Question

|x-7|>0; |x-4|<3; |2+x|<3; |x+3|>2; |x-4|>3; |x+2|>5​

Answer 1

|x-7|>0; х- любое, кроме х=7, т.к. в этом случае модуль равен нулю.

|x-4|<3; -3<х-4<3; 1<х<7

|2+x|<3; -3<х+2<3; -5<х<1

|x+3|>2; х+3>2 или  x+3<-2;  х>-1 или x<-5

|x-4|>3; х-4>3 или  x-4<-3; х>7 или x<1

Ix+2|>5​; х+2>5 или  x+2<-5; х>3 или x<-7

Answer 2

To solve these absolute value inequalities, we need to consider the cases when the expression inside the absolute value is positive and negative. Let's analyze each inequality separately:

1. |x - 7| > 0: This inequality is always true since the absolute value of any real number is non-negative. Therefore, all real numbers satisfy this inequality.

2. |x - 4| < 3: We consider two cases: Case 1: x - 4 > 0: In this case, the inequality becomes x - 4 < 3, which simplifies to x < 7. Case 2: x - 4 < 0: In this case, the inequality becomes -(x - 4) < 3, which simplifies to -x + 4 < 3. Solving for x gives x > 1. Combining the two cases, the solution is 1 < x < 7.

3. |2 + x| < 3: We consider two cases: Case 1: 2 + x > 0: In this case, the inequality becomes 2 + x < 3, which simplifies to x < 1. Case 2: 2 + x < 0: In this case, the inequality becomes -(2 + x) < 3, which simplifies to -x - 2 < 3. Solving for x gives x > -5. Combining the two cases, the solution is -5 < x < 1.

4. |x + 3| > 2: We consider two cases: Case 1: x + 3 > 0: In this case, the inequality becomes x + 3 > 2, which simplifies to x > -1. Case 2: x + 3 < 0: In this case, the inequality becomes -(x + 3) > 2, which simplifies to -x - 3 > 2. Solving for x gives x < -5. Combining the two cases, the solution is x < -5 or x > -1.

5. |x - 4| > 3: We consider two cases: Case 1: x - 4 > 0: In this case, the inequality becomes x - 4 > 3, which simplifies to x > 7. Case 2: x - 4 < 0: In this case, the inequality becomes -(x - 4) > 3, which simplifies to -x + 4 > 3. Solving for x gives x < 1. Combining the two cases, the solution is x < 1 or x > 7.

6. |x + 2| > 5: We consider two cases: Case 1: x + 2 > 0: In this case, the inequality becomes x + 2 > 5, which simplifies to x > 3. Case 2: x + 2 < 0: In this case, the inequality becomes -(x + 2) > 5, which simplifies to -x - 2 > 5. Solving for x gives x < -7. Combining the two cases, the solution is x < -7 or x > 3.

So, combining all the solutions, we have: -7 < x < -5, -1 < x < 1, 1 < x < 3, 7 < x.