Integer inequalities with absolute values

Learning Objectives Define absolute value as the distance of an integer from zero. Identify and interpret inequality symbols (<, >, ≤, ≥). Solve integer inequalities involving absolute values of the form |x| < a and |x| > a. Represent the solution sets of absolute value inequalities on a number line. Apply the concept of absolute value inequalities to simple real-world scenarios. Check if a given integer is a solution to an absolute value inequality. Imagine you're a weather reporter 🌡️! How would you describe temperatures that are 'within 5 degrees of freezing (0°C)'? This involves distance and limits! In this lesson, we'll learn about absolute values and how they help us describe distances and ranges using inequalities. Understanding these...

TermDefinitionExample IntegerWhole numbers and their opposites (..., -2, -1, 0, 1, 2, ...).The numbers -5, 0, and 12 are all integers. Absolute ValueThe distance of an integer from zero on a number line. It is always a non-negative value.|5| = 5 and |-5| = 5, because both 5 and -5 are 5 units away from zero. InequalityA mathematical statement comparing two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).x < 3 means 'x is less than 3', and y ≥ -2 means 'y is greater than or equal to -2'. Number LineA visual representation of numbers as points on a straight line, useful for graphing solutions to inequalities.A line with 0 in the middle, positive numbers to the right, and negative numbers...

Definition of Absolute Value $$|n| = \begin{cases} n & \text{if } n \ge 0 \\ -n & \text{if } n < 0 \end{cases}$$ The absolute value of a number is its distance from zero, always resulting in a positive value or zero. Solving Absolute Value Inequalities: Less Than If $a$ is a positive integer, then $$\text{if } |x| < a \text{, then } -a < x < a$$ If the absolute value of 'x' is less than 'a', it means 'x' is between -a and a. This represents numbers 'closer' to zero than 'a'. Solving Absolute Value Inequalities: Greater Than If $a$ is a positive integer, then $$\text{if } |x| > a \text{, then } x < -a \text{ or } x > a$$ If the absolute value of 'x' is greater than 'a',...