6. Linear Inequalities

Two real numbers or two algebraic expressions related by the symbols <, >, ≤ or ≥ form an inequality.

Equal numbers may be added to (or subtracted from ) both sides of an inequality.

Both sides of an inequality can be multiplied (or divided ) by the same positive number. But when both sides are multiplied (or divided) by a negative number, then the inequality is reversed.

The values of x, which make an inequality a true statement, are called solutions of the inequality.

To represent x < a (or x > a) on a number line, put a circle on the number a and dark line to the left (or right) of the number a.

To represent x ≤ a (or x ≥ a) on a number line, put a dark circle on the number a and dark the line to the left (or right) of the number x.

If an inequality is having ≤ or ≥ symbol, then the points on the line are also included in the solutions of the inequality and the graph of the inequality lies left (below) or right (above) of the graph of the equality represented by dark line that satisfies an arbitrary point in that part.

If an inequality is having < or > symbol, then the points on the line are not included in the solutions of the inequality and the graph of the inequality lies to the left (below) or right (above) of the graph of the corresponding equality represented by dotted line that satisfies an arbitrary point in that part.

The solution region of a system of inequalities is the region which satisfies all the given inequalities in the system simultaneously.