One Variable Linear Equations and Inequations
A linear equation is an equation of a straight line where the power of the variable is 1. It is expressed as ax + b = 0, where x is the variable and a and b are integers. Inequality is a statement of comparison between two expressions. Linear Inequations are two expressions where their values are compared by the inequality symbols such as <, >, ≤ or ≥. One variable linear equations and Inequations have only one solution or one root. Examples of one variable linear equations and inequations are x = 4, 2a + 3 = 9, 3x < 2 , 4y  5 > 6. etc.
What are One Variable Linear Equations and Inequations?
An algebraic equation is a statement that equates to two mathematical expressions. Linear equations are the firstdegree equations and they have the highest exponent of variable as 1. The standard form of a linear equation in one variable is ax + b = 0, where x is the variable. This means that the variables in a linear equation do not have exponents like squares or cubes. It has a single variable (unknown); it is linear i.e. the pattern it makes is a straight line and not a parabola or any nonstraight curve and it is an equation or an inequality.
Here is an example of a linear equation y = 2x + 3 plotted on the graph as a straight line.
Here is another graph that shows a graph of inequality. This graph shows x ≥  4.
Any mathematical equation or inequality has 2 sides to it. Left Hand Side (LHS) and Right Hand Side (RHS). In the case of an equation, the 2 sides are equal, that is, LeftHand Side is equal to the RightHand side. For example, 2 plus 4 is equal to six is an equation expressed in words. Examples of linear equations in one variable: x + 5 = 4; 2x + 9 = 23 The best way to visualize equations and inequations is to picture weighing balances.
Here is an example of what an equation looks like, where the RHS = LHS.
An algebraic linear inequality is similar to an algebraic linear equation wherein the equal sign is replaced by an inequality sign. In the case of inequality, instead of equality, some other relationship like less than or greater than exists between LHS and RHS. Example: x < 10, inequality prevails between the LeftHand side and the Right Hand Side. Here is an example of inequality where RHS ≠ LHS. In the below illustration we can see that the expression on the lefthand side, that is, 3x  4, is in fact lesser than the number on the righthand side, that is 20. Hence, we may express the inequality as: 3x  4 < 20 .
Solving One Variable Linear Equations and Inequations
When we substitute the variable with an integer, the resulting statement could be true or false. If the statement is true, then the integer is a solution to the equation or inequality. For solving a linear equation or inequality having only one variable, the following steps are followed, while still balancing the equation.
 Add or subtract like terms
 Isolate the variable
 Transpose or eliminate the terms
 Verify the answer.
Example: 1) Solve: 5x + 2 = 12
Keep the variable on the LHS and transpose all the other terms or the coefficient of x to the RHS.
5x = 12  2
5x = 10 ⇒ x = 2
Verify if x = 2 in the given linear equation.
Example: 2) Solve: 2(24x) >  6x + 7
4  8 x > 6x + 7
8 x + 6x > 7  4
2x > 3
x < 3/2
Related Topics
 Solutions of Linear Equations
 Practical Applications of Linear Equations
 Linear Inequalities
Important Notes
 The highest exponent of the variable in a linear equation or inequation is one.
 One variable linear equations are straight lines when plotted on a graph.
 All one variable linear equations will be of the form: ax + b = 0 where x is the variable.
Solved Examples

Example 1: The sum of 3 consecutive numbers is 24. Find the numbers.
Solution: Let x be the middle number out of the three consecutive numbers. Then the required numbers are x1, x, and x+1.
We have x1 + x + x +1 = 24
3x = 24 ⇒ x = 24/3 =8
Thus the required 3 numbers are 7, 8 and 9 
Example 2: What is the solution of \(\dfrac{2x5}{3}\) > \(\dfrac{3x+3}{4}\)?
Solution:\(\dfrac{2x5}{3}\) >\(\dfrac{3x+3}{4}\) ⇒ 4(2x 5) > 3(3x +3)
8x 20 > 9x +9
8x9x > 9+20
x > 29
⇒ x <  29 
Example 3: Mike traveled 4/9 of a certain distance by bus, 1/3 by car, and the remaining 2 miles on foot. Find the distance traveled by him on the bus.
Solution:Let the total distance covered by him be x miles.
\(\begin{align}\dfrac{4}{9}x + \dfrac{1}{3}x+2 &= x\\\\\dfrac{4x}{9} + \dfrac{x}{3}+2 &= x\\\\\dfrac{4x}{9} + \dfrac{3x}{9}+\dfrac{18}{9} &= x\\\\\dfrac{4x + 3x +18}{9} &=x\\\end{align}\)
⇒ 7x + 18 = 9x
7x  9x = 18
2 x = 18
x = 9
Therefore, distance covered by bus = \(\dfrac{4x}{9}\) = 4 miles
FAQs on One Variable Linear Equations and Inequations
What is the Degree of a Linear Inequality?
The highest power of the variable in the linear inequalities is 1.
What is Linear Equation in One Variable with Example?
A linear equation of the form ax + b = 0 where x is the only variable is the standard form of a linear equation. Example: 5 x + 3 = 8 is a linear equation having a single variable x in it. The only solution for this equation is x = 1.
What is the Difference Between Linear Equation and Linear Inequality?
A linear equation and the inequality differ only with the sign. Two expressions are equated in the linear equation, whereas two expressions are compared in the linear inequality.
What are the Symbols Used in Linear Inequalities?
The symbols used in linear inequalities are <, >, ≤ or ≥.
Can a Linear Inequality have One Solution?
A linear inequality can have one solution or no solution or infinite solutions.