Let's solve the inequality [tex]\(-2x - 3 \geq -1\)[/tex] step-by-step.
1. Isolate the variable term:
To isolate the term with [tex]\(x\)[/tex], start by adding 3 to both sides of the inequality:
[tex]\[
-2x - 3 + 3 \geq -1 + 3
\][/tex]
Simplifying this, we get:
[tex]\[
-2x \geq 2
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
Now, divide both sides by [tex]\(-2\)[/tex]. Remember, when you divide or multiply both sides of an inequality by a negative number, you must flip the inequality sign:
[tex]\[
x \leq \frac{2}{-2}
\][/tex]
Simplifying this, we get:
[tex]\[
x \leq -1
\][/tex]
So the solution to the inequality [tex]\(-2x - 3 \geq -1\)[/tex] is [tex]\(x \leq -1\)[/tex].
In interval notation, this is represented as:
[tex]\[
(-\infty, -1]
\][/tex]
Therefore, the complete solution encompasses all real numbers [tex]\(x\)[/tex] such that:
[tex]\[
-\infty < x \leq -1
\][/tex]
This means that [tex]\(x\)[/tex] can be any value that is less than or equal to [tex]\(-1\)[/tex].
