Linear Inequality in One Variable

A linear inequality in one variable is an equation that contains a variable and at least one inequality symbol such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).

To graph a linear inequality in one variable, first treat the inequality as an equation and graph the corresponding linear equation. Then, identify the section of the line that satisfies the inequality by shading the appropriate region (above or below the line) based on the direction of the inequality symbol.

The solution set of a linear inequality is the set of all possible values for the variable that make the inequality true. It can be represented using interval notation or set builder notation.

To solve a linear inequality algebraically, isolate the variable on one side of the inequality symbol. Flip the direction of the inequality if you multiply or divide both sides by a negative number. The resulting solution represents all values of the variable that satisfy the inequality.

When solving an equation, you find the exact value(s) of the variable that make the equation true. In contrast, when solving an inequality, you determine a range of values for the variable that satisfy the given condition.

Compound inequalities involve multiple inequalities combined using the logical operators 'and' or 'or'. For example, 'x > 2 and x < 5' denotes the intersection (overlap) of two inequalities, while 'x > 2 or x < 5' represents the union (combined range) of two inequalities.

Absolute value inequalities involve the absolute value of a variable and an inequality symbol. They typically contain expressions like |x| > 3 or |x - 2| ≤ 5. Solving absolute value inequalities often requires breaking the inequality into two separate cases.

Yes, a linear inequality can have no solution. This occurs when the solution set is an empty set, indicating that there are no values for the variable that satisfy the inequality.

Yes, there are general rules for solving linear inequalities. These include adding or subtracting the same number from both sides, multiplying or dividing both sides by a positive number without changing the inequality direction, and flipping the inequality direction when multiplying or dividing both sides by a negative number.

For further learning about linear inequalities in one variable, you can refer to math textbooks, online educational resources, or seek assistance from a math tutor or teacher.