To solve the inequality [tex]\(-4(x+3) \leq -2 - 2x\)[/tex], we need to follow a sequence of logical steps. Here is the detailed, step-by-step solution:
1. First, simplify and distribute the expression on the left-hand side:
[tex]\[
-4(x+3) \leq -2 - 2x
\][/tex]
Distributing [tex]\(-4\)[/tex] on the left-hand side:
[tex]\[
-4x - 12 \leq -2 - 2x
\][/tex]
2. Next, collect all the [tex]\(x\)[/tex] terms on one side of the inequality and the constant terms on the other side. To do this, we add [tex]\(4x\)[/tex] to both sides:
[tex]\[
-4x - 12 + 4x \leq -2 - 2x + 4x
\][/tex]
Simplifying, we get:
[tex]\[
-12 \leq -2 + 2x
\][/tex]
3. Isolate the [tex]\(x\)[/tex] term by adding 2 to both sides:
[tex]\[
-12 + 2 \leq 2x
\][/tex]
Simplifying, we get:
[tex]\[
-10 \leq 2x
\][/tex]
4. Solve for [tex]\(x\)[/tex] by dividing both sides by 2:
[tex]\[
\frac{-10}{2} \leq x
\][/tex]
Simplifying, we get:
[tex]\[
-5 \leq x
\][/tex]
Or written more conventionally,
[tex]\[
x \geq -5
\][/tex]
This can be expressed as:
[tex]\[
-5 \leq x < \infty
\][/tex]
Therefore, the solution set for the inequality is [tex]\(x \geq -5\)[/tex].
5. Representing this on a number line:
- Draw a number line.
- Locate [tex]\(-5\)[/tex] on the number line and draw a solid circle or a filled dot at [tex]\(-5\)[/tex] to indicate that [tex]\(-5\)[/tex] is included in the solution ([tex]\(\geq\)[/tex] indicates this).
- Shade the number line to the right of [tex]\(-5\)[/tex] extending infinitely, because our inequality is [tex]\(x \geq -5\)[/tex].
The final solution can be visualized on the number line as:
```
---+---+---+---+---+---+---+---+---+--->
-8 -7 -6 -5 -4 -3 -2 -1 0 1
<- shaded region indicates x-values starting from -5 and extending to infinity.
```
