Sure, I'll provide a detailed, step-by-step solution to the inequality [tex]\( -2(x + 3) \leq -12 \)[/tex] and illustrate its solution graphically.
1. Start with the given inequality:
[tex]\[
-2(x + 3) \leq -12
\][/tex]
2. Divide both sides of the inequality by -2.
Remember, when you divide or multiply both sides of an inequality by a negative number, you need to flip the direction of the inequality sign.
[tex]\[
\frac{-2(x + 3)}{-2} \geq \frac{-12}{-2}
\][/tex]
3. Simplify both sides:
[tex]\[
x + 3 \geq 6
\][/tex]
4. Subtract 3 from both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x + 3 - 3 \geq 6 - 3
\][/tex]
[tex]\[
x \geq 3
\][/tex]
The solution to the inequality is [tex]\( x \geq 3 \)[/tex].
### Graphical Representation:
To express [tex]\( x \geq 3 \)[/tex] graphically on a number line, follow these steps:
- Draw a number line.
- Locate the point [tex]\( 3 \)[/tex] on the number line and mark it distinctly.
- Since the inequality is [tex]\( x \geq 3 \)[/tex], you'll use a closed circle (or include the point) at [tex]\( 3 \)[/tex] to indicate that [tex]\( 3 \)[/tex] is part of the solution.
- Shade the number line to the right of [tex]\( 3 \)[/tex] to represent all values greater than or equal to [tex]\( 3 \)[/tex].
Here’s a simplified representation of the solution on the number line:
[tex]\[ \dots\ \ \ \ | \ \ \ \ \ 0 \ \ \ | \ \ \ \ \ 1 \ \ \ | \ \ \ \ \ 2 \ \ \ | \ \ \ \ \ 3 \ \ \ \blacksquare \ \ \ \rightarrow \ \ \ \][/tex]
The closed circle at [tex]\( 3 \)[/tex] indicates [tex]\( 3 \)[/tex] is included in the solution, and the arrow extending to the right represents all values [tex]\( x \geq 3 \)[/tex].
