Question

1) 3 |x| - 1 = 02) 2 |x| + 3 = 03) |3x + 2| - 4 = 04) |2x - 1| + 7 = 8

Answer 1

$1)3 |x| - 1 = 0 \\ |x| = \frac{1}{3} \\ x1 = \frac{1}{3} \\ x2 = - \frac{1}{3}$
$2)2 |x| + 3 = 0 \\ |x| = - \frac{3}{2} \\ x = - \frac{3}{2} \\ x = \frac{3}{2}$
$3) |3x + 2| - 4 = 0 \\ |3x + 2| = 4 \\ 1)3x + 2 = 4 \\ x = \frac{2}{3} \\ 2) |3x + 2| = - 4 \\ x = - 2$
$4) |2x - 1| + 7 = 8 \\ |2x - 1| = 1 \\ 1)2x - 1 = 1 \\ x = 1 \\ 2)2x - 1 = - 1 \\ x = 0$

Answer 2

To solve each of these equations, we need to isolate the absolute value term and then solve for the variable x. Remember that the absolute value of a number can be positive or negative, so we will have to consider both cases when solving these equations.

1. 3 |x| - 1 = 0

First, add 1 to both sides of the equation:

3 |x| = 1

Now, divide both sides by 3:

|x| = 1/3

Since the absolute value of a number cannot be negative, we can have two cases:

Case 1: x = 1/3 Case 2: x = -1/3

2. 2 |x| + 3 = 0

Subtract 3 from both sides of the equation:

2 |x| = -3

This equation has no solution because the absolute value of a number can never be negative, and the left side of the equation is positive or zero.

3. |3x + 2| - 4 = 0

First, add 4 to both sides of the equation:

|3x + 2| = 4

Now, we'll have two cases to consider:

Case 1: 3x + 2 = 4 Solve for x: 3x = 4 - 2 3x = 2 x = 2/3

Case 2: -(3x + 2) = 4 Solve for x: -3x - 2 = 4 -3x = 4 + 2 -3x = 6 x = -6/3 x = -2

So, the solutions are x = 2/3 and x = -2.

4. |2x - 1| + 7 = 8

Subtract 7 from both sides of the equation:

|2x - 1| = 1

Now, we'll have two cases:

Case 1: 2x - 1 = 1 Solve for x: 2x = 1 + 1 2x = 2 x = 2/2 x = 1

Case 2: -(2x - 1) = 1 Solve for x: -2x + 1 = 1 -2x = 1 - 1 -2x = 0 x = 0/(-2) x = 0

So, the solutions are x = 1 and x = 0.