Question

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

Answer 1

Ответ:

1) -4

2)3

Пошаговое объяснение:

1)Под модулем,2х и 3х положительные, следовательно,модули раскрываем,не меняя знаков.

решаем уравнение, известные в одну сторону, неизвестные в другую.

2х-3=3х-5

-х=4

х=4

2)Тоже самое)

х-2=2х-5

-х=-3

х=3

Answer 2

Let's solve each equation step by step:

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

To solve this equation, we need to consider two cases: one for when the expressions inside the absolute value signs are positive and another for when they are negative.

Case 1: (2x - 3) and (3x + 1) are both positive. In this case, the equation becomes: 2x - 3 = 3x + 1

Now, solve for x: 2x - 3x = 1 -x = 1 x = -1

However, we need to check if this solution satisfies the original equation when both expressions are positive: |2(-1) - 3| = |-2 - 3| = |-5| = 5 |3(-1) + 1| = |-3 + 1| = |-2| = 2

Since 5 does not equal 2, the solution x = -1 does not satisfy the original equation for Case 1.

Case 2: (2x - 3) is positive and (3x + 1) is negative. In this case, the equation becomes: 2x - 3 = -(3x + 1)

Now, solve for x: 2x - 3 = -3x - 1 2x + 3x = -1 + 3 5x = 2 x = 2/5

We need to check if this solution satisfies the original equation when (2x - 3) is positive and (3x + 1) is negative: |2(2/5) - 3| = |4/5 - 3| = |-11/5| = 11/5 |3(2/5) + 1| = |6/5 + 1| = |11/5| = 11/5

Since 11/5 equals 11/5, the solution x = 2/5 satisfies the original equation for Case 2.

Case 3: (2x - 3) is negative and (3x + 1) is positive. In this case, the equation becomes: -(2x - 3) = 3x + 1

Now, solve for x: -2x + 3 = 3x + 1 -2x - 3x = 1 - 3 -5x = -2 x = 2/5

We need to check if this solution satisfies the original equation when (2x - 3) is negative and (3x + 1) is positive: |-(2/5) - 3| = |-2/5 - 3| = |-17/5| = 17/5 |3(2/5) + 1| = |6/5 + 1| = |11/5| = 11/5

Since 17/5 does not equal 11/5, the solution x = 2/5 does not satisfy the original equation for Case 3.

Case 4: (2x - 3) and (3x + 1) are both negative. In this case, the equation becomes: -(2x - 3) = -(3x + 1)

Now, solve for x: -2x + 3 = -3x - 1 -2x + 3x = -1 - 3 x = -4

We need to check if this solution satisfies the original equation when both expressions are negative: |-(2(-4) - 3)| = |-(-8) - 3| = |5| = 5 |3(-4) + 1| = |-12 + 1| = |-11| = 11

Since 5 does not equal 11, the solution x = -4 does not satisfy the original equation for Case 4.

Final result: The only solution that satisfies the original equation is x = 2/5.

2. |x - 2| = 2x - 5

To solve this equation, we also consider two cases:

Case 1: (x - 2) is positive. In this case, the equation becomes: x - 2 = 2x - 5

Now, solve for x: x - 2x = -5 + 2 -x = -3 x = 3

Check if this solution satisfies the original equation when (x - 2) is positive: |3 - 2| = |1| = 1 |2(3) - 5| = |6 - 5| = |1| = 1

Since 1 equals 1, the solution x = 3 satisfies the original equation for Case 1.

Case 2: (x - 2) is negative. In this case, the equation becomes: -(x - 2) = 2x - 5

Now, solve for x: -x + 2 = 2x - 5 2 + 5 = 2x + x 7x = 7 x = 1

Check if this solution satisfies the original equation when (x - 2) is negative: |-(1 - 2)| = |-1| = 1 |2(1) - 5| = |2 - 5| = |-3| = 3

Since 1 does not equal 3, the solution x = 1 does not satisfy the original equation for Case 2.

Final result: The only solution that satisfies the original equation is x = 3.