Linear inequalities — KCSE Mathematics

In mathematics, compound inequalities involve two inequalities combined into one statement. To solve a compound inequality of the form a < x < b, follow these steps:

1. Solve each part of the inequality separately.
2. The solution is the intersection of the two inequalities.

For example, to solve 2 < x < 5:

• Start with 2 < x which means x > 2.
• Then solve x < 5.
• The solution is 2 < x < 5 or written in interval notation as (2, 5).

Inequalities involving absolute values require a different approach. For example, to solve |x - 3| < 2:

1. Rewrite the inequality as two separate inequalities: -2 < x - 3 < 2.
2. Solve the compound inequality:
   • Add 3 to all parts: 1 < x < 5.
3. The solution in interval notation is (1, 5).

Understanding these methods will help you tackle various problems involving inequalities effectively.

To solve linear inequalities in two variables, you first express the inequality in standard form, such as ( ax + by < c ) or ( ax + by \geq c ). Next, you find the boundary line by converting the inequality into an equation (e.g., ( ax + by = c )).

Steps to solve:

1. Graph the boundary line: Use a solid line for ( \geq ) or ( \leq ) and a dashed line for ( < ) or ( > ).
2. Test a point: Choose a test point not on the line (often (0,0) is easiest). Substitute it into the inequality.
3. Shade the appropriate region: If the test point satisfies the inequality, shade the region that includes the test point. If not, shade the opposite region.

Example: Solve and graph the inequality ( 2x + 3y < 6 ).

• Boundary line: Convert to ( 2x + 3y = 6 ) and graph it (dashed line).
• Test point (0,0): ( 2(0) + 3(0) < 6 ) is true, so shade the region containing (0,0).

Linear inequalities are mathematical expressions that involve a linear function and an inequality sign (>, <, ≥, ≤). To solve a linear inequality, follow these steps:

1. Isolate the variable: Just like solving an equation, aim to get the variable on one side.
2. Reverse the inequality sign: If you multiply or divide by a negative number, remember to flip the inequality sign.
3. Graph the solution: Use a number line to represent the solution set. Use open circles for < or > and closed circles for ≤ or ≥.

Example:

Solve the inequality: 3x - 5 < 4.

1. Add 5 to both sides: 3x < 9.
2. Divide by 3: x < 3.
3. Graph the solution: Draw a number line, place an open circle at 3, and shade to the left.

This means all numbers less than 3 are solutions to the inequality. Remember, understanding the graph helps visualize the solution set effectively.