What Is an ‘Inequality’?
Let’s consider two cases:
Case 1:
\(n = 2\)
Case 2:
\(n\le 4\)
In the first case the solution to the equation is that the value of n will always be equal to 2. But in the second case, known as an inequality there are a range of values that are associated with the value of n. The values could be 4, 3,-10 and so on. The second case leads us to the definition of an inequality.
In math, an inequality is a relation in which the two sides of the equation are not equal.
There are four symbols used in inequality relations:
Hence, there are four symbols that are used in inequality relations.
So,
- If, a＞b, this indicates that ‘a’ is greater than ‘b’.
- If, a＜b, this indicates that ‘a’ is less than ‘b’.
- If, a ≥ b, this indicates that ‘a’ is greater than and equal to ‘b’.
- If, a ≤ b, this indicates that ‘a’ is lesser than and equal to ‘b’.
Definition of Writing an Inequality
Now that we have defined an inequality as a relation that is not equal on both sides, let’s look at writing an inequality. What it means to write an inequality is that we express the result of an inequality relation on a number line. Do note that the solution for inequality relations are expressed as as, \(x \lt 3, 3 \gt e \gt 8, d \le 15, t\ge 23\) and so on.
We always follow a specific method of representing inequalities on a number line.
In the image, there are inequality relations expressed on individual number lines. What we can observe is that for ‘greater than’ and ‘less than’, the circle above the number is unfilled. But for ‘greater than and equal to’ and ‘less than and equal to’ the circles above the numbers are filled.
[Note: An inequality may have just one solution satisfying it, or a range of solutions all of which satisfy the inequality. In the case of a range of solutions, we can call the range of solutions as a ‘solution set’.]
Rapid Recall
Solved Examples
Example 1: Represent the following inequalities on a number line:
1. \(x \ge – 10\)
2. \(y \lt 3\)
3. \(x \le – \frac{1}{2}\)
4. \(r \ge -\text{ }5.75\)
Solution
1. \(x \ge – 10\)
2. \(y \lt 3\)
3. \(x \le – \frac{1}{2}\)
4. \(r \ge – 5.75\)
Example 2: Solve the following inequalities and represent the result on a number line.
1. \(x \lt 4 – 8\)
2. \(2x + 2 \gt 16\)
3. \(x + 3.15\text{ }\ge – 6.15\)
Solution
- x < 4 – 8
x < 4 – 8 [Write the inequality]
x < -4 [Simplify]
The solution written on a number line is:
2. 2x + 2 > 16
2x + 2 > 16 [Write the inequality]
2x + 2 – 2 > 16 – 2 [Subtract 2 from both sides]
2x > 14 [Simplify]
x > 7 [Divide both sides by 2]
The solution written on a number line is:
3. \(x + 3.15 \ge 6.15\)
\(x + 3.15 \ge 6.15\) [Write the inequality]
\(x + 3.15 – 3.15 \ge \text{ }6.15 \text{ }- 3.15\) [Subtract 3.15 from both sides]
\(x \ge 3\) [Simplify]
The solution written on a number line is:
In algebra, an equation is a mathematical statement that shows that two expressions are equal.
The symbol for ‘not equal to’ is the equal sign with a diagonal across it.
Inequalities do contain variables, coefficients and constants. Hence both sides of an inequality can be simplified further, on the same lines as an algebraic equation.
A group of inequalities called ‘quadratic inequalities’ consist of variables raised to an exponent.