Course: 6.3 多重不平等 | The Way To Learn

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多重不平等
Suppose you wanted to know if you had enough money to buy a movie ticket, but only knew that one friend bought two tickets and a popcorn, and that she spent more than another friend who purchased three tickets. Could you describe the price of popcorn compared to the price of a ticket? If you knew that the situation could be described with the inequality $\begin{aligned} 2 t + p > 3 t, \end{aligned}$ could you solve the inequality for $\begin{aligned} p \end{aligned}$ , the popcorn price?
::假设你想知道你是否有足够的钱买电影票,但只知道一个朋友买了两张票和爆米花,而且她花的钱比另一个朋友多,她买了三张票。你能描述一下爆米花的价格和票价的比较吗?如果你知道情况可以用不平等 2t+p>3t来描述,你能解决P的不平等,爆米花的价格吗?

Multi-Step Inequalities
::多重不平等

M ost inequalities require several steps to arrive at the solution. As with solving equations , we must use the to find the correct solution. In addition , remember that when we multiply or divide the inequality by a negative number, the direction of the inequality sign changes.
::多数不平等需要几个步骤才能找到解决方案。和解决方程式一样,我们必须利用方程式找到正确的解决方案。此外,记住当我们把不平等增加或除以负数时,不平等的方向就会改变。

The general procedure for solving multi-step inequalities is almost exactly like the procedure for solving :
::解决多阶段不平等的一般程序与解决以下问题的程序几乎一模一样:

Clear " data-term="Parentheses" role="term" tabindex="0"> parentheses on both sides of the inequality and collect like terms .
::在不平等的两侧划出清晰的括号,并收集类似术语。

Add or subtract terms so the variable is on one side and the constant is on the other side of the inequality sign.
::添加或减去条件,使变量位于不平等符号的一边,而常数位于不平等符号的另一方。

Multiply and divide by whatever constants are attached to the variable. Remember to change the direction of the inequality sign if you multiply or divide by a negative number.
::乘以和除以变量的任意常数。记住要改变不平等信号的方向, 如果您乘以负数或除以负数, 请记住要改变不平等信号的方向。

Solve the inequality $\begin{aligned} \frac{9 x}{5} - 7 \geq - 3 x + 12 \end{aligned}$ and graph the solution set .
::解决不平等问题 9x5-73x+12 并绘制解决方案集图。

Original problem: $\begin{aligned} \frac{9 x}{5} - 7 \geq - 3 x + 12 \end{aligned}$
::原名: 9x5-73x+12

Add $\begin{aligned} 3 x \end{aligned}$ to both sides: $\begin{aligned} \frac{9 x}{5} + 3 x - 7 \geq - 3 x + 3 x + 12 \end{aligned}$
::向两侧增加3x: 9x5+3x- 7 @%3x+3x+12

Simplify: $\begin{aligned} \frac{24 x}{5} - 7 \geq 12 \end{aligned}$
::简化: 24x5- 712

Add 7 to both sides: $\begin{aligned} \frac{24 x}{5} - 7 + 7 \geq 12 + 7 \end{aligned}$
::双方加7:24x5-7+7+12+7

Simplify: $\begin{aligned} \frac{24 x}{5} \geq 19 \end{aligned}$
::简化: 24x519

Multiply 5 to both sides: $\begin{aligned} 5 \cdot \frac{24 x}{5} \geq 5 \cdot 19 \end{aligned}$
::双方乘以5 : 5}24x5}5}19

Simplify: $\begin{aligned} 24 x \geq 95 \end{aligned}$
::简化: 24x95

Divide both sides by 24: $\begin{aligned} \frac{24 x}{24} \geq \frac{95}{24} \end{aligned}$
::将双方除以24:24x24_9524

Simplify: $\begin{aligned} x \geq \frac{95}{24} \end{aligned}$ Answer
::简化: x9524

Graph:
::图 :

Solve the inequality $\begin{aligned} - 25 x + 12 \leq - 10 x - 12 \end{aligned}$ and graph the solution set.
::解决不平等 - 25x+1210x-12 并绘制解决方案集图。

Original problem: $\begin{aligned} - 25 x + 12 \leq - 10 x - 12 \end{aligned}$
::原名: - 25x+1210x-12

Add $\begin{aligned} 10 x \end{aligned}$ to both sides: $\begin{aligned} - 25 x + 10 x + 12 \leq - 10 x + 10 x - 12 \end{aligned}$
::向两侧添加 10x : - 25x+10x+12+12+10x+10x+10x-12

Simplify: $\begin{aligned} - 15 x + 12 \leq - 12 \end{aligned}$
::简化: - 15x+12 @ 12

Subtract 12: $\begin{aligned} - 15 x + 12 - 12 \leq - 12 - 12 \end{aligned}$
::减号12: - 15x+12 - 12* 12 - 12

Simplify: $\begin{aligned} - 15 x \leq - 24 \end{aligned}$
::简化:- 15x% 24

Divide both sides by -15: $\begin{aligned} \frac{- 15 x}{- 15} \geq \frac{- 24}{- 15} \end{aligned}$ flip the inequality sign
::将两边除以 - 15: - 15x- 15\\\\\\\\\\\\\\\ 24- 15F/ 15 翻转不平等符号

Simplify: $\begin{aligned} x \geq \frac{8}{5} \end{aligned}$ Answer
::简化: x85Answer

Graph:
::图 :

Solve the inequality $\begin{aligned} 4 x - 2 (3 x - 9) \leq - 4 (2 x - 9) \end{aligned}$ .
::解决4x-2(3x-9) 4x(2x-9) 4x(2x-9) 不平等。

Original problem: $\begin{aligned} 4 x - 2 (3 x - 9) \leq - 4 (2 x - 9) \end{aligned}$
::原原问题: 4x-2(3x-9) 4x(2x-9)

Simplify parentheses: $\begin{aligned} 4 x - 6 x + 18 \leq - 8 x + 36 \end{aligned}$
::简化括号: 4x- 6x+18_ __ 8x+36

Collect like terms: $\begin{aligned} - 2 x + 18 \leq - 8 x + 36 \end{aligned}$
::收藏类似条件 : - 2x+18\\\\\\\\\\\\\\\\\3\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\36\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Add $\begin{aligned} 8 x \end{aligned}$ to both sides: $\begin{aligned} - 2 x + 8 x + 18 \leq - 8 x + 8 x + 36 \end{aligned}$
::向两侧增加8x: - 2x+8x+18+18+8x+8x+8x+36

Simplify: $\begin{aligned} 6 x + 18 \leq 36 \end{aligned}$
::简化: 6x+1836

Subtract 18: $\begin{aligned} 6 x + 18 - 18 \leq 36 - 18 \end{aligned}$
::减号18: 6x+18-18-18_18_36_18_18

Simplify: $\begin{aligned} 6 x \leq 18 \end{aligned}$
::简化: 6x18

Divide both sides by 6: $\begin{aligned} \frac{6 x}{6} \leq \frac{18}{6} \end{aligned}$
::将双方除以6:6:6x6186

Simplify: $\begin{aligned} x \leq 3 \end{aligned}$ Answer
::简化: x% 3 发报器

Example
::示例示例示例示例

Solve the inequality $\begin{aligned} \frac{5 x - 1}{4} > - 2 (x + 5) \end{aligned}$ .
::解决不平等5x-142(x+5)。

Original problem: $\begin{aligned} \frac{5 x - 1}{4} > - 2 (x + 5) \end{aligned}$
::原问题: 5x-142(x+5)

Simplify parenthesis: $\begin{aligned} \frac{5 x - 1}{4} > - 2 x - 10 \end{aligned}$
::简化括号: 5x- 14}%% 2x- 10

Multiply both sides by 4: $\begin{aligned} 4 \cdot \frac{5 x - 1}{4} > 4 (- 2 x - 10) \end{aligned}$
::将两边乘以 4: 4: 5x- 14> 4(-2x- 10)

Simplify: $\begin{aligned} 5 x - 1 > - 8 x - 40 \end{aligned}$
::简化: 5x- 1\\\\ _8x- 40

Add $\begin{aligned} 8 x \end{aligned}$ to both sides: $\begin{aligned} 5 x + 8 x - 1 > - 8 x + 8 x - 40 \end{aligned}$
::双方增加8x:5x+8x-1=8x+8x-40

Simplify: $\begin{aligned} 13 x - 1 > - 40 \end{aligned}$
::简化: 13x- 1 @%40

Add 1 to both sides: $\begin{aligned} 13 x - 1 + 1 > - 40 + 1 \end{aligned}$
::双方增加1份:13x-1+1+40+1

Simplify: $\begin{aligned} 13 x > - 39 \end{aligned}$
::简化: 13x39

Divide both sides by 13: $\begin{aligned} \frac{13 x}{13} > - \frac{39}{13} \end{aligned}$
::将双方除以13:13x13=3913

Simplify: $\begin{aligned} x > - 3 \end{aligned}$ Answer
::简化: x3Answer

Solve: $\begin{aligned} 3 x - 5 < x + 2 \end{aligned}$
::解决: 3x-5 <x+2

Review
::回顾

Solve each multi-step inequality.
::解决每个多阶段不平等。

$\begin{aligned} 3 x - 5 \geq x + 2 \end{aligned}$
::3x-5x+2 3x-5xx+2

$\begin{aligned} x - 5 > 2 x + 3 \end{aligned}$
::x-5>2x+3

$\begin{aligned} 2 (x - 3) \leq 3 x - 2 \end{aligned}$
::2(x-33)%3x-2

$\begin{aligned} 3 (x + 1) \geq 2 x + 5 \end{aligned}$
::3(x+1)2x+5

$\begin{aligned} 2 (x - 9) \geq - 1 (4 x + 7) \end{aligned}$
::2(x-9)%1(4x+7)

$\begin{aligned} \frac{x}{3} < x + 7 \end{aligned}$
::x3 < x+7 x3 < xx+7

$\begin{aligned} \frac{x}{4} < 2 x - 21 \end{aligned}$
::x4 < 2x- 21

$\begin{aligned} \frac{3 (x - 4)}{12} \leq \frac{2 x}{3} \end{aligned}$
::3(x-4)122x3

$\begin{aligned} 2 (\frac{x}{4} + 3) > 6 (x - 1) \end{aligned}$
::2(x4+3)>6(x-1)

$\begin{aligned} 9 x + 4 \leq - 2 (x + \frac{1}{2}) \end{aligned}$
::9x+42(x+12)

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

多重不平等

Multi-Step Inequalities ::多重不平等

Example ::示例示例示例示例

Review ::回顾

Review (Answers) ::回顾(答复)

Multi-Step Inequalities
::多重不平等

Example
::示例示例示例示例

Review
::回顾

Review (Answers)
::回顾(答复)