The Addition Property of Inequality states that if you add the same value to both sides of an inequality, the inequality remains true. Here are the key properties associated with the Addition Property of Inequality:
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Maintaining Inequality Direction: If \( a < b \), then for any real number \( c \): \[ a + c < b + c \] Similarly, this applies for greater than inequalities: \[ a > b \implies a + c > b + c \]
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Application to All Types of Inequalities: The property is applicable to all forms of inequalities, including:
- Less than (<)
- Greater than (>)
- Less than or equal to (≤)
- Greater than or equal to (≥)
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Combining Inequalities: You can apply this property to combine inequalities. For example, if \( x < y \) and \( z < w \), you can add \( c \) to both inequalities, resulting in: \[ x + c < y + c \quad \text{and} \quad z + c < w + c \]
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Solving Inequalities: The Addition Property of Inequality is a fundamental step in solving linear inequalities, allowing you to isolate the variable.
Overall, the Addition Property of Inequality is a crucial tool in understanding and manipulating inequalities in algebra.