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!