A solution is to be kept between ${68}^{\mathrm{o}}\mathrm{F}$ and ${77}^{\mathrm{o}}\mathrm{F}$. What is the range in temperature in degree Celsius $\left(\mathrm{C}\right)$ if the Celsius/Fahrenheit $\left(\mathrm{F}\right)$conversion formula is given by $\mathrm{F}=\frac{9}{5}\mathrm{C}+32 ?$

Two real numbers or two algebraic expressions connected by the symbol $<$ (less than) or $>$ (greater than) or $\le$ (less than or equal) or $\ge$ (greater than or equal) form an inequality or inequation.

2. Types of Inequalities:

(i) Strict inequalities:

The inequality containing $<$ or $>$ is called strict inequality.

(ii) Slack inequalities:

The inequality containing ≤ or ≥ is called slack inequality.

(iii) Double Inequalities:

The inequality having two simultaneous inequalities is called double inequality.

(iv) Numerical inequalities:

Inequalities which do not contain any variable is called numerical inequalities i.e, relationship between numbers. Example: $3<5$

(v) Literal or variable inequalities:

Inequalities which contains variables are called literal inequalities i.e, relationship between variables or variable and number. Example: $x<5$

(vi) Linear Inequation of One Variable:

An inequality of the form $ax+b<0$ or $ax+b\le 0$ or $ax+b>0$ or $ax+b \ge 0,$ where $a,b\in R\left(a\ne 0\right)$ and $x$ is a real variable, is called linear inequality in one variable $x$.

(vii) Linear Inequation of Two Variables:

An inequality of the form $ax+by+c<0$or  $ax+by+c \le 0$ or $ax+by+c<0$ or $ax+by+c\ge 0,$ where $a,b,c\in R (a\ne 0,b\ne 0)$ and $x, y$ are real variables, is called linear inequality in two variables $x,y$.

(viii) Quadratic inequalities:

An inequality of the form $a{x}^2+bx+c<0$ or $a{x}^2+bx+c\le 0$ or $a{x}^2+bx+c>0$ or $a{x}^2+bx+c\ge 0,$where $a, b, c\in R(a\ne 0)$and $x$ is a real variable, is called quadratic inequality in one variable $x$.

3. Solution of an Inequality:

The value(s) of the variable(s) that make the inequality a true statement is called a solution of the given inequality. The set of all possible solutions of an inequality is called the solution set of the given inequality.

4. Solving Linear Inequations in One Variable: 

(i) Equal numbers can be added to both sides of an inequality without affecting the sign of inequality.

(ii) Equal numbers can be subtracted from both sides of an inequality without affecting the sign of inequality.

(iii) Both sides of an inequality can be multiplied (or divided) by the same positive number, without affecting the sign of inequality. But, when both sides are multiplied (or divided) by a negative number, then the sign of inequality is reversed.

5. Representation of Solution of Linear Inequality in One Variable on a Number Line:

To represent the solution of a linear inequality in one variable on a number line. We use the following algorithm.

(i) If the inequality involves $‘>’$ or $‘<’$ we draw an open circle $\left(O\right)$ on the number line, which indicates that the number corresponding to the open circle is not included in the solution set.

(ii) If the inequality involves $‘\ge ’$ or $‘\le ’$ we draw a dark circle $\left(.\right)$ on the number line, which indicates the number corresponding to the dark circle is included in the solution set.

6. Graphical Representation of the Solution of Linear Inequality in One or Two Variables:

To represent the solution of a linear inequality in one or two variables graphically in a plane, we use the following algorithm.

(i) If the inequality involves $‘>’\text{or} ‘<’$ we draw the graph of the line as a dotted line to indicate that the points on the line are not included from the solution sets.

(ii) If the inequality involves $‘\ge ’$ or $‘\le ’$ we draw the graph of the line as a dark line to indicate the points on the line is included from the solution sets.

(iii) Solution of a linear inequality in one variable can be represented on the number line as well as in the plane but the solution of a linear inequality in two variables of the type $ax+by>c,ax+by\ge c,ax+by<c\text{ or }ax+by\le c \left(a\ne 0,b\ne 0\right)$ can be represented in the plane only.