"The world of inequalities"
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Kindly use your brain to solve the sums.
If you dunno them, flip your textbook and recall.
Testimonials
"It's simply brilliant!"
- by Dr. Madema Ticks,professor of Mat-Ee-Diats University
"The most wonderful invention in history - inequalities"
- by Mr Smurt Elak, teacher of Mads Nard Institute
"Most educational"
- by Ms Breiyn Las, top scholar in The Kandoo Betar School
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Everything you need to know 'bout inequalities!
What is an Inequality?
An inequality is a statement that shows the relationship between two (or more) expressions with one of the following five signs: <, ≤, >, ≥, ≠.
x < y means "x is less than y"
x ≤ y means "x is less than or equal to y"
x > y means "x is greater than y"
x ≥ y means "x is greater than or equal to y"
x ≠ y means "x is not equal to y"
Like an equation, an inequality can be true or false.
34 − 12 > 5 + 2 is a true statement.
1 + 3 < 6 − 2 is a false statement.
1 + 3 ≤ 6 − 2 is a true statement.
1 + 3 ≠ 6 − 2 is a false statement.
−20 < −18 is a true statement.
Properties of inequalities
Let a, b and c be real numbers.
Firstly, the notation a < b means that a is smaller than b,
and the notation a > b means that a is greater than b.
Secondly, a ≤ b means that a is less than or equal to b,
and a ≥ b means that a is less than or equal to b.
1. Trichotomy Property
The following are true:
a < b
a = b
a > b
2. Transitive Property
If a < b and b < c then a < c
3. Additional Property
If a < b then a + c < b + c
4. Subtraction Property
If a < b then a - c < b - c
5. Multiplication Property
I. If a < b and c is positive then c X a < c X b
II. If a < b and c is negative c X a > c X b
Note:
1.If each inequality sign is reversed in the above properties, we obtain similar properties.
2.If the inequality sign < is replaced by <= (less than or equal) or the sign > is replaced by >= (greater than or equal), we also obtain similar properties.
-3
< 5 - 2x AND
5
- 2x < 9
-8 < -2x
4 > x
(-inf, 4)
-2x < 4
x > -2
(-2, inf)
Try It Yourselfs - Answers are found below
1. Solve the inequality:
10x - 8 > 4x + 10
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2. Solve the inequality:
-3(4x + 1) + 10 > -4(x - 3)
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3. Solve the double inequality:
-1 < -2(x - 3) - 3 < 7
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1. 10x - 8 > 4x + 10
= 10x - 4x > 10 + 8
= 6x > 18
= x > 3 #
2. -3(4x + 1) + 10 > -4(x - 3)
= -12x - 3 + 10 > -4x + 3
= -12x + 4x > 3 + 3 - 10
= -8x > -4
= -x > -1/2 [Invert the signs]
= x < 1/2 #
3. -1 < -2(x - 3) - 3 < 7
= -1 + 3 < -2x + 6
= 2 - 6 < -2x
= -4 < -2x
= -2 < -x [Invert the signs]
= x < 2 #
Who's behind all this?Where did we get all this from?The originals