Question

|1+2x|<9 решите неравенства​

Answer 1

Ответ:

Пошаговое объяснение:

|1+2x|<9

Допустим: |1+2x|=9

При 1+2x≥0:

1+2x=9; x₁=(9-1)/2=4

При 1+2x<0:

-1-2x=9; -2x=9+1; x₂=10/(-2)=-5

Проверка при x₁>4; x₂>-5: |1+2·5|<9; |11|<9; 11>9 - неравенство не выполняется.

Проверка при x₁<4; x₂<-5: |1+2·(-6)|<9; |-11|<9; 11>9 - неравенство не выполняется.

Проверка при x₁<4; x₂>-5: |1+2·0|<9; |1|<9 - неравенство выполняется.

Проверка при x₁>4; x₂<-5: |1+2·5|<9; |11|<9; 11>9 - неравенство не выполняется; |1+2·(-6)|<9; |-11|<9; 11>9 - неравенство не выполняется.

Следовательно: -5<x<4

Ответ: x∈(-5; 4)

Answer 2

To solve the absolute value inequality |1 + 2x| < 9, we need to consider two cases, depending on the sign of the expression inside the absolute value.

Case 1: (1 + 2x) is positive or zero (1 + 2x ≥ 0): When (1 + 2x) is positive or zero, the absolute value simply becomes the expression itself. So we have:

1 + 2x < 9

Now, let's solve for x:

1 + 2x < 9 2x < 9 - 1 2x < 8 x < 8/2 x < 4

Case 2: (1 + 2x) is negative (1 + 2x < 0): When (1 + 2x) is negative, the absolute value becomes the negative of the expression inside the absolute value. So we have:

-(1 + 2x) < 9

Now, let's solve for x:

-(1 + 2x) < 9 -1 - 2x < 9 -2x < 9 + 1 -2x < 10

Since we have a negative value on the left side of the inequality, we need to flip the inequality sign when dividing by a negative number:

x > 10 / (-2) x > -5

So, the solution for the absolute value inequality is:

-5 < x < 4

In interval notation, the solution is (-5, 4).