Solutions of Linear Inequalities in One Variable: The mathematical concept used to achieve maximum efficiency in the manufacturing of objects is the same as that used to derive the apt combinations of drugs to treat specific medical conditions. These are achieved using systems of linear inequalities. Many other real-life situations demand us to solve inequalities.

Its applications are so common that we often do not realise that we are solving algebraic problems like, is it okay to get up \(10\) minutes late and still be on time for class? How much would farewell gifts for friends cost without exceeding a set budget? What is the total cost to fill the fuel tank in a car? In this article, let us learn in detail about inequalities in one variable and how to solve them.

What are Linear Inequalities?

Inequalities are algebraic expressions that have an inequality symbol. The symbols of inequalities are listed as shown below:

Symbol of Inequality	Meaning
\(<\)	Lesser than
\(>\)	Greater than
\( \leqslant \)	Less than or equal to
\( \geqslant \)	Greater than or equal to
\( \ne \)	Not equal to

Linear inequalities are polynomials that have a degree of one. Example: For the linear polynomial in one variable \(ax\), the inequalities may be \(ax < 0,\,ax > 0,\,ax \leqslant 0,\,ax \geqslant 0,\) or \(ax \ne 0.\)
Here, \(x\) is the variable, and \(a\) is the coefficient and a real number.

Such inequalities are called linear inequalities in one variable. These can also include a constant \(b\), as it will not affect the degree of the polynomial. The inequalities in one variable with a constant will look like \(ax < b,\,ax > b,\,ax \leqslant b,\,ax \geqslant b,\) or \(ax \ne b.\)
Note that \(a\) and \(b\) are real numbers.

Solutions of Linear Inequalities in One Variable

The solution of a linear inequality is the real number value of the variable that makes the statement true. In any case, a linear inequality has either an infinite number of solutions or none at all. Inequalities with infinitely many solutions are plotted on a graph, a number line or represented using interval notation.

How to Solve Linear Inequalities in One Variable?

The inequalities can be solved almost the same way as equalities. The keyword being ‘almost’. The properties that are used to find the solution are:

1. Addition property with positive and negative integers
(i) If \(x < a\), then \(x + b < a + b\)
(ii) If \(x < a\), then \(x – a < a – b\)
2. Multiplication property with positive integers
(i) If \(x < a\), then \(x \times b < a \times b\)
(ii) If \(x < a\), then \(\frac {x}{b} < \frac {a}{b}\)
The difference with inequalities is when we multiply a negative integer, the signs should be reversed.
3. Multiplication property with negative integers
(i) If \(x < a\), then \(x\left( { – b} \right) > a\left( { – b} \right)\)
(ii) If \(x < a\), then \(\frac{x}{{\left( { – b} \right)}} > \frac{a}{{\left( { – b} \right)}}\)

Solutions on a Number Line

Similar to equalities, solutions to linear inequalities can also be marked on a number line as it involves only one variable. Shown below is the equality \(x = 3\) represented on a number line.

Let’s use the same inequalities and see what they look like on a number line.

\(x < 3\)	\(x \leqslant 3\)
\(x > 3\)	\(x \geqslant 3\)

In these examples, observe that:

1. Open circles represent strict inequalities.
2. Closed circles represent slack inequalities.
3. Greater than inequalities are marked to the right.
4. Lesser than inequalities are marked to the left.

How Does the Solution Look Like on a Graph?

As the name suggests, the graphs of linear equalities are always a straight line. For example, consider this equality \(x = 3.\)

So what happens if it is an inequality? For inequalities in one variable, the solution is a region that defined a part of the coordinate plane. Observe the given inequalities and their solution graphs.

\(x < 3\)	\(x > 3\)
\(x \leqslant 3\)	\(x \geqslant 3\)

The following conditions are satisfied for solutions of linear inequalities in one variable.

1. The solution set of a linear inequality is a shaded half-plane.
(a) Greater than inequalities have solution sets in the region above the line.
(b) Lesser than inequalities have solution sets in the region below the line.
2. Notice that the line that serves as a boundary on one side of the infinitely shaded solution region. The line is either solid or dotted.

Type of Line	Boundary	Inequality	Symbols	Solution Set
Dotted line	Non-inclusive boundary	Strict inequality	\(<\) or \(>\)	Line is not included in the solution set
Solid line	Inclusive boundary	Slack inequality	\(\leqslant\) or \(\geqslant\)	Line is included in the solution set

System of Linear Inequalities in One Variable

A system of inequalities includes more than one inequality that has the same variable. A system of linear inequalities in one variable will look as shown below.

\(\begin{array}{*{20}{c}} {{a_1}x}& < &{{b_1}}\\ {{a_2}x}& < &{{b_2}}\\ {{a_3}x}& < &{{b_3}}\\ \vdots &{}& \vdots \\ {{a_n}x}& < &{{b_n}} \end{array}\)

Here,
\({a_1},\;{a_2}, \ldots {a_n} \to \) coefficients of the system of linear inequalities
\({b_1},\;{b_2}, \ldots {b_n} \to \) constants of the system of linear inequalities
\(x \to \) variable of the system of linear inequalities
The system is represented using a curly bracket “{“.

How to Solve a System of Linear Inequalities in One variable?

The steps to solve a system of linear inequalities are:
Step 1: Solve the inequalities in the system separately.
Step 2: Write the solution in interval notation.
Step 3: Combine the intervals of the given inequalities.
Step 4: Represent the solution using a number line or a graph.

The graphical solution of a system of linear inequalities is the region common to all the inequalities in that system.

Solved Problems – Solutions of Linear Inequalities in One Variable

Q.1. Solve and graph the solution set on a number line.
\(3x + 7 < 16\)
Ans: \(3x + 7 < 16\)
\(3x + 7 – 7 < 16 – 7\) (subtracting \(7\) on each side)
\(3x < 9\)
\(\frac{{3x}}{3} < \frac{9}{3}\) (dividing \(3\) on each side)
\(x < 3\)

Q.2. Solve: \(2x + 7 < 7x – 4\)
Ans: \(2x + 7 < 7x – 4\)
\(2x – 7 < – 4 – 7\)
\(- 5x < – 11\)
\(5x > 11\)
\(x > \frac{{11}}{5}\) or \(x > 2.2\).

Q.3. Solve the system of linear inequalities in interval notation.
\(\{2x + 3 \geqslant 1 – x + 2 > – 1\)
Ans: Step 1: Solve the inequalities in the system separately.

\(2x + 3 \geqslant 1\)	\(- x + 2 > – 1\)
\(2x \geqslant 1 – 3\) (subtracting \(3\) on each side) \(2x \geqslant – 2\) \(x \geqslant -1\) (dividing \(2\) on each side)	\(- x > – 1 – 2\) (subtracting \(2\) on each side) \( – x > – 3\) \( x > 3\) (dividing \(- 1\) on each side, reverse the sign)

Step 2: Write the solution in interval notation.

Given	\(2x + 3 \geqslant 1\)	\(- x + 2 > – 1\)
Simplified Form	\(x \geqslant – 1\)	\(x < 3\)
Interval Notation	\([-1,\,\infty)\)	\((\infty,\,3)\)

Step 3: Combine the intervals of the given inequalities.
\(\left[ { – 1,\,\infty } \right)\; \cup \;\left( { – \infty ,\,3} \right) = \left[ { – 1,\,3} \right)\)
Therefore, the solution set is \([-1,\,3)\).

Q.4. Solve and show the solution set on a graph:
\(\left\{ {5\left( { – 5x – 2} \right) – 102 <  – 4\left( { – x + 4} \right) + 3x + \frac{6}{2} + x \le \frac{{20}}{{12}}} \right\}\)
Ans:

\(5\left( { – 5x – 2} \right) – 102 < – 4\left( { – x + 4} \right) + 3x\) \( – 25x – 10 – 102 < 4x – 16 + 3x\) \( – 25x – 112 < 7x – 16\) \( – 25x – 7x < – 16 – 112\) \( – 32x < – 128\) \(x > 4\)

\(\frac{6}{2} + x \leqslant \frac{{20}}{{12}}\) \(x \leqslant \frac{{20}}{{12}} – \frac{6}{2}\) \(x \leqslant \frac{{20}}{{12}} – \frac{{36}}{{12}}\) \(x \leqslant \frac{{ – 16}}{{12}}\) \(x \leqslant \frac{{ – 4}}{3}\)

Graphical representation of the system of inequalities is

Observe that there is no common region for the two inequalities.
Hence, the given system of inequalities has no solution.

Q.5. In four exam papers, each of \(100\) marks, Rahul scored \(83,\,73,\,72,\,95\) marks. He intends to have an average above or equal to \(75\) marks and less than \(80\) marks. What should be the range of marks in his fifth exam paper for his desired average?
Ans: Let \(x\) be Rahul’s score in the fifth paper.
Average of \(5\) papers \( = \frac{{83 + 73 + 72 + 95 + x}}{5}\) ……(1)
Rahul’s desired range of average is defined as \(75 \leqslant\) average \(< 80\) ………(2)
Combining (1) and (2), we get,
\(75 \leqslant \frac{{323 + x}}{5} < 80\)
\(375 \leqslant 323 + x < 400\) (multiplying \(5\) on each side)
\(52 \leqslant x < 77\)
Hence, Rahul should score exactly or more than \(52\) and less than \(77\) marks in the fifth paper.

Summary

This article helps understand linear inequalities by listing the different symbols of inequalities. It defines and elaborates linear inequalities in one variable and a system of linear inequalities. The solution of a linear inequality is a half-plane, and the solution of a system is the intersecting regions of the inequalities in that system. We also learn how to plot the given inequalities on a number line, and using graphs.

Then, the article goes on to explain the steps involved to solve linear inequalities and a system of linear inequalities. We also learn some real-life applications and methods to solve linear inequalities using solved problems.

Solutions of Linear Inequalities in Two Variables

Frequently Asked Questions (FAQs)

Q.1. What is linear inequality and examples?
Ans: Linear inequalities are two or more expressions combined using inequality symbols such as \( > ,\; \geqslant ,\; < ,\, \leqslant \) or \(\ne\). Inequalities are said to be linear if the degree of the expression is one.
Example: \(5 < 15,\;a > 3,\;x + y \leqslant 3y\).

Q.2. What is a linear inequality in one variable?
Ans: Linear inequalities are polynomials that have a degree of one. Example: For the linear polynomial in one variable \(ax\), the inequalities may be \(ax < 0,\;ax > 0,\;ax \leqslant 0,\;ax \geqslant 0,\) or \( ax \ne 0.\)
Here, \(x\) is the variable, and \(a\) is the coefficient and a real number.
Such inequalities are called linear inequalities in one variable. These can also include a constant \(b\), as it will not affect the degree of the polynomial. The inequalities in one variable with a constant will look like \(ax < b,\;ax > b,\;ax \leqslant b,\;ax \geqslant b,\) or \(ax \ne b.\)
Note that \(a\) and \(b\) are real numbers.

Q.3. How do you solve linear inequalities with one variable?
Ans: The steps to solve a system of linear inequalities are:
Step 1: Solve the inequalities in the system separately.
Step 2: Write the solution in interval notation.
Step 3: Combine the intervals of the given inequalities.
Step 4: Represent the solution using a number line or a graph.
The graphical solution of a system of linear inequalities is the region common to all the inequalities in that system.

Q.4. What is the difference between linear equation and inequality in one variable?
Ans: A linear equation is two mathematical expressions combined by an equal \((=)\) sign. This represents that the left-hand side is equal to the right-hand side in that equation. In a linear inequality, although it has two expressions, an inequality symbol is used. The symbols of inequality are \( > ,\; \geqslant ,\; < , \leqslant ,\) and \(\ne.\)

Q.5. Which point would be a solution to the system of linear inequalities?
Ans: The solution to linear equality is an infinite number of ordered pairs. Likewise, the solution set is a region defined by a part of the coordinate plane to a linear inequality. This is also the half-plane. The solution set of a system of inequalities is the intersecting region of the solution sets of the inequalities in that system. Hence, any and all points that lie in this intersecting region could be a solution to the system of linear inequalities.
Example:

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