6.2 线性不平等

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  线性不平等

  Could you solve an inequality like $\begin{array}{r}x-12>-5\end{array}$ ? 

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  Linear Inequalities
  ::线性不平等

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  To solve an inequality we must isolate the variable on one side of the inequality sign. To isolate the variable, we use the same basic techniques used in solving equations .
  ::为了解决不平等问题,我们必须在不平等标志的一方孤立变量。 为了孤立变量,我们使用了解决方程式所使用的同样基本技术。

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  We can solve some inequalities by adding or subtracting a constant from one side of the inequality.
  ::我们可以通过从不平等的一方增加或减去一个常数来解决一些不平等。

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  Solve the inequality and graph the solution set.
  ::解决不平等问题,绘制解决方案的图表。

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  $\begin{array}{r}x-3<10\end{array}$
  ::x-3 < 10

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  Starting inequality: $\begin{array}{r}x-3<10\end{array}$
  ::起始不平等: x- 3 < 10

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  Add 3 to both sides of the inequality: $\begin{array}{r}x-3+3<10+3\end{array}$
  ::不平等的两面加3:x-3+3 <10+3

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  Simplify: $\begin{array}{r}x<13\end{array}$
  ::简化: x<13

  

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  Solve the inequality and graph the solution set.
  ::解决不平等问题,绘制解决方案的图表。

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  $\begin{array}{r}x-20\le 14\end{array}$
  ::x - 2014

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  Starting inequality: $\begin{array}{r}x-20\le 14\end{array}$
  ::起始不平等: x-2014

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  Add 20 to both sides of the inequality: $\begin{array}{r}x-20+20\le 14+20\end{array}$
  ::不平等的两面加20:x-20+2014+20

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  Simplify: $\begin{array}{r}x\le 34\end{array}$
  ::简化: x34

  

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  Solving Inequalities Using Multiplication and Division
  ::利用乘数和分数解决不平等问题

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  We can also solve inequalities by multiplying or dividing both sides by a constant. For example, to solve the inequality $\begin{array}{r}5x<3\end{array}$ , we would divide both sides by 5 to get $\begin{array}{r}x<\frac{3}{5}\end{array}$ .
  ::我们还可以通过将双方乘以常数或除以常数来解决不平等问题。 例如,为了解决不平等问题,我们将将双方除以5,再取x<35。

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  However, something different happens when we multiply or divide by a negative number. We know, for example, that 5 is greater than 3. But if we multiply both sides of the inequality $\begin{array}{r}5>3\end{array}$ by -2, we get $\begin{array}{r}-10>-6\end{array}$ . And we know that’s not true; -10 is less than -6.
  ::然而,当我们乘以负数或除以负数时,就会发生不同的事情。例如,我们知道,5比3大于3,但如果我们乘以不平等的两面5>3乘以 -2,我们就会得到-106。我们知道这不是真的; -10小于 -6。

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  This happens whenever we multiply or divide an inequality by a negative number, and so we have to flip the sign around to make the inequality true. For example, to multiply $\begin{array}{r}2<4\end{array}$ by -3, first we multiply the 2 and the 4 each by -3, and then we change the < sign to a > sign, so we end up with $\begin{array}{r}-6>-12\end{array}$ .
  ::当我们把不平等乘以负数或除以负数时,就会发生这种情况,因此我们必须翻转标志,使不平等变为真实。例如,将2 < 4乘以 - 3,首先将2和4乘以 - 3,然后将 " 符号 " 改为 " 符号 " ,我们最终将使用 - 6* 12。

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  The same principle applies when the inequality contains variables.
  ::当不平等包含变数时,同样的原则也适用。

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  Solve the inequality.
  ::解决不平等。

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  $\begin{array}{r}4x<24\end{array}$
  ::4x < 24

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  Original problem: $\begin{array}{r}4x<24\end{array}$
  ::原件: 4x<24

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  Divide both sides by 4: $\begin{array}{r}\frac{4x}{4}<\frac{24}{4}\end{array}$
  ::将双方除以4:4:4x4<244

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  Simplify: $\begin{array}{r}x<6\end{array}$
  ::简化: x <6

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  Solve the inequality.
  ::解决不平等。

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  $\begin{array}{r}-5x\le 21\end{array}$
  ::- 5x21 - 5x21

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  Original problem: $\begin{array}{r}-5x\le 21\end{array}$
  ::原件: - 5x21

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  Divide both sides by -5 : $\begin{array}{r}\frac{-5x}{-5}\ge \frac{21}{-5}\end{array}$ Flip the inequality sign.
  ::将两边除以 - 5 : - 5x- 521-5 翻转不平等标志 。

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  Simplify: $\begin{array}{r}x\ge -\frac{21}{5}\end{array}$
  ::简化: x% 215

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  Examples
  ::实例

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  Example 1
  ::例1

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  Solve the inequality.
  ::解决不平等。

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  $\begin{array}{r}x+8\le -7\end{array}$
  ::x+8+7

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  Starting inequality: $\begin{array}{r}x+8\le -7\end{array}$
  ::起始不平等: x+87

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  Subtract 8 from both sides of the inequality: $\begin{array}{r}x+8-8\le -7-8\end{array}$
  ::从不平等的两侧减去8:x+8-8-8+7-8

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  Simplify: $\begin{array}{r}x\le -15\end{array}$
  ::简化: x% 15

  

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  Example 2
  ::例2

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  $\begin{array}{r}x+4>13\end{array}$
  ::x+4>13

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  Starting inequality: $\begin{array}{r}x+4>13\end{array}$
  ::起始不平等: x+4>13

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  Subtract 4 from both sides of the inequality: $\begin{array}{r}x+4-4>13-4\end{array}$
  ::从不平等的两侧减去4:x+4-4>13-4

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  Simplify: $\begin{array}{r}x>9\end{array}$
  ::简化: x>9

  

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  Example 3
  ::例3

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  $\begin{array}{r}\frac{x}{25}<\frac{3}{2}\end{array}$
  ::x25<32

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  Original problem: $\begin{array}{r}\frac{x}{25}<\frac{3}{2}\end{array}$
  ::原件: x25<32

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  Multiply both sides by 25: $\begin{array}{r}25\cdot \frac{x}{25}<\frac{3}{2}\cdot 25\end{array}$
  ::将两边乘以 25: 25x25 < 3225

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  Simplify: $\begin{array}{r}x<\frac{75}{2}\end{array}$ or $\begin{array}{r}x<37.5\end{array}$
  ::简化: x<752 或 x<37.5

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  Example 4
  ::例4

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  $\begin{array}{r}\frac{x}{-7}\ge 9\end{array}$
  ::x79

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   Original problem: $\begin{array}{r}\frac{x}{-7}\ge 9\end{array}$
  ::原件: x-79

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  Multiply both sides by -7: $\begin{array}{r}-7\cdot \frac{x}{-7}\le 9\cdot (-7)\end{array}$ Flip the inequality sign.
  ::将两边乘以 - 7: - 7x- 79(-7) 翻转不平等标志 。

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  Simplify: $\begin{array}{r}x\le -63\end{array}$
  ::简化: x63

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  Review
  ::回顾

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  For 1-8, solve each inequality and graph the solution on the number line.
  ::对于 1-8, 解决每一个不平等问题, 并在数字线上绘制解决方案 。

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  1. $\begin{array}{r}x-5<35\end{array}$
     ::x-5<35
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  2. $\begin{array}{r}x+15\ge -60\end{array}$
     ::x+1560
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  3. $\begin{array}{r}x-2\le 1\end{array}$
     ::x-21
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  4. $\begin{array}{r}x-8>-20\end{array}$
     ::x-820
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  5. $\begin{array}{r}x+11>13\end{array}$
     ::x+11>13
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  6. $\begin{array}{r}x+65<100\end{array}$
     ::x+65 < 100
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  7. $\begin{array}{r}x-32\le 0\end{array}$
     ::x-320
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  8. $\begin{array}{r}x+68\ge 75\end{array}$
     ::x+68_75

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  For 9-12, solve each inequality. Write the solution as an inequality and graph it.
  ::9-12 解决每一个不平等问题。 将解决方案写成不平等问题, 并绘制图表 。

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  9. $\begin{array}{r}3x\le 6\end{array}$
     ::3x_6
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  10. $\begin{array}{r}\frac{x}{5}>-\frac{3}{10}\end{array}$
      ::x5310
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  11. $\begin{array}{r}-10x>250\end{array}$
      ::- 10x>250
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  12. $\begin{array}{r}\frac{x}{-7}\ge -5\end{array}$
      ::x7=5x7=5

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  Review (Answers)
  ::回顾(答复)

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  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 。