Certainly! Let's break down the solution to the inequality [tex]\( 1 \leq 4d + 9 < 21 \)[/tex] step by step and then graph it on the number line.
### Step-by-Step Solution:
1. Inequality Breakdown:
The inequality [tex]\( 1 \leq 4d + 9 < 21 \)[/tex] can be broken down into two separate inequalities:
- [tex]\( 1 \leq 4d + 9 \)[/tex]
- [tex]\( 4d + 9 < 21 \)[/tex]
2. Solving the First Inequality:
- [tex]\( 1 \leq 4d + 9 \)[/tex]
- Subtract 9 from both sides:
[tex]\[ 1 - 9 \leq 4d \][/tex]
[tex]\[ -8 \leq 4d \][/tex]
- Divide both sides by 4:
[tex]\[ \frac{-8}{4} \leq d \][/tex]
[tex]\[ -2 \leq d \][/tex]
3. Solving the Second Inequality:
- [tex]\( 4d + 9 < 21 \)[/tex]
- Subtract 9 from both sides:
[tex]\[ 4d < 21 - 9 \][/tex]
[tex]\[ 4d < 12 \][/tex]
- Divide both sides by 4:
[tex]\[ \frac{12}{4} > d \][/tex]
[tex]\[ d < 3 \][/tex]
4. Combining the Inequalities:
Combining both results, we get:
[tex]\[ -2 \leq d < 3 \][/tex]
### Graphing the Solution on the Number Line:
To graph the solution [tex]\(-2 \leq d < 3\)[/tex]:
- We use a closed dot (●) at [tex]\(d = -2\)[/tex] to indicate that [tex]\(-2\)[/tex] is included in the solution.
- We use an open dot (○) at [tex]\(d = 3\)[/tex] to indicate that [tex]\(3\)[/tex] is not included in the solution.
- We shade the number line between [tex]\(-2\)[/tex] and [tex]\(3\)[/tex] to represent all the values that [tex]\(d\)[/tex] can take in this interval.
### Number Line Representation:
```
●-----------------------○
-3 -2 -1 0 1 2 3 4
```
So, on the number line, the solution to the inequality [tex]\(1 \leq 4d + 9 < 21\)[/tex] is represented by the interval [tex]\([-2, 3)\)[/tex].
