Question

1)|х+3|<4; 2)|х-1|≤2; 3)|5+х|<8​

Answer 1

Ответ:

1) -4<x+3<4

-4-3<x<4-3

-7<x<1

2)-2<x-1<2

-2+1<x<2+1

-1<x<3

3) -8<5+x<8

-7-5<x<8-5

-12<x<-3

Answer 2

Let's solve each of these absolute value inequalities step by step:

1. |x + 3| < 4:

To solve this inequality, we can consider two cases:

Case 1: (x + 3) < 4 Solve for x: x + 3 < 4 x < 4 - 3 x < 1

Case 2: -(x + 3) < 4 Solve for x: -(x + 3) < 4 Multiply both sides by -1 (and reverse the inequality sign): x + 3 > -4 x > -4 - 3 x > -7

So, for case 1, we have x < 1, and for case 2, we have x > -7. The solution for the absolute value inequality is -7 < x < 1.

2. |x - 1| ≤ 2:

To solve this inequality, we can again consider two cases:

Case 1: (x - 1) ≤ 2 Solve for x: x - 1 ≤ 2 x ≤ 2 + 1 x ≤ 3

Case 2: -(x - 1) ≤ 2 Solve for x: -(x - 1) ≤ 2 Multiply both sides by -1 (and reverse the inequality sign): x - 1 ≥ -2 x ≥ -2 + 1 x ≥ -1

So, for case 1, we have x ≤ 3, and for case 2, we have x ≥ -1. The solution for the absolute value inequality is -1 ≤ x ≤ 3.

3. |5 + x| < 8:

To solve this inequality, we can again consider two cases:

Case 1: (5 + x) < 8 Solve for x: 5 + x < 8 x < 8 - 5 x < 3

Case 2: -(5 + x) < 8 Solve for x: -(5 + x) < 8 Multiply both sides by -1 (and reverse the inequality sign): 5 + x > -8 x > -8 - 5 x > -13

So, for case 1, we have x < 3, and for case 2, we have x > -13. The solution for the absolute value inequality is -13 < x < 3.