Solving Simultaneous Linear Inequalities in One Variable

Introduction

Linear inequality involves linear functions and algebraic expressions briefly representing certain inequality signs. These signs of linear inequalities include less than (<), greater than (>), less than or equal to (≤), greater than or equal to (≥), equal to (=), and not equal to (≠). These signs separate a linear inequality from a linear equation. When represented on the graph, a straight line is used to solve a linear inequality with two variables. On the other hand, linear inequality with just one variable is represented using a number line. We will discuss the method of solving simultaneous linear inequalities in one variable in the following details.

Many elements of linear equations and linear inequalities match solving methods and the end solutions. The representation of the same is done using a curly bracket. Although, the set of solutions of linear inequalities is significantly different from linear equations. Therefore, both these solution sets are represented in distinguished ways.

In the system of simultaneous linear inequalities, each part is resolved separately. After this, all those results are put together to draw a final solution to the overall inequality problem. Let’s move forward with the next section, where we will learn about different methods for solving simultaneous linear inequalities in one variable.

Example 1 – The following linear inequalities in class 11 contain just one variable. Let’s find out the solution and showcase it using an interval notation.

2x + 3 ≥ 1

-x + 2 ≥ -1

Solution

Step 1 – Let’s begin solving the first inequality like a linear equation using multiple steps:

2x + 3 ≥ 1

To algebraically solve the equation, we will subtract 3 from both sides and then divide those sides by 2.

2x + 3 – 3 ≥ 1 – 3

2x ≥ -2

After diving both sides by 2 to isolate x on one side, the inequality becomes

x ≥ -1 (This states that x is greater than or equal to -1.)

Now, we will solve the next linear inequality of the question.

-x + 2 ≥ -1

After subtracting 2 from both sides, inequality becomes -x ≥ -3. As per the rule of inequality, the sign is reversed when both sides of the linear inequalities are multiplied with a negative number. Hence, the result will be

x ≤ 3

Step 2 – To write the system of linear inequalities in an interval notation, we will need to write each separately in an interval notation. So, the representation of the results will be the following:

The interval notation for x ≥ -1 will be [-1, ∞) whereas the interval notation for x ≤ 3 will be (∞, 3].

Step 3 – Now, these solutions of the system of linear inequalities in one variable will be merged. The inequalities [-1, ∞) and (∞, 3] will unite as

[-1, ∞) U (∞, 3]

The conclusive interval notation of the above inequality will become [-1, 3], which is a closed interval.

Graphing linear inequalities can also be accomplished using a number line to represent the exact position of variables as real numbers.

Another example with different inequalities will simplify the solution of simultaneous linear inequalities in one variable!

Example 2 – Solve the given system of linear inequalities in one variable and represent the solution in an interval notation.

3x – 4 > 2x – 2

6x – 2 > 2x + 2

Solution

Step 1 – We will solve the first inequality by adding 4 to both sides and then subtracting 2x to get the value statement for x.

3x – 4 + 4 > 2x – 2 + 4

3x > 2x + 2

X > 2

Now, we will solve the second part of the problem!

6x – 2 > 2x + 2

The method for solving the above inequality will be just as simple as a linear equation.

6x – 2x > 2 + 2

4x > 4

After diving both sides by 4, we get x > 1

Step 2 – In this step, we will decide the interval notation of the outcomes we got in the earlier steps.

The interval notation for inequality 3x – 4 > 2x – 2 will be (2, ∞)

Similarly, the interval notation for the second inequality 6x – 2x > 2 + 2 will be (1, ∞)

Step 3 – This is the final step where we collectively set the outcome of linear inequalities based on previous results.

After combining the solution set in an interval notation of both these inequalities, we can write them as

(2, ∞) U (1, ∞)

As discussed in the earlier example, the definitive solution set in interval notation will be represented as a whole. We will pick the smaller beginning point and the larger endpoint from the united set of notations. In this case, the smaller number is 1, but the larger part is ∞. So, the conclusive interval notation will be (1, ∞), an open interval.

Conclusion

Linear inequalities can have numerous solutions, and all those solutions should be presented either on a number line or through an interval notation. The rules of solving a linear inequality are similar to linear equations except for one specific rule. When multiplied or divided by a negative number, you will need to reverse the sign of linear inequality . You must also know that compound inequalities with the ‘or’ word involve the union of each solution set by solving every inequality. On the contrary, compound inequalities with the word ‘and’ use the intersection of the solution set for each inequality.