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讓我們來學習解絕對值方程式

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先來複習下

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當計算一個數值的絕對值時

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以-1爲例 來計算絕對值

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你需要做的是

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計算出1與0之間的距離

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對-1來說 下面以數軸示例

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這個沒畫好

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這是數軸 這是0

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這是-1

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-1與0之間的距離是1

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那-1的絕對值就是1

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1和0之間的距離也是1

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所以1的絕對值同樣是1

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也就是說 絕對值爲與0之間的距離

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來換一種更簡單的方式來理解

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絕對值其實就是取數值的正值

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-7346的絕對值是7346.

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理解這點以後 下面

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開始學習解絕對值方程式

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來看這個方程式

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x-5的絕對值等於10

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可以這樣解讀這個等式

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也希望能夠這樣理解 這就是說

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x與5之間的距離爲10

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與5之間的距離是10的數值有幾個

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大家應該已經想到解法了

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不過下面要學習更係統的解法

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這個方程式有兩種情況

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一種是x-5等於10

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如果假定絕對值裏面的值爲10

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取絕對值後

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會得到10

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或者假設x-5等於-10

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如果x-5等於-10 取絕對值後

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也是10

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所以第二種情況是 x-5等於-10

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兩種情況同時滿足這個方程式

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先解這個

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等式兩邊加5

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得到x=15

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再解這個 等式兩邊加5

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x等於-5

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得出計算結果

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有兩個x值滿足這個方程式

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x可以是15

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15-5等於10 取絕對值後

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得到10 或者x爲-5

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-5減5等於-10

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取絕對值後 還是10

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注意 這兩個值

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與5之間的距離都是10

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再解一個

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再來一個

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現在有

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x+2的絕對值等於6

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這說明什麽呢

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也就是說 要麽x+2

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也就是絕對值符號裏的這個值 它等於6

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或者絕對值符號裏的這個值

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x+2 它等於-6

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假設等於-6時

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取絕對值後 得到6

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那麽 假設x+2等於-6

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然後 等式兩邊減2

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得到 x等於4

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再看這個方程式 兩邊減2

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得到x等於-8

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這就是方程式的兩種解法

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爲了加深理解

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對於絕對值 可以把它看成是距離

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這樣重寫這個絕對值方程式

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x減-2的絕對值等於6

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其實就是說

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哪些值與-2之間的距離剛好是6呢

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還記得嗎 在這裡講過

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哪些值與5之間的距離是10

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不管被5減的值是幾

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它們與5之間的距離肯定是10

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這邊呢

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哪些值與-2之間的距離是6

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那就是4 或-8

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自己試著驗證下結果

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再做一個

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再來一個 這次用紫色

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這有4x的絕對值

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來計算一個有難度的

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4x減1

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4x-1的絕對值 等於--

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這裡讓它等於19

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這樣 跟之前的解法一樣

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一種情況是4x減1等於19

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另一種情況是4x減1等於-19

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因爲當取絕對值以後

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都會得到19

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或者4x減1等於-19

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現在只需要解這兩個方程式

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等式兩邊加1

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兩個同時做

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兩邊加1後 得到4x等於20

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這兩邊加1後

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得到4x等於-18

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等式兩邊除4 得到x等於5

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這個等式兩邊除4 得到x等於-18/4

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也等於-9/2

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這兩個x值同時滿足這個絕對值方程式

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試試看

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-9/2乘4

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等於-18

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-18再減1就是-19

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取絕對值後 得到19

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等於5時 4乘5是20

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減1等於19

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再取絕對值以後

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還是等於19

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下面來畫絕對值方程式的圖形 增加趣味性

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現在有

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y等於x+3的絕對值

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這是個函數 或者說圖形

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並包含絕對值在裏面

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這要考慮兩種情況

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一種情況是

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絕對值符號裏的這個值爲正

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也就是x+3

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寫這邊 x+3>0

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另一種情況是x+3<0

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當x+3>0時

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這個圖形 或這條直線 這裡可能還不能稱爲一條直線

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這個函數 就等同於y=x+3

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如果這邊這個值是大於0的話

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那麽絕對值符號就形同虛設

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那麽這個就完全等同於

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y=x+3

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那麽什麽情況下x+3>0

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不等式兩邊減3

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得到x>-3

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那麽當x>-3時

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這個圖形與y=x+3是等同的

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當x+3<0時

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在這種情況下

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絕對值符號裏的值爲負

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在這種情況下等式就變成

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y等於-（x+3）

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爲什麽呢

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仔細聽 現在這裡是負值 如果

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x+3是負值

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在這個前提下 如果它是負值的話

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那麽當取負值的絕對值後

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它肯定是得到正值

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就相當於把它與-1相乘

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如果知道是取負值的絕對值

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就相當於用這個值乘-1

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因爲絕對值都是正值

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也就是這裡所討論的情況

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x+3少於0

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兩邊減3 當x少於-3

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當x少於-3

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那麽當x<-3時

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圖形應該是這樣

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當x>-3

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圖形是這樣

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下面一起來看

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完整的圖形

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先畫數軸

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這是x軸 這是y軸

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先把這個函數乘出來 這樣可以得到mx

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加b的函數格式

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這個等於-x-3

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先看看這種函數圖形

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的一般形式

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-x-3

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y軸截距是-3 所以1 2 3

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-x那就應該是向下傾斜 也就是

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向下斜率爲1

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看起來就是這樣

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x軸的截距應該是x等於

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當假設y等於0時 這邊x

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等於-3

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穿過這條線

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這個點

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這個圖形 如果沒有這裡這個限制條件

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應該是這樣

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如果沒有區間限制條件

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x軸的這個區間限制條件

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那麽這個圖形 應該是怎樣的

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來看看

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y軸截距是3

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像這樣

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x軸的截距呢

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當y等於0時 x等於-3

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還是一樣穿過這個點

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斜率爲1

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那就應該是這樣的

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這就是這個函數的圖形

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現在 可以得出結論 絕對值

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函數爲紫色直線

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當x<-3時

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當x<-3 這裡是x等於

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-3 當x<-3

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它就是這條紫線

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就是這條

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上面是當x<-3時的圖形

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但是如果x>-3時 就應該是

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這條綠線了

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就是這條

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圖形看起來很像奇怪的V

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當x>-3時 應該是正值

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所以這個圖形 斜率爲正

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但當x<-3時 本質上

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其實是取這個函數的負值 如果想要

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這麽理解的話 所以這裡是負斜率

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如果看到這種V型函數 或是

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V型圖 那就表明它是

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絕對值函數