Question

||2x+3|-2|=5 ||3x-2|+3|=7​

Answer 1

Пример 1

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Пример 2

$||3x-2|+3|=7\\\left \{ {{|3x-2|+3=7} \atop {|3x-2|+3=-7}} \right. \\\left \{ {{|3x-2|=4} \atop {|3x-2|=-10}} \right.$

|3x-2|=-10 не может быть по определению, так как модуль всегда либо равен нулю, либо больше нуля.

Следовательно:

$|3x-2|=4\\\left \{ {{3x-2=4} \atop {3x-2=-4}} \right. \\\left \{ {{3x=6} \atop {3x=-2}} \right. \\\left \{ {{x=2} \atop {x=-\frac{2}{3} }} \right.$

Ответ:

$x_1 = 2\\x_2 = -\frac{2}{3}$

Answer 2

To solve the absolute value equations:

1. |2x + 3| - 2 = 5
2. |3x - 2| + 3 = 7

You can solve them separately, one at a time.

Equation 1: |2x + 3| - 2 = 5

First, isolate the absolute value expression:

|2x + 3| = 5 + 2

|2x + 3| = 7

Now, you have two cases to consider, one when the expression inside the absolute value is positive, and one when it's negative.

Case 1: 2x + 3 is positive:

2x + 3 = 7

Subtract 3 from both sides:

2x = 7 - 3 2x = 4

Divide by 2:

x = 4 / 2 x = 2

Case 2: 2x + 3 is negative:

• (2x + 3) = 7

Now, multiply both sides by -1 to change the sign:

2x + 3 = -7

Subtract 3 from both sides:

2x = -7 - 3 2x = -10

Divide by 2:

x = -10 / 2 x = -5

So, for Equation 1, you have two solutions:

x = 2 (Case 1) x = -5 (Case 2)

Equation 2: |3x - 2| + 3 = 7

First, isolate the absolute value expression:

|3x - 2| = 7 - 3

|3x - 2| = 4

Now, as before, you have two cases to consider.

Case 1: 3x - 2 is positive:

3x - 2 = 4

Add 2 to both sides:

3x = 4 + 2 3x = 6

Divide by 3:

x = 6 / 3 x = 2

Case 2: 3x - 2 is negative:

• (3x - 2) = 4

Again, multiply both sides by -1 to change the sign:

3x - 2 = -4

Add 2 to both sides:

3x = -4 + 2 3x = -2

Divide by 3:

x = -2 / 3

So, for Equation 2, you have two solutions:

x = 2 (Case 1) x = -2/3 (Case 2)

In summary, the solutions to the given absolute value equations are:

For Equation 1: x = 2 and x = -5 For Equation 2: x = 2 and x = -2/3