Linear Inequalities
We are familiar with equations in which two sides of a statement (L.H.S. and R.H.S.) are connected by equality sign (=).
We may get certain statements involving a sign $<$ (less than), $>$ (greater than), $\leq$ (less than or equal), $\geq$ (greater than or equal) which are known as inequalities.
Example: The weight of all the students in my class is less than 45kg (weight<45)
Numerical inequalities: $2<3$, $7>4$
Literal inequalities: $x<9$, $y>12$, $x\geq5$, $y\leq12$
Double inequalities: $3<5<7$, $4\leq x<5$, $2<y\leq4$
Strict inequalities: $ax+b<0$, $ax+b>0$, $ax+by<c$, $ax+by>c$, $ax^{2}+bx+c>0$, $ax^{2}+bx+c<0$ [involves only $<$ and $>$]
Slack inequalities: $ax+b\leq0$, $ax+b\geq0$, $ax+by\leq c$, $ax+by\geq c$, $ax^{2}+bx+c\geq0$, $ax^{2}+bx+c\leq0$ [involves only $\leq$ and $\geq$]
Linear inequalities in one variable x when $a\neq0$: $ax+b<0$, $ax+b>0$, $ax+b\leq0$, $ax+b\geq0$
Linear inequalities in two variables x and y when $a\neq0$, $b\neq0$: $ax+by<c$, $ax+by>c$, $ax+by\leq c$, $ax+by\geq c$
Quadratic inequalities in one variable x when $a\neq0$: $ax^{2}+bx+c>0$, $ax^{2}+bx+c<0$, $ax^{2}+bx+c\geq0$, $ax^{2}+bx+c\leq0$
Solutions of Linear Inequalities in One Variable
Consider the situation: Ravi goes to market with Rs. 200 to buy rice, which is available in packets of 1kg. the price of one packet of rice is Rs 30. If x denotes the number of packets of rice, which he buys, then total amount spent by him is Rs 30x. Since, he has to buy rice in packets only, he may not be able to spend the entire amount of Rs 200. Hence 30x<200.
30x<200
For x=0, 30(0)=0<200, true
For x=1, 30(1)=30<200, true
For x=2, 30(2)=60<200, true
For x=3, 30(3)=90<200, true
For x=4, 30(4)=120<200, true
For x=5, 30(5)=150<200, true
For x=6, 30(6)=180<200, true
For x=7, 30(7)=210<200, false
The values of x (0,1,2,3,4,5,6), which makes above inequality a true statement, are called solutions of inequality and the set {0,1,2,3,4,5,6} is called its solution set.
Thus, any solution of a inequality in one variable is a value of the variable which makes it a true statement.
Algebra of Linear Inequalities
Let a and b be two real numbers.
1. Addition or Subtraction
Equal numbers may be added to (or subtracted from) both sides of an inequality.
If $a>b$, then
$a+c>b+c$
or $a-c>b-c$
2. Multiplication or Division
Both sides of an inequality can be multiplied (or divided) by the same positive number, the sign of inequality remains the same.
But when both sides are multiplied or divided by a negative number, then the sign of inequality is reversed.
(i) If $a>b$ and $c>0$, then
$ac>bc$ and $\frac{a}{c}> \frac{b}{c}$
(ii) If $a>b$ and $c<0$, then
$ac<bc$ and $\frac{a}{c}< \frac{b}{c}$
3. Transpose
Any number can be transposed from one side of an inequality to the other with the sign of the transposed number reversed.
$a+b>c$ ⇒ $a>c-b$.
Graphical Representation
(i) Closed Interval
If a and b are real numbers, then $a\leq x\leq b$ is called a closed interval and is denoted by $[a,b]$.
(ii) Open Interval
If a and b are real numbers, then $a<x<b$ is called an open interval and is denoted by $(a,b)$.
(iii) Semi-open or Semi-closed Interval
If a and b are real numbers, then $a<x\leq b$ is called semi-open, denoted by $(a,b]$ and $a\leq x<b$ is called semi-closed,denoted by $[a,b)$.
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We are familiar with equations in which two sides of a statement (L.H.S. and R.H.S.) are connected by equality sign (=).
We may get certain statements involving a sign $<$ (less than), $>$ (greater than), $\leq$ (less than or equal), $\geq$ (greater than or equal) which are known as inequalities.
Example: The weight of all the students in my class is less than 45kg (weight<45)
Numerical inequalities: $2<3$, $7>4$
Literal inequalities: $x<9$, $y>12$, $x\geq5$, $y\leq12$
Double inequalities: $3<5<7$, $4\leq x<5$, $2<y\leq4$
Strict inequalities: $ax+b<0$, $ax+b>0$, $ax+by<c$, $ax+by>c$, $ax^{2}+bx+c>0$, $ax^{2}+bx+c<0$ [involves only $<$ and $>$]
Slack inequalities: $ax+b\leq0$, $ax+b\geq0$, $ax+by\leq c$, $ax+by\geq c$, $ax^{2}+bx+c\geq0$, $ax^{2}+bx+c\leq0$ [involves only $\leq$ and $\geq$]
Linear inequalities in one variable x when $a\neq0$: $ax+b<0$, $ax+b>0$, $ax+b\leq0$, $ax+b\geq0$
Linear inequalities in two variables x and y when $a\neq0$, $b\neq0$: $ax+by<c$, $ax+by>c$, $ax+by\leq c$, $ax+by\geq c$
Quadratic inequalities in one variable x when $a\neq0$: $ax^{2}+bx+c>0$, $ax^{2}+bx+c<0$, $ax^{2}+bx+c\geq0$, $ax^{2}+bx+c\leq0$
Solutions of Linear Inequalities in One Variable
Consider the situation: Ravi goes to market with Rs. 200 to buy rice, which is available in packets of 1kg. the price of one packet of rice is Rs 30. If x denotes the number of packets of rice, which he buys, then total amount spent by him is Rs 30x. Since, he has to buy rice in packets only, he may not be able to spend the entire amount of Rs 200. Hence 30x<200.
30x<200
For x=0, 30(0)=0<200, true
For x=1, 30(1)=30<200, true
For x=2, 30(2)=60<200, true
For x=3, 30(3)=90<200, true
For x=4, 30(4)=120<200, true
For x=5, 30(5)=150<200, true
For x=6, 30(6)=180<200, true
For x=7, 30(7)=210<200, false
The values of x (0,1,2,3,4,5,6), which makes above inequality a true statement, are called solutions of inequality and the set {0,1,2,3,4,5,6} is called its solution set.
Thus, any solution of a inequality in one variable is a value of the variable which makes it a true statement.
Algebra of Linear Inequalities
Let a and b be two real numbers.
1. Addition or Subtraction
Equal numbers may be added to (or subtracted from) both sides of an inequality.
If $a>b$, then
$a+c>b+c$
or $a-c>b-c$
2. Multiplication or Division
Both sides of an inequality can be multiplied (or divided) by the same positive number, the sign of inequality remains the same.
But when both sides are multiplied or divided by a negative number, then the sign of inequality is reversed.
(i) If $a>b$ and $c>0$, then
$ac>bc$ and $\frac{a}{c}> \frac{b}{c}$
(ii) If $a>b$ and $c<0$, then
$ac<bc$ and $\frac{a}{c}< \frac{b}{c}$
3. Transpose
Any number can be transposed from one side of an inequality to the other with the sign of the transposed number reversed.
$a+b>c$ ⇒ $a>c-b$.
Graphical Representation
(i) Closed Interval
If a and b are real numbers, then $a\leq x\leq b$ is called a closed interval and is denoted by $[a,b]$.
If a and b are real numbers, then $a<x<b$ is called an open interval and is denoted by $(a,b)$.
If a and b are real numbers, then $a<x\leq b$ is called semi-open, denoted by $(a,b]$ and $a\leq x<b$ is called semi-closed,denoted by $[a,b)$.
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