Let's solve the inequality [tex]\(-4(x + 3) \leq -2 - 2x\)[/tex] step-by-step to determine the solution set and represent it on a number line.
### Step 1: Distribute [tex]\(-4\)[/tex] on the left side of the inequality
The given inequality is:
[tex]\[ -4(x + 3) \leq -2 - 2x \][/tex]
Distribute [tex]\(-4\)[/tex] inside the parenthesis:
[tex]\[ -4x - 12 \leq -2 - 2x \][/tex]
### Step 2: Add [tex]\(4x\)[/tex] to both sides to combine like terms
To isolate the term with [tex]\(x\)[/tex] on one side, add [tex]\(4x\)[/tex] to both sides:
[tex]\[ -4x - 12 + 4x \leq -2 - 2x + 4x \][/tex]
Simplify the equation:
[tex]\[ -12 \leq -2 + 2x \][/tex]
### Step 3: Add [tex]\(2\)[/tex] to both sides to further isolate the term with [tex]\(x\)[/tex]
Add [tex]\(2\)[/tex] to both sides:
[tex]\[ -12 + 2 \leq 2x \][/tex]
Simplify the equation:
[tex]\[ -10 \leq 2x \][/tex]
### Step 4: Divide both sides by [tex]\(2\)[/tex] to solve for [tex]\(x\)[/tex]
Divide by [tex]\(2\)[/tex]:
[tex]\[ \frac{-10}{2} \leq x \][/tex]
Simplify:
[tex]\[ -5 \leq x \][/tex]
This inequality can be written as:
[tex]\[ x \geq -5 \][/tex]
### Step 5: Represent the solution on the number line
The solution [tex]\(x \geq -5\)[/tex] implies that [tex]\( x \)[/tex] is any number greater than or equal to [tex]\(-5\)[/tex].
To represent this solution on the number line:
1. Draw a number line.
2. Locate the point [tex]\(-5\)[/tex] on the number line.
3. Draw a closed circle at [tex]\(-5\)[/tex] (because [tex]\(-5\)[/tex] is included in the solution set).
4. Shade the region to the right of [tex]\(-5\)[/tex] to indicate all numbers greater than or equal to [tex]\(-5\)[/tex].
The final number line representation for the solution set [tex]\(x \geq -5\)[/tex] looks like this:
```
<----|----|----|----|----|----|----|----|----|--->
-7 -6 -5 -4 -3 -2 -1 0 1
[=====================>
```
The closed circle at [tex]\(-5\)[/tex] and the shading to the right of [tex]\(-5\)[/tex] represent all values [tex]\(x\)[/tex] that are greater than or equal to [tex]\(-5\)[/tex].
