Question

1017. Решите неравенства: 1)| 9-x |< 2; 2) |x + 7 |> 8; 3) |10+ x |< 3; 4) |x-8| ≥ 9 ; 5) |x - 5| < 11;6)| 6-x|>7.Решите пожалуйста ​

Answer 1

Ответ:

$1) |9 - x| < 2 \\ - 9 + x < 2 \\ x < 11 \\ 2) |x + 7| > 8 \\ x + 7 > 8 \\ x > 1 \\ 3) |10 + x| < 3 \\ 10 + x < 3 \\ x < - 7$

$4) |x - 8 | \geqslant 9 \\ - x + 8 \geqslant 9 \\ - x \geqslant 1 \\ x \leqslant - 1 \\ 5) |x - 5| < 11 \\ - x + 5 < 11 \\ - x < 6 \\ x > - 6 \\ 6) |x - 6| < 7 \\ - x + 6 < 7 \\ - x < 1 \\ x > - 1$

Пошаговое объяснение:

1). х є (-нескінченність;11)

2) х є (1;+нескінченність)

3) х є (-нескінченність;-7)

4) х є (-нескінченність;-1]

5) х є (-6;+нескінченність)

6) х є (-1;+нескінченність)

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Answer 2

Solving the Inequalities

Let's solve the given inequalities step by step:

1) |9-x| < 2

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

Case 1: 9 - x > 0 In this case, the absolute value can be removed, and we have: 9 - x < 2

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

Case 2: 9 - x < 0 In this case, the absolute value can be removed, but the inequality sign needs to be reversed: -(9 - x) < 2

Solving for x: -x + 9 < 2 -x < 2 - 9 -x < -7

When multiplying or dividing by a negative number, the inequality sign needs to be reversed. So, multiplying both sides by -1: x > 7

Combining the solutions from both cases, we have: x < 7 or x > 7

2) |x + 7| > 8

Again, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.

Case 1: x + 7 > 0 In this case, the absolute value can be removed, and we have: x + 7 > 8

Solving for x: x > 8 - 7 x > 1

Case 2: x + 7 < 0 In this case, the absolute value can be removed, but the inequality sign needs to be reversed: -(x + 7) > 8

Solving for x: -x - 7 > 8 -x > 8 + 7 -x > 15

Multiplying both sides by -1: x < -15

Combining the solutions from both cases, we have: x < -15 or x > 1

3) |10 + x| < 3

Again, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.

Case 1: 10 + x > 0 In this case, the absolute value can be removed, and we have: 10 + x < 3

Solving for x: x < 3 - 10 x < -7

Case 2: 10 + x < 0 In this case, the absolute value can be removed, but the inequality sign needs to be reversed: -(10 + x) < 3

Solving for x: -x - 10 < 3 -x < 3 + 10 -x < 13

Multiplying both sides by -1: x > -13

Combining the solutions from both cases, we have: x < -7 or x > -13

4) |x - 8| ≥ 9

Again, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.

Case 1: x - 8 > 0 In this case, the absolute value can be removed, and we have: x - 8 ≥ 9

Solving for x: x ≥ 9 + 8 x ≥ 17

Case 2: x - 8 < 0 In this case, the absolute value can be removed, but the inequality sign needs to be reversed: -(x - 8) ≥ 9

Solving for x: -x + 8 ≥ 9 -x ≥ 9 - 8 -x ≥ 1

Multiplying both sides by -1: x ≤ -1

Combining the solutions from both cases, we have: x ≤ -1 or x ≥ 17

5) |x - 5| < 11

Again, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.

Case 1: x - 5 > 0 In this case, the absolute value can be removed, and we have: x - 5 < 11

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

Case 2: x - 5 < 0 In this case, the absolute value can be removed, but the inequality sign needs to be reversed: -(x - 5) < 11

Solving for x: -x + 5 < 11 -x < 11 - 5 -x < 6

Multiplying both sides by -1: x > -6

Combining the solutions from both cases, we have: x > -6 or x < 16

6) |6 - x| > 7

Again, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.

Case 1: 6 - x > 0 In this case, the absolute value can be removed, and we have: 6 - x > 7

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

Multiplying both sides by -1: x < -1

Case 2: 6 - x < 0 In this case, the absolute value can be removed, but the inequality sign needs to be reversed: -(6 - x) > 7

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

Multiplying both sides by -1: x < -1

Combining the solutions from both cases, we have: x < -1

Summary of Solutions:

1) |9-x| < 2: - Solution: x < 7 or x > 7

2) |x + 7| > 8: - Solution: x < -15 or x > 1

3) |10 + x| < 3: - Solution: x < -7 or x > -13

4) |x - 8| ≥ 9: - Solution: x ≤ -1 or x ≥ 17

5) |x - 5| < 11: - Solution: x > -6 or x < 16

6) |6 - x| > 7: - Solution: x < -1