What are Linear Inequalities

Linear functions include linear equations and inequalities. Linear inequalities are algebraic expressions which contain an inequality sign such as greater than ">", less than "<", greater than equal to "\geq" or less than equal to "\leq". Linear inequalities with one variable contain a single variable only. Unlike the solution set of a linear equation in two variables which is given in ordered pair that satisfies both inequalities, linear inequality with one variable is given in interval notation. 
We know that the graph of the linear inequality in two variables is a straight line. The region which represents the solution set is shaded. On the other hand, solution of the linear inequalities in one variable are represented on a number line. When the values from the solution set of the system of inequalities in one variable are substituted in each inequality, the solution set is true for each inequality.

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System of Linear Inequalities

We write systems of linear inequalities just like systems of equations using a curly bracket "{". The general startegy for solving inequalities is same as solving equations, however, the solution set of a system of inequalities is represented in a different way than the system of linear equations. Each linear inequality is solved separately and their solution sets are united to give the final solution of the entire system. The solution set is written either in parenthesis or square brackets. In this article, we will see how to solve the system of linear inequalities, write their solutions in interval notation and represent the solutions on the number line.

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Example 1

Solve the following system of linear inequalities with one variable. Write the solution set in interval notation and represent the system on a number line.

\begin{cases} 2x+3 \geq 1 \\ -x+2\geq -1 \end{cases}

Solution

To find the solution set, we will solve the above inequality in multiple steps.

Step 1-Solving the inequalities in the system separately

The first inequality is 2x+3 \geq 1. Solve it algebraically like you solve linear equation.

2x+3 \geq 1

Subtraction of 3 from both sides of the inequality will give the following expression:

2x+3-3 \geq 1-3

2x \geq -2

To isolate x on the left hand side of the inequality, both sides of the inequality will be divided by 2.

x \geq -1

In words, we can express this inequality as x is greater than or equal to -1. Let us solve the second inequality of the system now.

-x+2\geq-1

Subtract -2 from both sides of the inequality to get the following expression:

-x+2-2\geq-1-2

-x\geq-3

Now, do you remember the rule of the inequality which says that if both sides of the inequality are multiplied by negative numbers, the sign of the inequality is reversed. Since the variable x has a negative sign associated with it, so multiplying both sides of the inequality by -1 will give the following result:

x\leq 3

We can see the multiplication of both sides by negative 1 has flipped the sign of inequality to less than or equal to.

Step 2 - Writing the solution set of both the inequalities in interval notation separately

Before writing the system of linear inequalities in interval notation, first, write each inequality in interval notation separately. After simplifying the first inequality 2x+3 \geq 1 we got x \geq -1. In interval notation it will be written as:

[-1,\infty)

The simplified form of the second inequality -x+2\geq-1 was x\leq 3. In interval notation it will be written as:

(\infty,3]

Step 3- Combining the solution set in interval notation of both inequalities

Now, we will unite the solution set of both the inequalities. The solution sets of both the inequalities are [-1,\infty) and (\infty,3]. We will unite these like this:

[-1,\infty) U (\infty,3]

Now, we will take the smallest beginning point and largest endpoint from both the intervals. The smallest beginning point is -1 and the largest endpoint is 3, so in interval notation, the final solution will be written as:

[-1,3]

You can see that we have used square brackets on both sides of the beginning and endpoint because -1 and 3 were part of the solution set. It also shows that [-1,3] is a closed interval.

Step 4-Representing the solution set on the number line

Now, the final step is to represent the solution set on the number line. You can either directly represent the solution set on the number line, or first, represent both inequalities separately as shown below. You can see that we have used closed circles for -1 and 3 because these two numbers were part of the solution set.

Example 1 - Solutions set on a number line

 

Example 2

Solve the following system of linear inequalities in one variable. Write the solution set in interval notation and represent the system on a number line.

\begin{cases} 2x+3 < 1 \\ -x+6< 3 \end{cases}

Solution

Like example 1, we will solve the above inequality in multiple steps.

Step 1-Solving the inequalities in the system separately

First, solve the first inequality in the system like this:

2x+3<1

By subtracting 3 from both sides of the inequality we will get the following expression:

2x+3-3<1-3

2x<-2

To isolate the variable x on the left-hand side of the inequality, divide both sides by 2:

x<-1

Now, solve the second part of the inequality -x+6<3

Subtract 6 from both sides of the inequality:

-x+6-6<3-6

-x<-3

Since, the variable in the resulting inequality -x<-3 is negative, therefore we will multiply both sides by -1. This will also flip the inequality sign and change the sign of -3 on the right hand side.

x>3

Step 2 - Writing the solution set of both the inequalities in interval notation separately

After simplifying the first inequality 2x+3<1, we got x<-1. In interval notation, it will be written as:

(\infty,-1)

You can see that we have used parentheses on both sides of the inequality because -1 is not part of the solution.

The simplified form of the second inequality -x+6<3 was x>3. In interval notation, it will be written as:

(3,\infty)

Again we have not used square brackets before or after beginning and endpoints because 3 is not included in the solution set.

Step 3- Combining the solution set in interval notation of both inequalities

Now, we will combine the solution set in interval notation of both the inequalities. The interval notation of the first inequality is (\infty,-1) and the solution set in interval notation of the second inequality is (3,\infty).

Uniting these inequalities gives us no solution at all.

Step 4-Representing the solution set on the number line

Since the system of inequality has no solution at all, so on the number line there will no common values represented which are part of both the inequalities.

Example 2 - Solution set on a number line

Example 3

Solve the following system of linear inequalities with one variable. Write the solution set in interval notation and represent the system on a number line.

\begin{cases} 3x-4 >2x-2 \\ 6x-2> 2x+2 \end{cases}

Solution

To find the solution set, we will solve the above inequality in multiple steps.

Step 1-Solving the inequalities in the system separately

First, solve the first inequality in the system like this:

3x-4 >2x-2

Apply basic operations of addition and subtraction to simplify the above inequality:

Adding 4 to both sides of the inequality will give:

3x-4+4>2x-2+4

3x>2x+2

Subtract 2x from both sides of the inequality to isolate the variable x on the left hand side:

x>2

Now, for solving the second part of the inequality 6x-2> 2x+2, combine the like terms like you do while solving linear equations:

6x-2x>2+2

4x>4

Divide both sides by 4 to get the resultant inequality in simplified form:

x>1

Step 2 - Writing the solution set of both the inequalities in interval notation separately

Simplifying the first inequality 3x-4 >2x-2 gave us x>2. The interval notation of the first inequality will be written like this:

(2,\infty)

Similarly, by simplifying the second inequality 6x-2> 2x+2 we got x>1 . The interval notation of the second inequality will be written like this:

(1,\infty)

Step 3- Combining the solution set in interval notation of both inequalities

Now, we will combine the solution set in interval notation of both the inequalities. The interval notation of the first inequality is (2,\infty) and the solution set in interval notation of the second inequality is (1,\infty). Unite these two inequalities like this:

(2,\infty) U (1,\infty)

The final solution set in interval notation will be written by taking the smaller beginning point and larger endpoints from the intervals of both inequalities. Since the smaller beginning point is 1 and the greater endpoint is \infinity, so the interval notation of the system will be:

(1,\infty)

The final solution is an open interval.

Step 4-Representing the solution set on the number line

The interval notation of the system of inequalities is (1,\infty). It shows that the interval has infinitely many solutions. On the number line, it will look like this:

Example 3 - Solution set on a number line

You can see that since 1 is not included in the interval, so we have used the open circle to represent it on the number line.

Example 4

Solve the following system of linear inequalities with one variable. Write the solution set in interval notation and represent the system on a number line.

\begin{cases} 5x-4 \leq 4x-3 \\ 3x-2 \geq 2x-1 \end{cases}

Solution

Follow the following steps to solve the above inequality.

Step 1-Solving the inequalities in the system separately

Take the first inequality from the system and combine like terms by taking coefficients on the left hand side and constants on the right hand side of the inequality:

5x-4 \leq 4x-3

5x-4x \leq 4-3

x \leq 1

Now, solve the second inequality like this:

3x-2 \geq 2x-1

Subtract 2x and add 2 on both sides of the inequality:

x \geq 1

Step 2 - Writing the solution set of both the inequalities in interval notation separately

By simplifying the first inequality 5x-4 \leq 4x-3 we get x \leq 1. The interval notation of the first inequality will be written like this:

(\infty,1]

Similarly, by simplifying the second inequality 3x-2 \geq 2x-1 we got x \geq 1. The interval notation of the second inequality will be written like this:

[1,\infty)

 

Step 3- Combining the solution set in interval notation of both inequalities

Now, we will combine the solution set in interval notation of both the inequalities. The interval notation of the first inequality is (\infty,1]  and the solution set in interval notation of the second inequality is [1,\infty). Unite these two inequalities like this:

(\infty,1]U[1,\infty)

The final solution set in interval notation will be written by taking the smallest beginning point and largest endpoint from the intervals of both inequalities. Since the smallest beginning point is 1 and the greater endpoint is also 1, so the interval notation of the system will be:

[1,1]

Hence, the system has only one solution. We have enclosed the only solution is square brackets because the interval is closed.

Step 4-Representing the solution set on the number line

The interval notation of the system of inequalities is [1]. It shows that the interval has only one solution. On the number line, it will look like this:

Example 4 - Solution set on a number line

You can see that since 1 is included in the interval, so we have used the closed circle to represent it on the number line.

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.