Question

|4x+5|=5 |x|+8=6 ||x|=10

Answer 1

$|4x + 5| = 5 \\ \\ - 4x - 5 = 5 \\ 4x + 5 = 5 \\ \\ - 4x = 10 \\ 4x = 0 \\ \\ x = 0 \\ x = - 2.5$

$|x| + 8 = 6 \\ \\ - x = - 4 \\ x = - 4 \\ \\ x = 4 \\ x = - 4$

x=+-10

Answer 2

To solve absolute value equations, we need to consider both the positive and negative cases for the expression inside the absolute value bars. Let's solve each equation step by step:

1. |4x + 5| = 5:

Case 1: 4x + 5 = 5 Subtract 5 from both sides: 4x = 0 Divide both sides by 4: x = 0

Case 2: -(4x + 5) = 5 Distribute the negative sign: -4x - 5 = 5 Add 5 to both sides: -4x = 10 Divide both sides by -4: x = -10/4 or x = -5/2

So, the solutions for the equation |4x + 5| = 5 are x = 0 and x = -5/2.

2. |x| + 8 = 6:

Subtract 8 from both sides: |x| = -2

For real numbers, the absolute value cannot be negative, so there are no solutions for this equation. Thus, the equation |x| + 8 = 6 has no solution.

3. ||x|| = 10:

Since there are double absolute value signs, we'll consider two cases:

Case 1: |x| = 10

Case 1.1: x = 10 Case 1.2: x = -10

Case 2: -|x| = 10

Case 2.1: -x = 10 Case 2.2: -x = -10

For Case 2.1, multiply both sides by -1: x = -10

For Case 2.2, multiply both sides by -1: x = 10

The solutions for the equation ||x|| = 10 are x = -10 and x = 10.

To summarize:

• For |4x + 5| = 5, the solutions are x = 0 and x = -5/2.
• For |x| + 8 = 6, there are no solutions.
• For ||x|| = 10, the solutions are x = -10 and x = 10.