Sure, let's solve the inequality [tex]\(|y - 5| < 2\)[/tex] step-by-step and graph the solution on the number line.
1. Understanding the Absolute Value Inequality:
The inequality [tex]\(|y - 5| < 2\)[/tex] means that the expression inside the absolute value (i.e., [tex]\(y - 5\)[/tex]) is within 2 units of 0 on the number line. This can be translated into two separate inequalities:
[tex]\[
-2 < y - 5 < 2
\][/tex]
2. Solving the Inequalities:
We now need to isolate [tex]\(y\)[/tex] in each part of the inequality.
- For the left part:
[tex]\[
-2 < y - 5
\][/tex]
Add 5 to both sides:
[tex]\[
-2 + 5 < y \implies 3 < y
\][/tex]
- For the right part:
[tex]\[
y - 5 < 2
\][/tex]
Add 5 to both sides:
[tex]\[
y < 2 + 5 \implies y < 7
\][/tex]
3. Combining the Results:
Combining the inequalities, we get:
[tex]\[
3 < y < 7
\][/tex]
4. Graphing the Solution:
To graph this solution on the number line, we plot an open interval between 3 and 7, indicating that [tex]\(y\)[/tex] can take any value between 3 and 7, but not including 3 and 7 themselves.
- Place an open circle at [tex]\(y = 3\)[/tex].
- Place an open circle at [tex]\(y = 7\)[/tex].
- Shade the region between 3 and 7.
Here is the graphical representation:
[tex]\[
\text{---} \circ \text{-------------------} \circ \text{---}
3 7
\][/tex]
This graphical representation indicates that [tex]\(y\)[/tex] can take any value between 3 and 7, not including the endpoints.
