Inequalities Revision Notes

An inequality compares two values or expressions, showing that one is larger, smaller, or possibly equal to the other. The main types of inequalities are strict and inclusive. Strict inequalities use the symbols '<' (less than) and '>' (greater than), meaning the value does not include the boundary point. Inclusive inequalities use '≤' (less than or equal to) and '≥' (greater than or equal to), meaning the value can include the boundary point.

For example, the inequality x > 2 means x can be any number greater than 2 but not 2 itself, while x ≥ 2 means x can be 2 or any number greater than 2. Understanding this difference is essential when representing inequalities visually.

• Strict inequalities: '<' and '>' exclude the boundary value.
• Inclusive inequalities: '≤' and '≥' include the boundary value.

A number line is a straight horizontal line marked with numbers at equal intervals. To represent inequalities on a number line, we indicate the range of values that satisfy the inequality by shading the appropriate region and marking boundary points with circles.

The key convention is: use an open circle at a boundary point if the inequality is strict (the point is not included), and use a closed (filled) circle if the inequality is inclusive (the point is included). The shaded region extends from the boundary point in the direction indicated by the inequality.

For example, to represent x > 3, place an open circle at 3 and shade the line to the right, indicating all numbers greater than 3. For x ≤ 5, place a closed circle at 5 and shade the line to the left, indicating all numbers less than or equal to 5.

• Open circle: boundary point not included (strict inequality).
• Closed circle: boundary point included (inclusive inequality).
• Shading shows all values satisfying the inequality.

Consider the inequality -3 < x ≤ 1. This means x is greater than -3 but less than or equal to 1. To represent this on a number line:

1. Draw a horizontal number line with marks at -3 and 1.

2. At -3, draw an open circle because the inequality is strict (x is greater than -3, not equal). This shows -3 is not included in the solution set. 3. At 1, draw a closed circle because the inequality is inclusive (x can be equal to 1). 4. Shade the region between -3 and 1 to indicate all values x can take between these points, including 1 but excluding -3. This visual representation helps to quickly identify the solution set and understand the inequality's meaning.

• Open circle at -3 (strict inequality).
• Closed circle at 1 (inclusive inequality).
• Shade between -3 and 1 to show all valid x values.

When given a number line with shaded regions and circles, you can write the corresponding inequality by observing the following:

Identify the boundary points where circles are drawn.

Determine if each circle is open or closed to decide if the boundary is included or excluded. Note the direction of shading to understand whether the inequality is 'less than' or 'greater than'. For example, a number line with an open circle at 2 and shading to the right represents x > 2. A closed circle at -1 with shading to the left represents x ≤ -1.

• Open circle means strict inequality at that point.
• Closed circle means inclusive inequality at that point.
• Shading direction indicates inequality direction.

Always use open circles for '<' or '>' and closed circles for '≤' or '≥'. Clearly shade the correct region to avoid ambiguity. Label boundary points on the number line for clarity. When inequalities combine two conditions (e.g., -3 < x ≤ 1), represent both boundaries correctly with appropriate circles and shading between them. Practice sketching inequalities from algebraic expressions and writing inequalities from number line diagrams to build confidence.

• Use correct circle types to earn full marks.
• Label boundary points clearly.
• Shade accurately between or beyond boundary points.
• Check inequality symbols carefully when interpreting or drawing.