Course: 1.14 解决绝对价值不平等

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity 新闻通告
  
  新闻通告 Forum
Select section 解决绝对价值不平等

Collapse Expand
解决绝对价值不平等
The tolerance for the weight of a volleyball is 2.6 grams. If the average volleyball weighs 260 grams, what is the range of weights for a volleyball?

Absolute Value Inequalities
::绝对价值不平等

Like absolute value equations, absolute value inequalities also will have two answers. However, they will have a range of answers, just like .
::绝对价值等式一样,绝对价值不平等也有两种答案。但是,它们也有一系列答案,就像。

Consider the inequality $\begin{align*}|x|>1.\end{align*}$ If this were an equation , it would have two answers: one when $\begin{align*}x\end{align*}$ is 1 and the other when $\begin{align*}\text{-}x\end{align*}$ is 1 , but how does the inequality sign affect it? As an inequality, t he two possibilities would be:
::考虑不平等 x11. 如果这是一个等式,它会有两个答案:一个是x是1,另一个是-x是1,但不平等标志如何影响它?作为不平等,两种可能性是:

Notice in the second inequality, we did not write $\begin{align*}x>-1\end{align*}$ . This is because what is inside the absolute value sign can be positive or negative. Therefore , if $\begin{align*}x\end{align*}$ is negative, then $\begin{align*}-x>1\end{align*}$ . It is a very important difference between the two inequalities. Therefore, for the first solution, we leave the inequality sign the same and for the second solution we need to change the sign of the answer AND flip the inequality sign.
::在第二个不平等的警告中,我们没有写 x\\\ 1 。这是因为绝对值符号内的内容可以是正的或负的。因此, 如果 x 是负的, 那么 - - x> 1 。这是两种不平等之间的一个非常重要的区别。因此,对于第一个解决方案,我们让不平等的标志相同,对于第二个解决方案,我们需要改变答案的标志并翻转不平等标志。

Let's solve the following absolute value inequalities.
::让我们解决以下的绝对价值不平等问题。

Solve $\begin{align*}|x+2| \le 10.\end{align*}$
::解决 x+2 10 。

There will be two solutions, one with the answer and sign unchanged and the other with the inequality sign flipped and the answer with the opposite sign.
::会有两种解决方案, 一种是答案和签名不变, 另一种是不平等标志翻转, 答案与相反的标志。

$\begin{aligned} | x + 2 | \leq 10 \\ ↙↘ \\ x + 2 \leq 10 x + 2 \geq - 10 \\ x \leq 8 x \geq - 12 \end{aligned}$
$\begin{align*}& \qquad \quad |x+2| \le 10\\ & \qquad \qquad \swarrow \searrow\\ & x+2 \le 10 \qquad x+2 \ge -10\\ & \quad \ \ x \le 8 \qquad \qquad \ x \ge -12\end{align*}$
::x+210x+210x+210x+210 x_8 x12

Test a solution, $\begin{align*}x = 0:\end{align*}$
::测试溶液, x=0 :

$\begin{aligned} | 0 + 2 | & \leq 10 \\ | 2 | & \leq 10 \end{aligned}$
$\begin{align*}|0+2| & \le 10\\ |2| & \le 10 \end{align*}$

When graphing this inequality, we have
::在绘制这种不平等的图表时,我们

Notice that this particular absolute value inequality has a solution that is an “and” inequality because the solution is between two numbers.
::注意到这种特定的绝对价值不平等有一个“和”不平等的解决办法,因为解决办法在两个数字之间。

If $\begin{align*}|ax+b| < c\end{align*}$ , then $\begin{align*}-c < ax+b < c\end{align*}$ .
::如果+ax+bc,则-c<ax+b>c。

If $\begin{align*}|ax+b| \le c\end{align*}$ , then $\begin{align*}-c \le ax+b \le c\end{align*}$ .
::如果ax+bc,则-cx+bc。

If $\begin{align*}|ax+b| > c\end{align*}$ , then $\begin{align*}ax+b<-c\end{align*}$ or $\begin{align*}ax+b>c\end{align*}$ .
::如果+ax+bc,则ax+bc或ax+b>c。

If $\begin{align*}|ax+b| \ge c\end{align*}$ , then $\begin{align*}ax+b \le -c\end{align*}$ or $\begin{align*}ax+b \ge c\end{align*}$ .
::如果+ax+bc,则ax+bc或ax+bc。

If you are ever confused by the rules above, you can always test one or two solutions and graph it.
::如果你被上面的规则弄混了, 你总是可以测试一个或两个解决方案, 并用图表绘制它。

Solve and graph $\begin{align*}|4x-3|>9.\end{align*}$
::解析和图表 #% 4x - 3\\\ 9 。

Break apart the absolute value inequality to find the two solutions.
::打破绝对价值不平等,找到两种解决办法。

$\begin{aligned} | 4 x - 3 | > 9 \\ ↙↘ \\ 4 x - 3 > 9 4 x - 3 < - 9 \\ 4 x > 12 4 x < - 6 \\ x > 3 x < - \frac{3}{2} \end{aligned}$
$\begin{align*} & \qquad \ \ \ |4x-3|>9\\ & \qquad \qquad \swarrow \searrow\\ & 4x-3>9 \quad \ 4x-3<-9\\ & \quad \ \ 4x>12 \qquad \ 4x<-6\\ & \qquad x>3 \qquad \quad \ \ x<-\frac{3}{2}\end{align*}$
::4x- 394x- 39 9 9 4x>12 4x6x3 x32

Test a solution, $\begin{align*}x = 5:\end{align*}$
::测试溶液, x=5:

$\begin{aligned} | 4 (5) - 3 | & > 9 \\ | 20 - 3 | & > 9 \\ 17 & > 9 \end{aligned}$
$\begin{align*}|4(5)-3|& >9\\ |20-3| & >9 \\ 17 & >9\end{align*}$

The graph is:
::图表是:

Solve $\begin{align*}|-2x+5|<11.\end{align*}$
::解决% 2x+5+% 11 。

Given the rules above, this will become an “and” inequality, so the solution will be a range between two values.
::鉴于上述规则,这将成为一种“和”的不平等,因此解决办法将在两种价值观之间找到。

$\begin{aligned} | - 2 x + 5 | < 11 \\ ↙↘ \\ - 2 x + 5 < 11 - 2 x + 5 > - 11 \\ - 2 x < 6 - 2 x > - 16 \\ x > - 3 x < 8 \end{aligned}$
$\begin{align*}& \qquad \quad \ |-2x+5|<11\\ & \qquad \qquad \quad \ \ \swarrow \searrow\\ & -2x+5<11 \quad -2x+5>-11\\ & \quad \ \ -2x<6 \qquad \quad \ -2x>-16\\ & \qquad \quad \ x>-3 \qquad \qquad \ x<8\end{align*}$
::@ 2x+5 @ 5 @ 11 @ 2x+5 < 11- 2x+5 @ 11- 2x < 6- 2x_ 16 x 3 x < 8

The solution is $\begin{align*}x\end{align*}$ is greater than $\begin{align*}-3\end{align*}$ and less than $\begin{align*}8\end{align*}$ . In other words, the solution is $\begin{align*}-3 < x < 8\end{align*}$ .
::溶液x大于-3,小于8。换句话说,溶液为-3<x<8。

The graph is:
::图表是:

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the range of weights for a volleyball.
::早些时候,有人要求你找到排球的重量范围

Set up an absolute value inequality where $\begin{align*}w\end{align*}$ is the range of weights of the volleyball.
::在排球重量范围内建立绝对价值不平等

$\begin{aligned} | w - 260 | \leq 2.6 \\ ↙↘ \\ w - 260 \leq 2.6 w - 260 \geq - 2.6 \\ w \leq 262.6 w \geq 257.4 \end{aligned}$
$\begin{align*}& \qquad \qquad |w-260| \le 2.6\\ & \qquad \qquad \quad \swarrow \searrow\\ & w-260 \le 2.6 \qquad w-260 \ge -2.6\\ & \quad \ \ w \le 262.6 \qquad \qquad \ w \ge 257.4\end{align*}$
::- 2602.6 - 2602.6 - 2602.6 w 262.6 w 257.4

So, the range of the weight of a volleyball is $\begin{align*} 257.4 \le w \le 262.6 \end{align*}$ grams.
::排球的重量范围是257.4瓦262.6克

Example 2
::例2

Is $\begin{align*}x = \text{-}4\end{align*}$ a solution to $\begin{align*}|15-2x|>9?\end{align*}$
::x=4 是15 -2x9的解决方案吗?

Plug in -4 for $\begin{align*}x\end{align*}$ to see if it works.
::x 插插到 - 4 中以查看它是否有效。

$\begin{aligned} | 15 - 2 (- 4) | > 9 \\ | 15 + 8 | > 9 \\ | 23 | > 9 \\ 23 > 9 \end{aligned}$
$\begin{align*}|15-2(-4)| > 9\\ |15+8| > 9\\ |23| > 9\\ 23 > 9\end{align*}$

Yes, -4 works, so it is a solution to this absolute value inequality.
::是的,4有效,所以这是解决这种绝对价值不平等的办法。

Example 3
::例3

Solve and graph $\begin{align*}\bigg |\frac{2}{3}x+5 \bigg | \le 17.\end{align*}$
::解析和图表 #% 23x+5+% 17 。

Split apart the inequality to find the two answers.
::将不平等分割开来找到两个答案。

$\begin{aligned} | \frac{2}{3} x + 5 | \leq 17 \\ ↙↘ \\ | \frac{2}{3} x + 5 | \leq 17 \frac{2}{3} x + 5 \geq - 17 \\ \frac{2}{3} x \leq 12 \frac{2}{3} x \geq - 22 \\ x \leq 12 \cdot \frac{3}{2} x \geq - 22 \cdot \frac{3}{2} \\ x \leq 18 x \geq - 33 \end{aligned}$
$\begin{align*}& \qquad \quad \ \bigg | \frac{2}{3}x+5\bigg | \le 17\\ & \qquad \qquad \ \ \ \swarrow \searrow\\ & \bigg | \frac{2}{3}x+5\bigg | \le 17 \qquad \frac{2}{3}x+5 \ge -17\\ & \qquad \ \frac{2}{3}x \le 12 \qquad \quad \ \ \frac{2}{3}x \ge -22\\ & \qquad \quad x \le 12 \cdot \frac{3}{2} \qquad \quad x \ge -22 \cdot \frac{3}{2}\\ & \qquad \quad x \le 18 \qquad \qquad \ x \ge -33\end{align*}$
::23x+5}17 23x+5 1723x+5 1723x5 23x12 23x22x12x32x22x32x18x_33

Test a solution, $\begin{align*}x = 0:\end{align*}$
::测试溶液, x=0 :

$\begin{aligned} | \frac{2}{3} (0) + 5 | & \leq 17 \\ | 5 | & \leq 17 \\ 5 & \leq 17 \end{aligned}$
$\begin{align*}\bigg | \frac{2}{3}(0)+5\bigg | & \le 17\\ |5| & \le 17 \\ 5 & \le 17\end{align*}$

Review
::回顾

Determine if the following numbers are solutions to the given absolute value inequalities.
::确定以下数字是否是特定绝对值不平等的解决办法。

$\begin{align*}|x-9|>4;10\end{align*}$
::X- 94;10

$\begin{align*}\bigg | \frac{1}{2} x-5 \bigg | \le 1;8\end{align*}$
::12x-51;8

$\begin{align*}|5x+14| \ge 29;-8\end{align*}$
::5x+14=29;-8

Solve and graph the following absolute value inequalities.
::解决以下绝对价值不平等问题并绘制图表。

$\begin{align*}|x+6|>12\end{align*}$
::X+612

$\begin{align*}|9-x| \le 16\end{align*}$
::9 -x16

$\begin{align*}|2x-7| \ge 3\end{align*}$
::2x-73

$\begin{align*}|8x-5|<27\end{align*}$
::8x-527

$\begin{align*}\bigg | \frac{5}{6}x+1 \bigg |>6\end{align*}$
::56x+16

$\begin{align*}|18-4x| \le 2\end{align*}$
::18 -4x2

$\begin{align*}\bigg | \frac{3}{4}x-8 \bigg |>13\end{align*}$
::34x-813

$\begin{align*}|6-7x| \le 34\end{align*}$
::6 -7x34

$\begin{align*}|19+3x| \ge 46\end{align*}$
::19+3x*46

Solve the following absolute value inequalities. $\begin{align*}a\end{align*}$ is greater than zero.
::解决以下绝对值不平等。 a大于零。

$\begin{align*}|x-a|>a\end{align*}$
::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

$\begin{align*}|x+a| \le a\end{align*}$
::X+aa

$\begin{align*}|a-x| \le a\end{align*}$
:a) -xa

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Section outline

General

解决绝对价值不平等

Absolute Value Inequalities ::绝对价值不平等

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Review ::回顾

Review (Answers) ::回顾(答复)

Absolute Value Inequalities
::绝对价值不平等

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Review
::回顾

Review (Answers)
::回顾(答复)