Math Is Fun Forum

paulb203

Member

Registered: 2023-02-24

Posts: 345

Why are inequalities so called?

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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Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,830

Re: Why are inequalities so called?

Hi paulb203,

See the link below:

Solving Inequalities

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paulb203

Member

Registered: 2023-02-24

Posts: 345

Re: Why are inequalities so called?

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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Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: Why are inequalities so called?

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

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paulb203

Member

Registered: 2023-02-24

Posts: 345

Re: Why are inequalities so called?

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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paulb203

Member

Registered: 2023-02-24

Posts: 345

Re: Why are inequalities so called?

Ha! My emoji was supposed to be a smiley face

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