Inequalities express mathematical relationships where two values are not equal and are compared using symbols such as , ≤, or ≥. These expressions define a range of possible solutions rather than a single value. Interval notation provides a concise way to express these solution sets, especially when the variable spans a continuous range. An open interval, written as (a, b), excludes the endpoints, while a closed interval [a, b] includes them. There are also half-open intervals, such as (a, b] and [a, b), which include only one endpoint. The notations (a, ∞) and (−∞, b] are used when a set extends indefinitely in one direction, with infinity symbols never being included as actual values.
Solving inequalities requires applying operations while adhering to specific rules that maintain or reverse the direction of the inequality. When a constant is added or subtracted from both sides, the inequality’s direction remains unchanged. This is formalized in the addition property: if
a < b, then a + c < b + c. Similarly, multiplying or dividing both sides by a positive number retains the inequality's direction, whereas using a negative number reverses it. For example, multiplying both sides of 3 < 7 by 2 results in 6 < 14, but multiplying both sides of 3 < 7 by −2 gives -6 > -14.
The reciprocal property of inequalities also plays a critical role. If a > 0, then 1/a > 0, and taking reciprocals of both sides of an inequality involving positive values reverses the inequality’s direction. For instance, if a 1/b.
Inequalities compare two values to show their relationship using symbols like less than, greater than, less than or equal to, and greater than or equal to.
Interval notations represent the range of values that satisfy an inequality as a concise form of solution sets.
A closed interval includes the endpoints and is shown using square brackets, while an open interval does not include the endpoints and is shown using parentheses.
The properties of inequalities include the rule that adding a value or subtracting the same value on both sides keeps the direction of inequality the same.
Multiplying both sides by a positive number or dividing both sides by the same number also keeps the direction the same, but multiplying or dividing by a negative number reverses the inequality.
When both sides of an inequality are positive, taking reciprocals reverses the inequality direction. This rule does not apply when values are negative.
Simplifying an inequality involves removing the constant and then dividing both sides with a constant to isolate the variable.
These inequalities appear in real life. For example, a ride’s ‘over 100 cm tall’ rule allows only those taller than 100 cm.
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