Absolute Value Inequalities 3642De

1. The problem is to understand how to use absolute value in inequalities. 2. The absolute value of a number $x$, denoted $|x|$, is the distance of $x$ from zero on the number line, always non-negative. 3. For an inequality involving absolute value, such as $|x| < a$ where $a > 0$, it means $-a < x < a$. 4. Similarly, $|x| > a$ means $x < -a$ or $x > a$. 5. For example, to solve $|x - 3| \\leq 5$, rewrite as $-5 \\leq x - 3 \\leq 5$. 6. Add $3$ to all parts: $-5 + 3 \\leq x \\leq 5 + 3$ which simplifies to $-2 \\leq x \\leq 8$. 7. This means $x$ is between $-2$ and $8$ inclusive. 8. Always remember to consider the direction of inequalities and the sign of the number on the right side when dealing with absolute values.