To solve the inequality [tex]\( -\frac{1}{2}x - 3 \leq -2.5 \)[/tex], let's proceed step by step.
### Step 1: Applying the Addition Property of Inequality
First, we want to isolate the variable term on one side of the inequality. To do this, we need to eliminate the constant term, which is [tex]\(-3\)[/tex]. We achieve this by adding 3 to both sides of the inequality:
[tex]\[
-\frac{1}{2}x - 3 + 3 \leq -2.5 + 3
\][/tex]
Simplifying both sides, we get:
[tex]\[
-\frac{1}{2}x \leq 0.5
\][/tex]
### Step 2: Applying the Multiplication Property of Inequality
Next, we need to isolate [tex]\( x \)[/tex]. Currently, [tex]\( x \)[/tex] is multiplied by [tex]\(-\frac{1}{2}\)[/tex]. We can isolate [tex]\( x \)[/tex] by multiplying both sides of the inequality by the reciprocal of [tex]\(-\frac{1}{2}\)[/tex], which is [tex]\(-2\)[/tex]. Note that when we multiply or divide both sides of an inequality by a negative number, the direction of the inequality reverses.
So, we now multiply both sides by [tex]\(-2\)[/tex]:
[tex]\[
-\frac{1}{2}x \times (-2) \geq 0.5 \times (-2)
\][/tex]
Simplifying, we get:
[tex]\[
x \geq -1
\][/tex]
### Step 3: Writing the Final Solution
The solution to the inequality [tex]\( -\frac{1}{2}x - 3 \leq -2.5 \)[/tex] is:
[tex]\[
x \geq -1
\][/tex]
To express this in interval notation, the solution can be written as:
[tex]\[
[-1, \infty)
\][/tex]
This means that [tex]\( x \)[/tex] can be any value greater than or equal to [tex]\(-1\)[/tex].
