System of Inequalities: Definition, Types, Solution

System of inequalities: A system of inequalities is a collection of two or more inequalities in one or more variables. When a situation necessitates various solutions, and there are multiple constraints on those solutions, systems of inequalities are used.

A system of linear inequalities is typically used to find the optimum solution. This answer could be as straightforward as estimating how many units of a product should be manufactured to maximise profit or as complex as determining the best medicine combination to give a patient. This article will discuss the types of inequalities and methods to find their solutions.

What is an Inequality?

Any statement involving variable(s) and signs of inequality such as \(>,<, \geq\) and \(\leq\) is called an inequality or an inequation.

Example: \(3 x-2<0,5 x-3>10\) and \(x^{2}-3 x+2 \leq 0\) are inequalities or inequations.

Linear Inequality

An inequality that involves a linear function i.e., a function of the form, \(f(x)=a+b x\), where \(b \neq 0\) and \(a, b \in R\) or a function of the form, \(g(x, y)=a x+b y+c\) where both \(a\) and \(b\) together cannot be zero, and \(a, b \in R\) is called a linear inequality. Note that an inequality contains one of the signs of inequality.

Example: \(3 x+2 y+6>0,2 x+5>7 x+8\)

Linear Inequalities in One Variable

Let \(a\) be a non-zero real number, and \(x\) be a variable. Then inequalities of the form \(ax + b < 0,ax + b \le 0,ax + b \ge 0\) and \(a x+b>0\) are linear inequalities in one variable.

Linear Inequalities in Two Variables

Let \(a,b\) are two non-zero real numbers and \(x,y\) are variables. Then, inequalities of the form \(ax + by + c < 0,ax + by + c \le 0,ax + by + c > 0,ax + by + c \ge 0\) are linear inequalities in two variables \(x\) and \(y.\)

Quadratic Inequalities in One Variable

Let \(a\) be a non-zero real number, and \(x\) be a variable. Then inequalities of the form \(a{x^2} + bx + c < 0,a{x^2} + bx + c > 0,a{x^2} + bx + c \le 0\) and \(a x^{2}+b x+c \geq 0\) are known as quadratic inequalities in one variable \(x\).

Solution of an Inequality

Solution of inequality is the value(s) of the variables(s) that makes it a true statement For example consider \(x+3>5\)

This inequality will become a true statement if we give any real value greater than \(2\) for \(x\).

\(\therefore\) The solution of this inequality is \(x>2\)

System of Linear Inequalities

A system of inequalities is a collection of two or more inequalities with one or more variables.

A system of linear inequalities appears like this:

\(\begin{array}{{c}}
{{a_{11}}{x_1}} & + & {{a_{12}}{x_2}} & + & {{a_{13}}{x_3}} & + & \ldots & + & {{a_{1n}}{x_n}} & < & {{b_1}} \\
{{a_{21}}{x_1}} & + & {{a_{22}}{x_2}} & + & {{a_{23}}{x_3}} & + & \ldots & + & {{a_{2n}}{x_n}} & < & {{b_2}} \\
{{a_{31}}{x_1}} & + & {{a_{32}}{x_2}} & + & {{a_{33}}{x_3}} & + & \ldots & + & {{a_{3n}}{x_n}} & < & {{b_3}} \\
\vdots & {} & \vdots & {} & \vdots & {} & {} & {} & \vdots & {} & \vdots \\
{{a_{m1}}{x_1}} & + & {{a_{m2}}{x_2}} & + & {{a_{m3}}{x_3}} & + & \ldots & + & {{a_{mn}}{x_n}} & < & {{b_m}} \\
\end{array} \)

Here

\(a_{11}, a_{12}, a_{13}, \ldots\) are coefficients of the linear inequality system

\(x_{1}, x_{2}, x_{3}, \ldots\) are variables of the linear inequality system

\(b_{1}, b_{2}, b_{3}, \ldots\) are constants of the linear inequality system.

Solution of System of Linear Inequalities in One Variable

The intersection of the solutions of each inequality is the solution of the system of inequalities.

For Example: Let us find the solution of the following system of inequalities

\(x>3\)

\(x<6\)

Solution: The solution set of inequality \(x>3\) is the set of all the real numbers greater than \(3\) i.e….\((3, \infty)\). It can be represented in the number line by a ray as follows:

And the solution set of system of inequality, \(x<6\) is the set of all real numbers less than \(6\) i.e.., \((-\infty, 6)\) We can represent it in the number line by a ray as shown below

We can observe that the region included in both the above number line is \(3<x<6\) i.e.., numbers satisfying both the given inequalities lies in the interval \((3,6)\).

Hence, the solution of the given system of inequalities is \((3,6)\),

Steps to Solve a System of Linear Inequalities in One Variable

Step1: Find the solution of each inequality.

While solving the linear inequalities, we use simplifications which are governed by the following rules:

• If we add or subtract both sides of an inequality by the same number, the inequality will not be changed.
• If we multiply both sides of the inequality by the same number, the inequality will not be changed.
• The sign of the inequality will be reversed when both sides of the inequality are multiplied or divided by the same negative number. 
• Any term of the inequality can be taken to other side with its sign changed, without affecting the inequality.

Step2: Represent the solutions on a number line.

The solution of inequalities in one variable can be represented as rays on the number line.

• We use an open dot to show that the ray’s endpoint is not part of the solution if the inequality is “strict” \((<\) or \(>)\)
• If inequalities are slack \((\leq\) and \(\geq)\) we use a closed dot to indicate that the endpoint of the ray is a part of the solution

Step3: Find the intersection of the solutions.

The intersection is the solution of the system of inequalities.

Graphical Solutions

The region of the plane containing all the points whose coordinates satisfy a given inequality is called the solution region (or graph) of the inequality.

Representation of a System of Linear Inequalities in One Variable

The following steps are followed to represent the system of linear inequalities in one variable.

Step 1: Represent each inequality as rays on the number line.

Step 2: We use an open dot to show that the ray’s endpoint is not part of the solution if the inequality is “strict” \((<\) or \(>)\).

If inequalities are slack \((\leq\) and \(\geq)\) we use a closed dot to indicate that the endpoint of the ray is a part of the solution.

Step 3: Darkened the region representing the solution of each inequality.

Step 4: Identify the region satisfied by all inequalities or identify the region common in all the number lines.

Step 5: The region satisfied by all the inequalities is the required solution of the given system of inequalities.

Representations of System of Linear Inequalities Involving Two Variables

We know that a line divides the Cartesian plane into two halves, each of which is referred to as a half plane. A vertical line divides the plane into left and right planes and a non-vertical line divides a plane into lower and upper plane.

A point in the Cartesian plane will lie either on the line or on the half-planes. We can understand this with the help of the following example.

The graph of the equation \(x+y=3\) is given below

We can check that any point on the line \(x+y=3\) satisfies the equation.

Now, let us consider the cases of inequalities

\(x+y>3\)	\(x+y<3\)
\(x+y \geq 3\)	\(x+y \leq 3\)

The solution set is a shaded half-plane. We can observe that the example has an inequality of form, \(a x+b y>c, a x+b y \geq c, a x+b y<c, a x+b y \leq c\) where, \(a, b>0.\) And, for the signs of inequality \(>, \geq\), the solution region is the above the line. And, for the sign of inequality \(<, \leq\) the solution region is below the line.

Observe that we have used either solid or dotted lines. This line serves as a boundary on one side, while the region extends infinitely on other sides. We can also check that any point lying on these shaded regions satisfies the corresponding inequalities.

In order to find the solution region of the system of inequalities, first, we convert the inequalities into equations of the form \(a x+b y=c\). The graph of the equations is a line.

 If the inequality contains \(\leq\) or \(\geq\) draw the solid line for the equation. In the case of strict inequality \((<\) or \(>)\), draw the dotted line for the equations representing the given inequality. Further, find the region represented by the given inequalities by substituting the coordinates of any point not lying on the line, if the inequality is satisfied, shade the region containing that point, otherwise shade the opposite region. Repeat this for each inequalities, and the common shaded region will be the required solution.

Steps Involved

The algorithm for solving a system of linear inequalities involving two variables on a graph is as follows.

Step 1: Convert all inequalities into equations of the form \(a x+b y=c\)

Step 2: The graph of the equations is a line.

Step 3: Find two points on each line and mark them in the Cartesian plane.

Step 4: If the inequalities contains \(\leq\) or \(\geq\), join the points by a solid line. In the case of strict inequality \((<\) or \(>)\), join the points by dotted line.

Step 5: If possible, choose a point, not on the line preferably \((0,0)\)

Step 6: In the inequalities, substitute this point \((0,0)\) Shade the half-plane region containing the point \((0,0)\) if the inequality is satisfied  Otherwise, shade the area that does not include the specified point.

Step 7: Find the common part of the coordinate plane which satisfies all the given linear inequalities.

Step 8: This common part of the coordinate plane is the required solution of the given inequalities.

Solved Examples – System of Inequalities

Q.1. Solve the following system of linear inequalities:
\(3 x-6 \geq 0\)
\(4 x-10 \leq 6\)
Ans: Given:
\(3 x-6 \geq 0\) ……(i)
\(4 x-10 \leq 6\) ……(ii)
Now, \(3 x-6 \geq 0\)
\(\Rightarrow 3 x \geq 6\)
\(\Rightarrow \frac{3 x}{3} \geq \frac{6}{3}\)
\(\Rightarrow x \geq 2\)
\(\therefore\) Solution set of inequality (i) is \([2, \infty)\).

Now, consider
\(4 x-10 \leq 6\)
\(\Rightarrow 4 x \leq 16\)
\(\Rightarrow x \leq 4\)
\(\therefore\) Solution set of inequality (ii) is \((-\infty, 4]\)

The intersection of these solution sets is the set \([2,4]\)

Hence, the solution set of the given system of inequalities is the interval \([2,4]\)

Q.2. Solve the following system of inequalities.
\(\frac{5 x}{4}+\frac{3 x}{8}>\frac{39}{8}\)
\(\frac{2 x-1}{12}-\frac{x-1}{3}<\frac{3 x+1}{4}\)
Ans: The given system of inequalities is
\(\frac{5 x}{4}+\frac{3 x}{8}>\frac{39}{8}\) …..(i)
\(\frac{2 x-1}{12}-\frac{x-1}{3}<\frac{3 x+1}{4}\)……(ii)
Now, \(\frac{5 x}{4}+\frac{3 x}{8}>\frac{39}{8}\)
\(\Rightarrow \frac{10 x+3 x}{8}>\frac{39}{8}\)
\(\Rightarrow 13 x>39\)
\(\Rightarrow x>3\)
So, the solution set of inequality (i) is the interval \((3, \infty)\)

Now,  consider
\(\frac{2 x-1}{12}-\frac{x-1}{3}<\frac{3 x+1}{4}\)
\(\Rightarrow \frac{(2 x-1)-4(x-1)}{12}<\frac{3 x+1}{4}\)
\(\Rightarrow \frac{-2 x+3}{12}<\frac{3 x+1}{4}\)
\(\Rightarrow-2 x+3<3(3 x+1)\) [Multiplying both sides by \(12\) ]
\(\Rightarrow-2 x+3<9 x+3\)
\(\Rightarrow-2 x-9 x<3-3\)
\(\Rightarrow-11 x<0\)
\(\Rightarrow x>0\) [Dividing both sides by \(-11\) ]
\(\Rightarrow x \in(0, \infty)\)
Thus, the solution set of inequality (ii) is the interval \((0, \infty)\)

Observe that the intersection of the solution sets of inequalities (i) and (ii) is the interval \((3, \infty)\)
Hence, the solution set of the given system of inequalities is the interval \((3, \infty)\).

Q.3. Exhibit graphically the solution set of the linear inequalities
\(3 x+4 y \leq 12,4 x+3 y \leq 12, x \geq 0, y \geq 0\)
Ans: Converting the inequalities into equations, we get
\(3x + 4y = 12,\;\,4x + 3y = 12,\,x = 0\) and \(y = 0\)
Region Represented by \(3 x+4 y \leq 12\):
The portion containing the origin represents the solution set of the inequality \(3 x+4 y \leq 12\)

Region Represented by \(4 x+3 y \leq 12\) :
The region containing the origin is represented by the inequality \(4 x+3 y \leq 12\)

Region Represented by \(\boldsymbol{x} \geq \mathbf{0}\) and \(y \geq 0\)
\(x \geq 0\) and \(y \geq 0\) represent the first quadrant.

Hence, the shaded region in the figure below represents the solution set of the given linear inequations.

Q.4. Draw the diagram of the solution set of the linear inequalities
\(3 x+4 y \geq 12\)
\(y \geq 1\)
\(x \geq 0\)
Ans: Converting the given inequalities into equations, we get
\(3 x+4 y=12\)
\(y=1\)
\(x=0\)
Region Represented by \(3 x+4 y \geq 12\):
Since \((0,0)\) does not satisfy the inequality \(3 x+4 y \geq 12\) the portion of the graph that does not contain the origin is represented by the inequality \(3 x+4 y \geq 12\).

Region Represented by \(y \geq 1:\)

Region Represented by \(x \geq 0\):

The solution set of the given linear constraints is the intersection of the above regions as shown in the figure given below.

Q.5. Exhibit graphically the solution set of the linear inequalities
\(x+y \leq 5\)
\(4 x+y \geq 4\)
\(x+5 y \geq 5\)
\(x \leq 4\)
\(y \leq 3\)
Ans: Converting the inequations into equations, we have
\(x+y=5\)
\(4 x+y=4\)
\(x+5 y=5\)
\(x=4\)
\(y=3\)
Region Represented by \(x+y \leq 5\):
The solution set of the inequation \(x+y \leq 5\) is represented by the region containing the origin.

Region Represented by \(4 x+y \geq 4\):
Since \((0,0)\) does not satisfy the inequation \(4 x+y \geq 4\), the portion of the graph that does not contain the origin is represented by the inequation \(4 x+y \geq 4\)

Region Represented by \(x+5 y \geq 5\).
Notice that \((0,0)\) does not satisfy the inequation \(x+5 y \geq 5\). Hence, the solution set of the given inequation is represented by the region not containing the origin.

Region Represented by \(x \leq 4\):
The region represented by \(x \leq 4\) is the portion on the left side of \(x=4\).

Region Represented by \(y \leq 3\):
The portion containing the origin is represented by the  given inequation.

The solution set of the given linear inequalities is represented by the common region of the above five regions, as illustrated in the picture below.

Summary

A set of two or more inequalities in one or more variables is known as a system of inequalities. The intersection of the solutions of each inequality is the solution of the system of inequalities. We follow the graphical method to find the solution of a system of linear inequalities in two variables. The solution of linear inequalities in one variable is obtained by simplifying each inequality by using its rules to solve it and then using a graphical method to find the interval in which the variable lies. Recall that if the inequality contains \(\leq\) or \(\geq\). Join the points of the required line by a solid line. In the case of strict inequality \((<\) or \(>)\), join the points by a dotted line. In the inequality, substitute this point \((0,0)\). Shade the half-plane region containing the point \((0,0)\) if the inequality is satisfied. Otherwise, shade the area that does not include the specified point. The shaded region is the required solution region.

Frequently Asked Questions (FAQs)

Q.1. How do you solve systems of inequalities?
Ans: When solving a system of linear inequalities graphically we will follow these steps: 
Step 1: For \(y\) solve the inequality.
Step 2: Turn the inequality into a linear equation and graph the line as a solid or dashed line, depending on the inequality sign. 
Step 3: Fill in the shaded area with the region that best satisfies the inequality.
Step 4: For each inequality, repeat steps \(1-3\)
Step 5: The solution set will be the region where all inequalities are overlapped.

Q.2. What is a system of equations or a system of inequalities?
Ans: A set of equations with the same variables is a system of equations. A system of inequalities is nearly identical to an equation system. Only you are dealing with inequalities rather than equations.

Q.3. How many solutions are there to a system of inequalities?
Ans: There are many solutions to a linear system of inequalities. When graphing a linear inequality, keep in mind that the solution is a shaded region of the graph that contains all of the inequality’s possible solutions. There are two or more linear inequalities in a system.

Q.4. Do all systems of inequalities have solutions? 
Ans: Sometimes, there are no solutions to a system of inequalities. When the borderlines are parallel, this can happen.
Consider the following system:
\(y \geq 2.5 x+4\)
\(y<2.5 x\)
A graph of this system is shown below.

There is no solution set since the bounded regions of the two inequalities do not coincide.

Q.5. What is an example of a system of inequalities in real life?
Ans: Consider the following scenarios: highway speed restrictions, minimum credit card payments, the number of text messages you can send per month from your phone, and the time it will take you to get from home to school. Mathematical inequalities can be used to represent all of them.

Learn about Properties of Inequalities

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