Certainly! Let's solve the inequality [tex]\(|y - 7| \leq 3\)[/tex] step-by-step.
### Step 1: Understand the Absolute Value Inequality
The inequality [tex]\(|y - 7| \leq 3\)[/tex] means that the expression inside the absolute value, [tex]\(y - 7\)[/tex], must lie between [tex]\(-3\)[/tex] and [tex]\(3\)[/tex]. In mathematical terms, we can write this as:
[tex]\[ -3 \leq y - 7 \leq 3 \][/tex]
### Step 2: Break Down the Compound Inequality
To solve for [tex]\(y\)[/tex], we need to break this compound inequality into two separate inequalities:
1. [tex]\( y - 7 \geq -3 \)[/tex]
2. [tex]\( y - 7 \leq 3 \)[/tex]
### Step 3: Solve Each Inequality
Let's solve each part separately:
1. [tex]\( y - 7 \geq -3 \)[/tex]:
[tex]\[ y \geq -3 + 7 \][/tex]
[tex]\[ y \geq 4 \][/tex]
2. [tex]\( y - 7 \leq 3 \)[/tex]:
[tex]\[ y \leq 3 + 7 \][/tex]
[tex]\[ y \leq 10 \][/tex]
### Step 4: Combine the Results
Now, combining these two results, we get:
[tex]\[ 4 \leq y \leq 10 \][/tex]
### Step 5: Graph the Solution
We need to represent this solution interval [tex]\([4, 10]\)[/tex] on a number line.
1. Draw a number line.
2. Mark the points [tex]\(4\)[/tex] and [tex]\(10\)[/tex] on the number line.
3. Since the inequality is inclusive (it includes 4 and 10), we use solid circles to represent these points.
4. Shade the region between 4 and 10 to represent all numbers [tex]\(y\)[/tex] that satisfy [tex]\(4 \leq y \leq 10\)[/tex].
Here is a graphical representation:
```
---|---------|---------|---------|---------|-------|---------|-------
3 4 5 6 7 8 9 10
[=================================================]
```
The shaded region represents the solution to the inequality [tex]\(|y - 7| \leq 3\)[/tex].
### Conclusion
The solution to the inequality [tex]\(|y - 7| \leq 3\)[/tex] is the interval [tex]\([4, 10]\)[/tex], which includes all the values of [tex]\(y\)[/tex] between 4 and 10, inclusive.
