- Order relation is an important comparing property of ordering real numbers.
- This order relation helps us to compare two real numbers ‘a’ and ‘b’ when a ≠ b.
- This comparability is of primary importance in many applications.
- We may compare prices, heights, weights, temperatures, distances, costs of manufacturing, distances, time etc.
- The inequality symbols < and > were introduced by an English mathematician Thomas Harriot (1560 — 1621)
Defining inequalities:
- Let a, b be real numbers. Then a is greater than b if the difference a – b is positive and we denote this order relation by the inequality a> b.
- An equivalent statement is that in which b is less than a, symbolized by b< a Similarly, if a – b is negative, then a is less than b and expressed in symbols as a < b.
- Sometimes we know that one number is either less than another number or equal to it. But we do not know which one is the case.
- In such a situation we use the symbol which is read as “less than or equal to”. Likewise, the symbol is used to mean “greater than or equal to”.
- The symbols < , >, and > are also called inequality signs.
- The inequalities x > y and x < y are known as strict (or strong) whereas the inequalities where as x y and y x are called non-strict (or weak).
- If we combine a < b and b < c we get a double inequality written in a compact form as a < b < c which means “b lies between a and c” and read as “a is less than b less than c” Similarly, “a < b < c” is read as “b is between a and c, inclusive.”
- A linear inequality in one variable x is an inequality in which the variable x occurs only to the first power and has the standard form ax + b < 0, a ≠ 0 where a and b are real numbers. We may replace the symbol < by >, < or > also.
Solving Linear Inequalities
You need to know the rules of solving inequalities and learn how to solve by looking at these examples:
Example 1: Solve 9 – 7x > 19 – 2x, where x ∈R.
9 – 7x > 19 – 2x
9–5x>19 …… (Adding 2x to each side)
–5x > 10 …… (Adding –9 to each side)
x < – 2 …… (Multiplying each side by -1/5)
Hence the solution set = {x | x < – 2}
Example 2: Solve: 7x+3<5x+9 where x ∈R.
7x+3-5x < 5x+9-5x
2x+3 <9
2x < 9-3
2x < 6
x < 3
Hence the solution set = {x | x < 3}
Example 3: Solve the inequality 4x – 1 < 3 < 7 + 2x, where x ∈ R.
The given inequality holds if and only if both the separate inequalities 4x – 1 < 3 and 3 < 7 + 2x hold. We solve each of these inequalities separately and get the solution set = {x | -2 ≤ x ≤ 1}.
Questions for review:
- Define linear inequalities with examples.
- What is a double inequality?
- What are trichotomy and transitive properties of inequalities?
- What are additive closure and multiplicative properties?
- What do you need to know to solve inequalities?
- Solve the inequality:
- 3x+1<5x–4
- 4x–10.3<21x–1.8
- 3(2x+1)–2(2x+5)<5(3x–2)
Need more help with solving inequalities?
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