To solve the inequality [tex]\( |2x - 6| \geq 3 \)[/tex], we need to consider the definition of absolute value and how it can lead to two separate inequalities. Specifically, for [tex]\( |A| \geq B \)[/tex], it means [tex]\( A \geq B \)[/tex] or [tex]\( A \leq -B \)[/tex].
1. Break down the inequality:
[tex]\[
|2x - 6| \geq 3
\][/tex]
This can be split into two cases:
[tex]\[
2x - 6 \geq 3 \quad \text{or} \quad 2x - 6 \leq -3
\][/tex]
2. Solve each case:
- Case 1:
[tex]\[
2x - 6 \geq 3
\][/tex]
Add 6 to both sides:
[tex]\[
2x \geq 9
\][/tex]
Divide both sides by 2:
[tex]\[
x \geq \frac{9}{2}
\][/tex]
- Case 2:
[tex]\[
2x - 6 \leq -3
\][/tex]
Add 6 to both sides:
[tex]\[
2x \leq 3
\][/tex]
Divide both sides by 2:
[tex]\[
x \leq \frac{3}{2}
\][/tex]
3. Combine the solution sets:
Thus, the combined solution set is:
[tex]\[
x \geq \frac{9}{2} \quad \text{or} \quad x \leq \frac{3}{2}
\][/tex]
In interval notation, this is:
[tex]\[
(-\infty, \frac{3}{2}] \cup [\frac{9}{2}, \infty)
\][/tex]
Therefore, the correct choice is (A) "There are infinitely many solutions. The solution set is [tex]\( \boxed{(-\infty, \frac{3}{2}] \cup [\frac{9}{2}, \infty)} \)[/tex]."
Graphing the solution set:
On the number line, you would have:
- A closed circle at [tex]\( x = \frac{3}{2} \)[/tex] with a line extending to the left, indicating all values less than or equal to [tex]\( \frac{3}{2} \)[/tex].
- Another closed circle at [tex]\( x = \frac{9}{2} \)[/tex] with a line extending to the right, indicating all values greater than or equal to [tex]\( \frac{9}{2} \)[/tex].
Graphs:
Look for the graph that contains shading on the number line from [tex]\( -\infty \)[/tex] to [tex]\( \frac{3}{2} \)[/tex] (inclusive) and from [tex]\( \frac{9}{2} \)[/tex] to [tex]\( \infty \)[/tex] (inclusive).
The correct graph is:
[tex]\[
\text{(a graph showing two intervals: } (-\infty, \frac{3}{2}] \text{ and } [\frac{9}{2}, \infty) \text{)}
\][/tex]
Choose the correct graph that represents the solution set as described.
