One Variable Linear Equations and Inequations

An algebraic equation is a statement that equates to two mathematical expressions. Linear equations are the first-degree 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 non-straight 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 in the blue line 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, Left-Hand Side is equal to the Right-Hand 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 Left-Hand 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 left-hand side, that is, 3x - 4, is in fact lesser than the number on the right-hand side, that is 20. Hence, we may express the inequality as: 3x - 4 < 20 . 

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.

1. Add or subtract like terms
2. Isolate the variable
3. Transpose or eliminate the terms
4. 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(2-4x) > - 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.