Inequalities

Inequalities

Basic Concepts

Inequalities are notations used to compare the relative size or value of different numbers or expressions.
An inequality sign shows that one value is not equal to another value. The signs used in inequalities are > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to).
Inequality notation can also include two complementary ideas: strict inequality (> or <) and non-strict inequality (≥ or ≤).

Solving Inequalities

To solve an inequality, apply the same operations to both sides, just like in equations. For example, if x + 3 ≤ 7, by subtracting 3 from both sides, we find x ≤ 4.
When multipling or dividing an inequality by a negative number, reverse the inequality symbol. For example, if -2x ≥ 6, by dividing both sides by -2, we get x ≤ -3.

Graph Representation

Graphs can represent inequalities on a number line or a coordinate plane.
For single inequalities, use a closed circle to represent ‘greater than or equal to’ (≥) or ‘less than or equal to’ (≤) and open circles for ‘greater than’ (>) or ‘less than’ (<).
For coordinate plane, shaded regions represent areas where inequalities hold true.

Compound Inequalities

Compound inequalities involve more than one inequality. They can be ‘and’ inequalities or ‘or’ inequalities.
‘And’ inequalities (x > a and x < b) indicate that x is between two values a and b.
‘Or’ inequalities indicate that the value of x can be in one of two separate ranges.

Manipulating Inequalities

When given an inequality, the solution is not unique; there exists a range of values.
Always check your solution by substituting the value back into the original inequality.
Inequalities can be used to solve word problems that involve limitations or restrictions on possible values.

Remember, understanding and correctly using inequalities will allow you to accurately express a range of possible solutions to a problem. With practice, you can feel confident handling these vital tools in algebra!