绝对值表示一个数到原点的距离。对于任意数a，如果a大于等于0，绝对值就是a本身；如果a小于0，绝对值就是负a。让我们在数轴上看看这是如何工作的。
当绝对值方程含有变量时，我们需要分情况讨论。对于方程|x-3|=5，我们要找到与3的距离为5的所有点。第一种情况是x-3等于5，解得x等于8。第二种情况是x-3等于负5，解得x等于负2。在数轴上可以看到，8和负2都距离3恰好5个单位。
多重绝对值的计算需要按步骤进行。首先计算每个绝对值符号内的表达式：3减5等于负2，负2减4等于负6，6减2等于4。然后应用绝对值定义：负2的绝对值是2，负6的绝对值是6，4的绝对值是4。最后进行算术运算：2加6减4等于4。
绝对值不等式的求解需要转化为普通不等式。对于|x-2|小于3，意味着x到2的距离小于3。这等价于负3小于x-2小于3。两边同时加2，得到负1小于x小于5。在数轴上，解集是开区间(-1,5)。
这是一个综合应用题。已知两个绝对值的和等于0，要求a加b的值。关键是理解：由于绝对值总是非负的，两个非负数的和为0，当且仅当每个数都为0。因此，a加3的绝对值等于0且b减2的绝对值等于0。这意味着a加3等于0且b减2等于0，所以a等于负3，b等于2。最终答案是a加b等于负1。
When absolute value equations contain variables, we need to consider different cases. For the equation |x-3|=5, we need to find all points that are at a distance of 5 from 3. The first case is when x-3 equals 5, giving us x equals 8. The second case is when x-3 equals negative 5, giving us x equals negative 2. On the number line, we can see that both 8 and negative 2 are exactly 5 units away from 3.
When solving equations with multiple absolute values, we use interval analysis. For |x+1| + |x-2| = 5, we first find critical points where expressions inside absolute values equal zero: x = -1 and x = 2. These divide the number line into three intervals. In interval 1 where x < -1, both expressions are negative, giving us x = -2. In interval 2 where -1 ≤ x < 2, we get 3 = 5 which has no solution. In interval 3 where x ≥ 2, both expressions are positive, giving us x = 3.
Absolute value inequalities require transformation to regular inequalities. For |x-2| < 3, this means the distance from x to 2 is less than 3. This is equivalent to -3 < x-2 < 3. Adding 2 to all parts gives us -1 < x < 5. On the number line, the solution set is the open interval (-1, 5). Remember the rules: |A| < k means -k < A < k, while |A| > k means A < -k or A > k.
This is a comprehensive application problem. Given that the sum of two absolute values equals zero, we need to find a plus b. The key insight is that since absolute values are always non-negative, the sum of two non-negative numbers equals zero if and only if each number is zero. Therefore, |a+3| = 0 and |b-2| = 0. This means a+3 = 0 and b-2 = 0, so a = -3 and b = 2. The final answer is a+b = -1.