Class 11 Linear Inequalities notes
Linear Inequalities
# Equation
An equation is a statement involving variable or variable and the sign of equality(=).
For Example: 2x+4y=7, 7x-2= 5x+3
# Inequality
An inequality is a statement involving variable or variables and any one of the following signs of inequality:
(i) < (less than), (ii) > (Greater than); (iii) ≤ (Less than or equal to); (iv) ≥ (Greater than or equal to)
For Example: 3 < 5, 7/2x > 10, 5 ≤ y ≤ 10 etc.
# Numerical Inequality
An inequality involving numbers only is known as numerical inequality.
For Example: -4 < -3, 3 > 2 etc.
# Literal Inequality
An inequality involving one or more variables is known as literal inequality.
For Example: 4x ≥ 7, -3x <10 etc.
# Double Inequality
A system of inequalities, consisting of a pair of inequalities is known as a double inequality.
For Example: -3 ≤ 4x ≤ 9, 5 ≤ 7/2x ≤ 18 etc.
# Strict Inequality
An inequality Involving only '>' or '<' sign in known as strict inequality.
For example: 3x > 1, - 7y < 8 etc.
# Slack Inequality
An inequality involving only '≥ ' or '≤ ' sign is known as slack inequality.
For example: 5y ≥ 9, -8x ≤ -9 etc.
# Point to Remember
1. Two real numbers or two algebraic expressions related by the symbols <, >, ≥, ≤ a form an inequality.
2. Equal numbers may be added to (or subtracted from) both sides of an inequality.
3. Both sides of an inequality can be multiplied (or divided) by the same positive number. But when both the sides of an inequality are multiplied (or divided) by a negative number, then the sign of the inequality is reversed.
4. The value(s) of x which makes an inequality a true statement is/are called the solution(s) of the inequality.
5. if a, b in R and b ≠ 0 then
(i) ab > 0 or a/b >0 → a and b are of the same sign.
(ii) ab < 0 or a/b <0 → a and b are of opposite signs.
6. If a is any positive real number i.e. a > 0, then
(1) (i) |x| < a ↔ -a < x < a
(ii) |x| ≤ a ↔ -a ≤x ≤ a
(2) (i) |x| > a ↔ x < -a or x > a
(ii) |x| ≥ a ↔ x ≤ -a or x ≥ a
7. The region containing all the solutions of an inequality is called the solution region.
8. In order to identify the half plane represented by an inequality, it is sufficient to take any point (a, b) (not on the line) and check whether it satisfies the inequality or not. If it satisfies, then the inequality represents the corresponding half plane and shade the region which contains the point, otherwise, the inequality represents that half plane which does not contain the point within it. For convenience, (0, 0) is preferred.
9. In an inequality of the type ax + by ≤ c or ax + by ≥ c, points on the line ax + by = c are also included in the solution region. So, we draw a dark line in the solution region
10. In an inequality of the form ax + by > c or ax + by < c, the points on the line ax + by = c are not included in the solution region. So, we draw a broken or dotted line in the solution region.
For example: 3x - 4y < 12
