Course: 3.10 解决双重不平等-interactive

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解决双步不平等

In This Lesson
::在本课程中

Solving inequalities is a very similar process to solving equations . I solate the variable by undo-ing the operations in the inequality and check the solution. Often, you will also need to graph the solution on a number line. Here is an example of solving an inequality:
::解决不平等是一个与解决方程式非常相似的过程。通过撤销不平等操作来孤立变量, 并检查解决方案。通常, 您还需要用数字线来绘制解决方案。以下是解决不平等的例子 :

\begin{aligned} \frac{1}{2} x & + 9 < 4 isolate x \\ - 9 - 9 subtract 9 from both sides \\ \frac{1}{2} x < - 5 simplify \\ 2 \cdot & \frac{1}{2} x < - 5 \cdot 2 multiply both sides by 2 \\ x < - 10 \end{aligned}

$\begin{align*}\frac{1}{2}x&+9<4 ~~~~~~~~~~~~~\text{ isolate }x\\ &\textcolor{red}{ ~-9~~~-9} ~~~~~~~~~~\text{subtract 9 from both sides}\\ &\frac{1}{2}x<-5 \space\space\space\space\space\space\space\space\space\space\space\space\text{simplify}\\ \textcolor{red}{\cancel{2}}\cdot& \frac{1}{\textcolor{red}{\cancel{2}}}x<-5\cdot \textcolor{red}{2} ~~~~~\text{multiply both sides by 2}\\ &x<-10\end{align*}$
::12+9 < 4 分离 x- 9 - 9 减去两侧的 9 12 x @ @ 5 简化2\ 12x @ 52 将两侧乘以 2x @ 10

To graph this inequality, plot a point at -10. Because the sign is only less than , the point will be open, to indicate that -10 is not a part of the solution set . Then the shading will be on the left side of the point.
::要绘制这种不平等图, 请在 - 10 处绘制一个点。因为标记小于 - 10 。因为标记将打开点, 显示 - 10 不是设定的解决方案的一部分。然后阴影将位于点的左侧。

Number line showing an open point at -10 with shading to the left.

Now, select a point that is in this range to test the solution. Any value less than -10 should work. Let's pick -12.
::现在, 选择一个在此范围内的点来测试解决方案。任何值小于 - 10 应该有效。让我们选择 - 12 。

\begin{aligned} \frac{1}{2} (- 12) + 9 & < 4 \\ - 6 + 9 & < 4 \\ 3 & < 4 ✓ \end{aligned}

$\begin{align*}\frac12 \left( -12\right) + 9 &< 4\\ -6 + 9 &< 4 \\ 3 &< 4 ~~ \checkmark \\\end{align*}$

This is a true inequality and therefore our solution is correct.
::这是真正的不平等,因此我们的解决办法是正确的。

W hat if the coefficient is negative? What do you think will happen to the inequality?
::如果系数是负的呢?你认为不平等会怎样?

Flip My Sign
::翻转我的标志

Fill out the table below.
::表格如下。

Discussion Question
::讨论问题

Discuss with a classmate the steps to solving $\begin{align*}4−\frac13 t<2.\end{align*}$ Make sure to check a number in the solution range and graph it on a number line.
::与同学讨论解决 4 - 13t<2 的步骤。务必在解决方案范围内检查一个数字, 并在数字行上绘制图表。

In this interactive, you learned that whenever you multiply or divide by a negative number, you must flip the inequality sign to so that the statement remains true. Other than this important detail, solving an inequality is exactly like solving an equation . In addition , when checking your solution, always use a value from the shaded part of the graph.
::在此互动中, 您学会了, 当您以负数乘或除以负数时, 您必须翻转不平等符号, 以使声明保持真实。除了这个重要细节外, 解决不平等问题完全就像解决方程式一样。此外, 在检查您的解决方案时, 您总是使用图表阴影部分的值。

When graphing an inequality, you need to determine if the value that was solved for is in the solution set or not and which way to shade. If the sign is < or >, the value is not in the solution set and has an open circle to indicate that. If the sign is ≤ or ≥, the value is included and will have a closed, or filled in, circle. See the table for guidelines.
::当绘制不平等图时,您需要确定所解决的值是否在解决方案设置中,以及阴影方式。如果符号 < 或 > ,则该值不在解决方案设置中,并有一个可显示该值的开阔圆。如果符号为 _ 或 _ ,则包含该值并将有一个封闭的圆或填充在圆中。请参见指南表格。

Inequality	Circle is...	Shading is...	Graph
$\begin{aligned} x & < a \\ or \\ a & > x \end{aligned}$ $\begin{align}x&<a \\ &\text{or}\\ a&>x\end{align}$	open	to the left
$\begin{aligned} x & \leq a \\ or \\ a & \geq x \end{aligned}$ $\begin{align}x&\le a \\ &\text{or}\\ a &\ge x\end{align}$	closed	to the left
$\begin{aligned} x & > a \\ or \\ a & < x \end{aligned}$ $\begin{align}x&>a \\ &\text{or}\\ a&<x\end{align}$	open	to the right
$\begin{aligned} x & \geq a \\ or \\ a & \geq x \end{aligned}$ $\begin{align}x& \ge a \\ &\text{or}\\ a& \ge x\end{align}$	closed	to the right

Solution Burst
::解决方案布局

In this interactive, you are given an inequality to solve at the top. Possible answers will "fly" by and you will need to click on any that are correct . Correct answers receive one point, but if you click on an incorrect answer, you will lose a point.
::在此互动中, 您可以在顶部解决不平等问题。可能的答案会“ 飞过” , 您需要点击任何正确的答案。正确的答案会收到一个点, 但如果您点击错误的答案, 您将会失去一个点。

Discussion Question
::讨论问题

Mekhi claims that there are no values of $\begin{align*}n\end{align*}$ that make the inequality $\begin{align*}n+1 < n\end{align*}$ true. Do you agree or disagree? Use evidence to support your argument.
::Mekhi 声称没有 n 的值可以证明不平等 n+1<n 。您同意还是不同意? 使用证据支持您的论点。

Summary
::摘要

When multiplying or dividing both sides by a negative number, you need to flip the inequality sign so that the statement remains true.
::当将双方乘以负数或除以负数时,你需要翻转不平等标志,以使声明保持真实。

When graphing the solution of an inequality :
::当绘制不平等解决方案的图表时:

$\begin{align*}\geq\end{align*}$ and $\begin{align*}\leq\end{align*}$ will have closed circles
::. . . 和 . . . . . . . . . .

$\begin{align*}>\end{align*}$ and $\begin{align*}<\end{align*}$ will have open circles
::和 < 将开圆圈

Section outline

General

解决双步不平等

In This Lesson ::在本课程中

Flip My Sign ::翻转我的标志

Solution Burst ::解决方案布局

Summary ::摘要

In This Lesson
::在本课程中

Flip My Sign
::翻转我的标志

Solution Burst
::解决方案布局

Summary
::摘要