LINEAR INEQUALITIES
Frequently Asked Questions
A linear inequality is an inequality of the form \(ax + b < 0\), \(ax + b \le 0\), \(ax + b > 0\), or \(ax + b \ge 0\), where \(a\) and \(b\) are real numbers and \(a \ne 0\).
A linear equation uses an equality sign \(=\) and has a unique solution, while a linear inequality uses \(<, \le, >, \ge\) and has a range of solutions.
The symbols are less than \((<)\), less than or equal to \((\le)\), greater than \((>)\), and greater than or equal to \((\ge)\).
The solution set is the collection of all real numbers that satisfy the given inequality.
It is solved by isolating the variable using standard algebraic operations while maintaining the inequality sign.
The inequality sign is reversed when both sides are multiplied or divided by a negative number.
For \(2x - 5 < 3\), we get \(2x < 8\) and hence \(x < 4\).
It is a graphical method where solutions are shown as points or intervals on the number line.
Strict inequalities \((<, >)\) are represented using open circles to exclude the boundary point.
Inclusive inequalities \((\le, \ge)\) are represented using closed circles to include the boundary point.
Compound linear inequalities involve two inequalities connected by “and” or “or”.
“And” means the intersection of solution sets, where both inequalities must be satisfied simultaneously.
“Or” means the union of solution sets, where at least one inequality must be satisfied.
For \(1 < x < 5\), the solution is all real numbers between 1 and 5.
For \(x < -2\) or \(x > 3\), the solution includes numbers less than \(-2\) and greater than \(3\).