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Integer Inequalities with Absolute Values – Integers: Positive & Negative – Integers include whole numbers, their negatives, and zero. – Recap: Integers on Number Line – Remember how integers are ordered from least to greatest? – Absolute Values Defined – Absolute value is the distance from zero, always positive. – Solving Inequalities with Absolutes – Apply rules of absolute values to find solution sets. | This slide introduces the concept of integer inequalities with absolute values, building on the students’ prior knowledge of integers. Start by ensuring that students recall what integers are, including both positive and negative numbers, and how they are represented on a number line. Emphasize the importance of understanding absolute values as the distance from zero, which is always a positive number, regardless of the original integer’s sign. Then, guide students through the process of solving inequalities that involve absolute values, demonstrating how to interpret and solve these types of problems. Provide examples and encourage students to think about how absolute values can affect the solutions to inequalities.

Understanding Absolute Values – Define absolute value – The absolute value of a number is its distance from 0 on a number line, regardless of direction. – Absolute value as zero distance – Think of it as how far a number is from zero: |3| is 3 steps away. – Absolute value examples – |7| = 7, |-3| = 3, and |0| = 0. The absolute value is always non-negative. – Comparing absolute values – |5| > |-2| because 5 is farther from zero than 2. | This slide introduces the concept of absolute value, which is a fundamental part of understanding integer inequalities. Start by defining absolute value as the distance a number is from zero on the number line, emphasizing that it’s always a positive distance, even for negative numbers. Provide clear examples to illustrate this point. Encourage students to visualize the number line and use it to compare absolute values, reinforcing the idea that absolute value represents a magnitude without direction. This will set the groundwork for solving inequalities involving absolute values.

Understanding Integer Inequalities with Absolute Values – Define inequalities – Inequalities show how numbers compare, like 5 > 3 (5 is greater than 3). – Learn inequality symbols – Symbols: > (greater than), < (less than), e (greater than or equal to), d (less than or equal to). – How to read inequalities – '3 7 means x is any number greater than 7. | This slide introduces the concept of inequalities, which are statements about the relative size or order of two numbers. It’s crucial for students to understand the symbols and their meanings, as they are the foundation for writing and interpreting inequalities. Provide examples of each symbol in use and ensure students can read them aloud correctly. Encourage practice by writing inequalities to represent real-world situations, such as minimum age requirements or scores needed to pass a test. This will help solidify their understanding and prepare them for working with absolute values in inequalities.

Combining Absolute Values and Inequalities – Solve absolute value inequalities – Use |x| < a to represent distances less than a from 0 – Split inequality into two cases – Case 1: x -a (for |x| < a) – Solve each case separately – Find solutions for x in each scenario – Check solutions on a number line – Plot solutions to visualize the answer range | When solving absolute value inequalities, it’s crucial to understand that absolute value represents a distance from zero on the number line. Therefore, an inequality involving absolute value, like |x| < a, can be thought of as a distance problem. To solve, we consider two separate cases: one where the value inside the absolute value is positive (less than a) and one where it's negative (greater than -a). Solve each case as a separate inequality and then represent the solution set on a number line. This visual aid helps students to better grasp the concept of solution ranges for inequalities. Encourage students to practice with different inequalities and to always check their solutions by substituting back into the original inequality.