这部分需要会什么？
- 将绝对值理解为到某点的距离
- 按右侧正负拆分支求解绝对值方程
- 用距离区间表示绝对值不等式
核心概念与公式
\(|x-a|\) 表示 \(x\) 与 \(a\) 的距离。\(|x-a|=d\) 在 \(d>0\) 时有 \(x=a\pm d\)，\(d=0\) 时只有 \(x=a\)，\(d<0\) 时无解。
\(|x-a|\le d\)（\(d\ge0\)）表示 \(a-d\le x\le a+d\)；\(|x-a|\ge d\) 表示 \(x\le a-d\) 或 \(x\ge a+d\)。先把绝对值单独移到一边，再判断另一边的符号。
做题时怎样使用
- 先把绝对值整体隔离到等号或不等号一边
- 判断另一边的正负，再拆成对应的分支或距离区间
- 把候选解代回原式检查，确认分支条件成立
容易混淆的地方
绝对值方程有时只有一个解或无解，不能一律说有两个。绝对值外还带系数时不要直接拆分支，先化成 \(|…|=d\) 的形式。
用一个例题串起方法
知识点教学例题解 \(|2x-3|=5\)。
展开解题思路 ＋
分成 \(2x-3=5\) 或 \(2x-3=-5\)。因此 \(x=4\) 或 \(x=-1\)。两者代回绝对值均为 5。
本知识点题库
先独立作答，再查看答案与解析。
绝对值方程与不等式进阶
\( \frac{|4x-48|+4}{8}=6 \)
What is the sum of the solutions to the given equation?
原库参考解析
绝对值方程与不等式挑战
The equation \( 5|\ x-8\ |=k \) , where k is a constant, has exactly one solution. Which of the following could be the value of \( \frac{k}{5} \)?
原库参考解析
绝对值方程与不等式挑战
The equation\( 5|\ x-6\ |=k \) , where k is a constant, has exactly one solution. Which of the following could be the value of \( \frac{k}{5} \)?
原库参考解析
绝对值方程与不等式基础
\( 5|\ x\ |=35 \)
What is the positive solution to the given equation?
原库参考解析
绝对值方程与不等式进阶
The graph of a system of an absolute value function and a linear function is shown. What is the solution \( (x,y) \) to this system of two equations?
原库参考解析
绝对值方程与不等式挑战
The equation \( 5|x-9|=k \) , where k is a constant, has exactly one solution. Which of the following could be the value of k?
原库参考解析
绝对值方程与不等式进阶
\( 6|x−6|=24 \)
What is the sum of the solutions to the given equation?
原库参考解析
绝对值方程与不等式挑战
If \( |\ 12-2x\ |=48 \), which of the following is a value of \( 6−x \)?
原库参考解析
绝对值方程与不等式进阶
\( \frac{|\ 3x-54\ |+3}{6}=9 \)
What is the sum of the solutions to the given equation?
参考答案
绝对值方程与不等式进阶
\( \frac{|4x-48|+4}{6}=8 \)
What is the sum of the solutions to the given equation?
参考答案
绝对值方程与不等式进阶
\( ∣2x−3∣=−8 \)
How many solutions does the given equation have?
参考答案
绝对值方程与不等式挑战
\( |x−9|=14 \)
What is the sum of the solutions to the given equation?
What is the sum of the solutions to the given equation?
参考答案
