(x-4)/(2x+4) is greater than or equal to 1.

How do i express solutions to inequalities in interval notation without using the calculator? please help.

To express the solutions to an inequality in interval notation without using a calculator, you need to follow these steps:

1. Solve the inequality:
In this case, the inequality is (x-4)/(2x+4) ≥ 1.

To solve it, you can start by multiplying both sides of the inequality by (2x+4) to get rid of the denominator. However, you need to be careful because if (2x+4) is negative, you will need to reverse the direction of the inequality.

So, let's consider two cases:
Case 1: (2x+4) > 0:
In this case, you can multiply both sides of the inequality by (2x+4) without reversing the direction of the inequality.
(x-4)/(2x+4) ≥ 1 ⇒ (x-4) ≥ (2x+4)

Case 2: (2x+4) < 0:
In this case, since multiplying by a negative number reverses the direction of the inequality, you have to change the direction of the inequality when multiplying both sides by (2x+4).
(x-4)/(2x+4) ≥ 1 ⇒ (x-4) ≤ (2x+4)

2. Solve the resulting inequalities:
To solve the inequalities (x-4) ≥ (2x+4) and (x-4) ≤ (2x+4), you need to isolate the variable x by moving the terms involving x to one side and the constant terms to the other side.

Case 1: (2x+4) > 0:
(x-4) ≥ (2x+4)
x - 2x ≥ 4 + 4
-x ≥ 8
x ≤ -8

Case 2: (2x+4) < 0:
(x-4) ≤ (2x+4)
x - 2x ≤ 4 + 4
-x ≤ 8
x ≥ -8

3. Express the solutions in interval notation:
Based on the solutions from step 2, you can express the solutions in interval notation.

Case 1: (2x+4) > 0:
The solution is x ≤ -8, which can be expressed as (-∞, -8].

Case 2: (2x+4) < 0:
The solution is x ≥ -8, which can be expressed as [-8, +∞).

Therefore, the solutions to the inequality (x-4)/(2x+4) ≥ 1, expressed in interval notation without using a calculator are (-∞, -8] ∪ [-8, +∞).