Question

1017. Решите неравенства: 1) |9 – х| = 2;5) |x – 5| <11;6) |6 – x| > 7.ДАЮ 20 БАЛЛОВ​

Answer 1

$\displaytyle \tt 1) \: |9-x|=2\\\\ \displaytyle \tt 9-x=2\\\displaytyle \tt 9-x=-2\\\\\displaytyle \tt -x=-7\\\displaytyle \tt -x=-11\\\\ \displaytyle \tt \boxed{\bold{x_1=7}}\\\displaytyle \tt \boxed{\bold{x_2=11}}$

$\displaytyle \tt 5)\: |x-5|$

$\displaytyle \tt 6) \: |6-x|>7\\\\ \displaytyle \tt 6-x>7, \: \: 6-x\geq 0\\\displaytyle \tt -(6-x)>7, \: \: 6-x6\\\\ \displaytyle \tt x\in(-\infty; \: -1)\\\displaytyle \tt x\in(13; \: +\infty)\\\\ \displaytyle \tt \boxed{\bold{x\in(-\infty; \: -1)\cup (13; \: +\infty)}}$

Answer 2

1. |9 - x| = 2

To solve this inequality, we'll consider two cases: when the expression inside the absolute value is positive and when it is negative.

Case 1: 9 - x > 0 (positive) In this case, we have |9 - x| = 9 - x. Therefore, we can rewrite the equation as follows: 9 - x = 2

Solving for x: 9 - 2 = x 7 = x

So x = 7 is a solution.

Case 2: 9 - x < 0 (negative) In this case, we have |9 - x| = -(9 - x). Therefore, we can rewrite the equation as follows: -(9 - x) = 2

Solving for x: -9 + x = 2 x = 2 + 9 x = 11

So x = 11 is another solution.

Therefore, the solutions to the inequality |9 - x| = 2 are x = 7 and x = 11.

5. |x - 5| < 11

To solve this inequality, we'll consider two cases: when the expression inside the absolute value is positive and when it is negative.

Case 1: x - 5 > 0 (positive) In this case, we have |x - 5| = x - 5. Therefore, we can rewrite the inequality as follows: x - 5 < 11

Solving for x: x < 11 + 5 x < 16

So x < 16 is a solution.

Case 2: x - 5 < 0 (negative) In this case, we have |x - 5| = -(x - 5). Therefore, we can rewrite the inequality as follows: -(x - 5) < 11

Solving for x:

• x + 5 < 11
• x < 11 - 5
• x < 6

Dividing both sides by -1 and flipping the inequality sign: x > -6

So x > -6 is another solution.

Therefore, the solutions to the inequality |x - 5| < 11 are x < 16 and x > -6.

6. |6 - x| > 7

To solve this inequality, we'll consider two cases: when the expression inside the absolute value is positive and when it is negative.

Case 1: 6 - x > 0 (positive) In this case, we have |6 - x| = 6 - x. Therefore, we can rewrite the inequality as follows: 6 - x > 7

Solving for x: -x > 7 - 6 -x > 1

Multiplying both sides by -1 and flipping the inequality sign: x < -1

So x < -1 is a solution.

Case 2: 6 - x < 0 (negative) In this case, we have |6 - x| = -(6 - x). Therefore, we can rewrite the inequality as follows: -(6 - x) > 7

Solving for x: -x + 6 > 7 -x > 7 - 6 -x > 1

Multiplying both sides by -1 and flipping the inequality sign: x < -1

So x < -1 is another solution.

Therefore, the solution to the inequality |6 - x| > 7 is x < -1.

The solutions for the given inequalities are:

1. x = 7, 11
2. x < 16, x > -6
3. x < -1