up with the expression 10+5n is greater than 100 )
I had noted the warning about changing the sign from the examples on the main website but your examples were good revision. Thanks.
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Ha! My emoji was supposed to be a smiley face
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