Inequalities and Real-World Applications

A linear inequality is an inequality that involves a linear function. It can be expressed in the form $$ax + by < c$$ , $$ax + by > c$$ , $$ax + by \leq c$$ , or $$ax + by \geq c$$ , where a, b, and c are constants.

To graph a linear inequality, first graph the corresponding linear equation as a dashed line (for < or >) or a solid line (for ≤ or ≥). Then, shade the region that satisfies the inequality.

'At least' means that the value can be equal to or greater than the specified amount. For example, if a person needs at least $200, they can have $200 or more.

To write an inequality for savings, identify the initial amount, the amount withdrawn, and the minimum amount desired. For example, if you start with $500 and withdraw $25 each week, the inequality would be $$500 - 25x \geq 200$$ , where x is the number of weeks.

A system of inequalities is a set of two or more inequalities with the same variables. The solution is the region where the graphs of the inequalities overlap.

To determine the feasible region, graph each inequality on the same coordinate plane and identify the area where all shaded regions overlap.

The slope indicates the steepness of the line and the direction of the inequality. A positive slope means the line rises, while a negative slope means it falls.

The solution of a linear inequality represents all the possible values of the variables that satisfy the inequality. It can be expressed as a range of values.

A solid line indicates that the points on the line are included in the solution (≤ or ≥), while a dashed line indicates that the points on the line are not included (< or >).

To represent a real-world scenario with inequalities, identify the constraints and objectives, then translate them into mathematical expressions using inequalities.